On 11/8/2024 6:33 PM, Richard Damon wrote:Gödel showed otherwise.
On 11/8/24 6:36 PM, olcott wrote:
On 11/8/2024 3:59 PM, Richard Damon wrote:
On 11/8/24 4:17 PM, olcott wrote:
On 11/8/2024 12:31 PM, Richard Damon wrote:
On 11/8/24 1:08 PM, olcott wrote:
On 11/8/2024 12:02 PM, Richard Damon wrote:
On 11/8/24 12:25 PM, olcott wrote:
That formal systems that only apply truth preserving operations >>>>>>>>> to expressions of their formal language that have been
stipulated to be true cannot possibly be undecidable is proven >>>>>>>>> to be over-your-head on the basis that you have no actual
reasoning as a rebuttal.
MM doesn’t even contain the same sentences as PA.IT SHOWS THAT I KNOW IT IS STUPID TO CONSTRUE TRUE IN META-MATH ASNo, it just shows you don't understand how meta-systems work.No, all you have done is shown that you don't undertstand what >>>>>>>> you are talking about.The equivocation of switching formal systems from PA to meta-math. There’s no such thing happening. They are very clearly separated.
Godel PROVED that the FORMAL SYSTEM that his proof started in, is >>>>>>>> unable to PROVE that the statement G, being "that no Natural
Number g, that satifies a particularly designed Primitive
Recursive Relationship" is true, but also shows (using the Meta- >>>>>>>> Mathematics that derived the PRR for the original Formal System) >>>>>>>> that no such number can exist.
TRUE IN PA.
Yes it is. If MM proves that a sentence is true in PA, that sentenceBut, as I pointed out, the way Meta-Math is derived from PA,Meta-math <IS NOT> PA.
True in meta-math <IS NOT> True in PA.
It’s a perfectly wellformed sentence.This sentence is not true: "This sentence is not true"
is only true because the inner sentence is bullshit gibberish.
What is "the liar paradox applied to itself"?But MM has exactly the same axioms and rules as PA, so anythingOne single level of indirect reference CHANGES EVERYTHING.
established by that set of axioms and rules in MM is established in PA
too.
There are additional axioms in MM, but the rules are built specifically
PA speaks PA. Meta-math speaks ABOUT PA.
The liar paradox is nonsense gibberish except when applied to itself,
then it becomes true.
On 11/8/2024 5:58 AM, Alan Mackenzie wrote:
In comp.theory olcott <polcott333@gmail.com> wrote:
On 11/6/2024 2:34 PM, Alan Mackenzie wrote:
In comp.theory olcott <polcott333@gmail.com> wrote:
On 11/6/2024 10:45 AM, Alan Mackenzie wrote:
[ .... ]
There is another sense in which something could be a lie. If, for >>>>>> example, I emphatically asserted some view about the minutiae of
medical surgery, in opposition to the standard view accepted by
practicing surgeons, no matter how sincere I might be in that
belief, I would be lying. Lying by ignorance.
That is a lie unless you qualify your statement with X is a
lie(unintentional false statement). It is more truthful to
say that statement X is rejected as untrue by a consensus of
medical opinion.
No, as so often, you've missed the nuances. The essence of the
scenario is making emphatic statements in a topic which requires
expertise, but that expertise is missing. Such as me laying down the
law about surgery or you doing the same in mathematical logic.
It is not at all my lack of expertise on mathematical logic
it is your ignorance of philosophy of logic as shown by you
lack of understanding of the difference between "a priori"
and "a posteriori" knowledge.
Garbage.
Surgical procedures and mathematical logic are in fundamentally
different classes of knowledge.
But the necessity of expertise is present in both, equally. Emphatically
to assert falsehoods when expertise is lacking is a form of lying. That
is what you do.
This allows for the possibility that the consensus is not
infallible. No one here allows for the possibility that the
current received view is not infallible. Textbooks on the
theory of computation are NOT the INFALLIBLE word of God.
Gods have got nothing to do with it. 2 + 2 = 4, the fact that the
world is a ball, not flat, Gödel's theorem, and the halting problem,
have all been demonstrated beyond any doubt whatsoever.
Regarding the last two they would have said the same thing about
Russell's Paradox and what is now known as naive set theory at the
time.
There's no "would have said" regarding Russell's paradox. Nobody would
have asserted the correctness of naive set theory, a part of mathematics
then at the forefront of research and still in flux. We've moved beyond
that point in the last hundred years.
And you are continually stating that theorems like 2 + 2 = 4 are false.
That is a lie. I never said anything like that and you know it.
Here is what I actually said:
When the operations are limited to applying truth preserving
operations to expressions of language that are stipulated to
be true then
True(L,x) ≡ (L ⊢ x) and False(L, x) ≡ (L ⊢ ~x)
Then
(Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)))
becomes
(¬TruthBearer(L,x) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x))) Incompleteness utterly ceases to exist
In C we can have a pointer to a character string
and a pointer to a pointer to a character string.
But in Formal System, the definition ARE "infallibe".
Not when they contradict other definitions.
On 11/8/2024 10:02 AM, Alan Mackenzie wrote:
As I said, it's not a matter of "belief". It's a matter of certain
knowledge stemming from having studied for and having a degree in maths.
You understand what the received view is.
My view is inconsistent with the received view therefore
(when one assumes that the received view is infallible)
I must be wrong.
I reject what you say because it's objectively wrong. Just as if you
said 2 + 2 = 5.
What I said about is a semantic tautology just like
2 + 3 = 5. Formal systems are only incomplete when
the term "incomplete" is a euphemism for the inability
of formal systems to correctly determine the truth
value of non-truth-bearers.
No. You lack the expertise.
I know how the current systems work and I disagree
that they are correct. This is not any lack of expertise.
On 11/9/2024 5:01 AM, joes wrote:
Am Fri, 08 Nov 2024 18:39:34 -0600 schrieb olcott:
On 11/8/2024 6:33 PM, Richard Damon wrote:Gödel showed otherwise.
On 11/8/24 6:36 PM, olcott wrote:
On 11/8/2024 3:59 PM, Richard Damon wrote:
On 11/8/24 4:17 PM, olcott wrote:
On 11/8/2024 12:31 PM, Richard Damon wrote:
On 11/8/24 1:08 PM, olcott wrote:
On 11/8/2024 12:02 PM, Richard Damon wrote:
On 11/8/24 12:25 PM, olcott wrote:
That formal systems that only apply truth preserving operations >>>>>>>>>>> to expressions of their formal language that have been
stipulated to be true cannot possibly be undecidable is proven >>>>>>>>>>> to be over-your-head on the basis that you have no actual >>>>>>>>>>> reasoning as a rebuttal.
That is counter-factual within my precise specification.
When truth is only derived by starting with
truth and applying truth preserving operations
then unprovable in PA becomes untrue in PA.
Everyone is so sure that whatever I say must be wrong
that they don't pay any f-cking attention to what I say.
The above paragraph <is> infallibly correct.
There’s no such thing happening. They are very clearly separated.No, all you have done is shown that you don't undertstand what >>>>>>>>>> you are talking about.The equivocation of switching formal systems from PA to meta-math.
Godel PROVED that the FORMAL SYSTEM that his proof started in, is >>>>>>>>>> unable to PROVE that the statement G, being "that no Natural >>>>>>>>>> Number g, that satifies a particularly designed Primitive
Recursive Relationship" is true, but also shows (using the Meta- >>>>>>>>>> Mathematics that derived the PRR for the original Formal System) >>>>>>>>>> that no such number can exist.
MM doesn’t even contain the same sentences as PA.No, it just shows you don't understand how meta-systems work.IT SHOWS THAT I KNOW IT IS STUPID TO CONSTRUE TRUE IN META-MATH AS >>>>>>> TRUE IN PA.
Yes it is. If MM proves that a sentence is true in PA, that sentenceBut, as I pointed out, the way Meta-Math is derived from PA,Meta-math <IS NOT> PA.
True in meta-math <IS NOT> True in PA.
is true in PA.
Within my model: Only PA can prove what is true in PA.
It’s a perfectly wellformed sentence.This sentence is not true: "This sentence is not true"
is only true because the inner sentence is bullshit gibberish.
What is "the liar paradox applied to itself"?But MM has exactly the same axioms and rules as PA, so anythingPA speaks PA. Meta-math speaks ABOUT PA.
established by that set of axioms and rules in MM is established in PA >>>> too.
There are additional axioms in MM, but the rules are built specifically >>> One single level of indirect reference CHANGES EVERYTHING.
The liar paradox is nonsense gibberish except when applied to itself,
then it becomes true.
Can yo please add a newline so that
you comments are no buried in my comments?
This sentence is not true: "This sentence is not true"
is true because the inner sentence is nonsense gibberish.
On 11/9/2024 5:01 AM, joes wrote:
On 11/8/24 12:25 PM, olcott wrote:
Gödel showed otherwise.That formal systems that only apply truth preserving operations >>>>>>>>>>> to expressions of their formal language that have been
stipulated to be true cannot possibly be undecidable is proven >>>>>>>>>>> to be over-your-head on the basis that you have no actual >>>>>>>>>>> reasoning as a rebuttal.
That is counter-factual within my precise specification.
When truth is only derived by starting with
truth and applying truth preserving operations
then unprovable in PA becomes untrue in PA.
Everyone is so sure that whatever I say must be wrong
that they don't pay any f-cking attention to what I say.
The above paragraph <is> infallibly correct.
--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
On 11/9/2024 5:01 AM, joes wrote:What Gödel did is a fact.
Am Fri, 08 Nov 2024 18:39:34 -0600 schrieb olcott:That is counter-factual within my precise specification.
On 11/8/2024 6:33 PM, Richard Damon wrote:Gödel showed otherwise.
On 11/8/24 6:36 PM, olcott wrote:
On 11/8/2024 3:59 PM, Richard Damon wrote:
On 11/8/24 4:17 PM, olcott wrote:
On 11/8/2024 12:31 PM, Richard Damon wrote:
On 11/8/24 1:08 PM, olcott wrote:
On 11/8/2024 12:02 PM, Richard Damon wrote:
On 11/8/24 12:25 PM, olcott wrote:
That formal systems that only apply truth preserving
operations to expressions of their formal language that have >>>>>>>>>>> been stipulated to be true cannot possibly be undecidable is >>>>>>>>>>> proven to be over-your-head on the basis that you have no >>>>>>>>>>> actual reasoning as a rebuttal.
When truth is only derived by starting with truth and applying truth preserving operations then unprovable in PA becomes untrue in PA.No, unless your system is less powerful than PA.
PA can’t prove anything about itself.There’s no such thing happening. They are very clearly separated.No, all you have done is shown that you don't undertstand what >>>>>>>>>> you are talking about.The equivocation of switching formal systems from PA to
Godel PROVED that the FORMAL SYSTEM that his proof started in, >>>>>>>>>> is unable to PROVE that the statement G, being "that no Natural >>>>>>>>>> Number g, that satifies a particularly designed Primitive
Recursive Relationship" is true, but also shows (using the >>>>>>>>>> Meta- Mathematics that derived the PRR for the original Formal >>>>>>>>>> System) that no such number can exist.
meta-math.
MM doesn’t even contain the same sentences as PA.No, it just shows you don't understand how meta-systems work.IT SHOWS THAT I KNOW IT IS STUPID TO CONSTRUE TRUE IN META-MATH AS >>>>>>> TRUE IN PA.
Within my model: Only PA can prove what is true in PA.Yes it is. If MM proves that a sentence is true in PA, that sentence isBut, as I pointed out, the way Meta-Math is derived from PA,Meta-math <IS NOT> PA.
True in meta-math <IS NOT> True in PA.
true in PA.
How does your newsreader mark quotes?It’s a perfectly wellformed sentence.This sentence is not true: "This sentence is not true"
is only true because the inner sentence is bullshit gibberish.
Can yo please add a newline so that you comments are no buried in my comments?What is "the liar paradox applied to itself"?But MM has exactly the same axioms and rules as PA, so anythingOne single level of indirect reference CHANGES EVERYTHING.
established by that set of axioms and rules in MM is established in
PA too.
There are additional axioms in MM, but the rules are built
specifically
PA speaks PA. Meta-math speaks ABOUT PA.
The liar paradox is nonsense gibberish except when applied to itself,
then it becomes true.
This sentence is not true: "This sentence is not true" is true becauseI think you missed some quotation marks there. The outer sentence is true,
the inner sentence is nonsense gibberish.
On 11/9/2024 10:03 AM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/9/2024 5:01 AM, joes wrote:
On 11/8/24 12:25 PM, olcott wrote:
Gödel showed otherwise.That formal systems that only apply truth preserving operations >>>>>>>>>>>>> to expressions of their formal language that have been >>>>>>>>>>>>> stipulated to be true cannot possibly be undecidable is proven >>>>>>>>>>>>> to be over-your-head on the basis that you have no actual >>>>>>>>>>>>> reasoning as a rebuttal.
That is counter-factual within my precise specification.
That's untrue - you don't have a precise specification. And even if you
did, Gödel's theorem would still hold.
When truth is only derived by starting with
truth and applying truth preserving operations
then unprovable in PA becomes untrue in PA.
No. Unprovable will remain.
*Like I said you don't pay f-cking attention*
Unprovable(L,x) means Untrue(L,x)
Unprovable(L,~x) means Unfalse(L,x)
~True(L,x) ^ ~True(L, ~x) means ~Truth-Bearer(L,x)
--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
On 11/9/2024 10:04 AM, joes wrote:Where is the difference?
Am Sat, 09 Nov 2024 08:45:12 -0600 schrieb olcott:Gödel had a different f-cking basis.
On 11/9/2024 5:01 AM, joes wrote:
Am Fri, 08 Nov 2024 18:39:34 -0600 schrieb olcott:
On 11/8/2024 6:33 PM, Richard Damon wrote:Gödel showed otherwise.
On 11/8/24 6:36 PM, olcott wrote:
On 11/8/2024 3:59 PM, Richard Damon wrote:
On 11/8/24 4:17 PM, olcott wrote:
On 11/8/2024 12:31 PM, Richard Damon wrote:
On 11/8/24 1:08 PM, olcott wrote:
On 11/8/2024 12:02 PM, Richard Damon wrote:
On 11/8/24 12:25 PM, olcott wrote:
That formal systems that only apply truth preserving >>>>>>>>>>>>> operations to expressions of their formal language that have >>>>>>>>>>>>> been stipulated to be true cannot possibly be undecidable is >>>>>>>>>>>>> proven to be over-your-head on the basis that you have no >>>>>>>>>>>>> actual reasoning as a rebuttal.
Then it is incomplete or inconsistent.That is counter-factual within my precise specification.What Gödel did is a fact.
It is not any less powerful than PA in the same f-cking way that ZFC isWhen truth is only derived by starting with truth and applying truthNo, unless your system is less powerful than PA.
preserving operations then unprovable in PA becomes untrue in PA.
Untrue means the negation is true, but ~G is also unprovable.
not less powerful than naive set theory.
PA can’t prove anything about itself.Within my model: Only PA can prove what is true in PA.Yes it is. If MM proves that a sentence is true in PA, that sentenceBut, as I pointed out, the way Meta-Math is derived from PA,Meta-math <IS NOT> PA.
True in meta-math <IS NOT> True in PA.
is true in PA.
What is "the liar paradox applied to itself"?But MM has exactly the same axioms and rules as PA, so anythingOne single level of indirect reference CHANGES EVERYTHING.
established by that set of axioms and rules in MM is established in >>>>>> PA too.
There are additional axioms in MM, but the rules are built
specifically
PA speaks PA. Meta-math speaks ABOUT PA.
The liar paradox is nonsense gibberish except when applied to
itself, then it becomes true.
Does your reader not mark quotes?Instead of replying immediately after my comment, skip a line. Leave a freaking blank line inbetween.Can yo please add a newline so that you comments are no buried in myHow does your newsreader mark quotes?
comments?
"This sentence is true" however has a welldefined meaning.https://en.wikipedia.org/wiki/Colorless_green_ideas_sleep_furiously isThis sentence is not true: "This sentence is not true" is true becauseI think you missed some quotation marks there. The outer sentence is
the inner sentence is nonsense gibberish.
true, but the inner is perfectly wellformed and syntactically correct.
also syntactically well formed and semantic gibberish.
--Gödels sentence isn’t exactly this, because formal systems don’t speak. >> It is just a number that happens to encode itself.
On 11/9/2024 11:58 AM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/9/2024 10:03 AM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/9/2024 5:01 AM, joes wrote:
On 11/8/24 12:25 PM, olcott wrote:
Gödel showed otherwise.That formal systems that only apply truth preserving >>>>>>>>>>>>>>> operations to expressions of their formal language that >>>>>>>>>>>>>>> have been stipulated to be true cannot possibly be >>>>>>>>>>>>>>> undecidable is proven to be over-your-head on the basis >>>>>>>>>>>>>>> that you have no actual reasoning as a rebuttal.
That is counter-factual within my precise specification.
That's untrue - you don't have a precise specification. And even if you >>>> did, Gödel's theorem would still hold.
When truth is only derived by starting with
truth and applying truth preserving operations
then unprovable in PA becomes untrue in PA.
No. Unprovable will remain.
*Like I said you don't pay f-cking attention*
Stop swearing. I don't pay much attention to your provably false
utterances, no. Life is too short.
That you denigrate what I say without paying attention to what
I say <is> the definition of reckless disregard for the truth
that loses defamation cases.
Hint: Gödel's theorem applies in any sufficiently powerful logical
system, and the bar for "sufficiently powerful" is not high.
Unless it is stipulated at the foundation of the notion of
formal systems that ~Provable(PA, g) simply means ~True(PA, g).
Unprovable(L,x) means Untrue(L,x)
Unprovable(L,~x) means Unfalse(L,x)
~True(L,x) ^ ~True(L, ~x) means ~Truth-Bearer(L,x)
If you're going to change the standard meaning of standard words, you'll
find communicating with other people somewhat strained and difficult.
ZFC did the same thing and that was the ONLY way
that Russell's Paradox was resolved.
When ~Provable(PA,g) means ~True(PA,g) then
incompleteness cannot exist.
[ .... ]
--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
On 11/9/2024 11:27 AM, joes wrote:The metatheory proves otherwise. If G were not true, ~G would need to
Am Sat, 09 Nov 2024 11:09:02 -0600 schrieb olcott:True is only provable from axioms thus ~Provable(PA, G) == ~True(PA, G)
On 11/9/2024 10:04 AM, joes wrote:Where is the difference?
Am Sat, 09 Nov 2024 08:45:12 -0600 schrieb olcott:Gödel had a different f-cking basis.
On 11/9/2024 5:01 AM, joes wrote:
Am Fri, 08 Nov 2024 18:39:34 -0600 schrieb olcott:
On 11/8/2024 6:33 PM, Richard Damon wrote:Gödel showed otherwise.
On 11/8/24 6:36 PM, olcott wrote:
On 11/8/2024 3:59 PM, Richard Damon wrote:
On 11/8/24 4:17 PM, olcott wrote:
On 11/8/2024 12:31 PM, Richard Damon wrote:
On 11/8/24 1:08 PM, olcott wrote:
On 11/8/2024 12:02 PM, Richard Damon wrote:
On 11/8/24 12:25 PM, olcott wrote:
That formal systems that only apply truth preserving >>>>>>>>>>>>>>> operations to expressions of their formal language that >>>>>>>>>>>>>>> have been stipulated to be true cannot possibly be >>>>>>>>>>>>>>> undecidable is proven to be over-your-head on the basis >>>>>>>>>>>>>>> that you have no actual reasoning as a rebuttal.
It is not any less powerful than PA in the same f-cking way that ZFCWhen truth is only derived by starting with truth and applying truth >>>>> preserving operations then unprovable in PA becomes untrue in PA.No, unless your system is less powerful than PA.
Untrue means the negation is true, but ~G is also unprovable.
is not less powerful than naive set theory.
Within my model: Only PA can prove what is true in PA.Yes it is. If MM proves that a sentence is true in PA, thatBut, as I pointed out, the way Meta-Math is derived from PA, >>>>>>>>> Meta-math <IS NOT> PA.True in meta-math <IS NOT> True in PA.
sentence is true in PA.
But MM has exactly the same axioms and rules as PA, so anything >>>>>>>> established by that set of axioms and rules in MM is established >>>>>>>> in PA too.One single level of indirect reference CHANGES EVERYTHING. PA
There are additional axioms in MM, but the rules are built
specifically
speaks PA. Meta-math speaks ABOUT PA.
The liar paradox is nonsense gibberish except when applied to
itself, then it becomes true.
Are you reading in plaintext?Does your reader not mark quotes?Instead of replying immediately after my comment, skip a line. Leave aCan yo please add a newline so that you comments are no buried in my >>>>> comments?How does your newsreader mark quotes?
freaking blank line inbetween.
No it does f-cking not. WTF is it true about?"This sentence is not true" however has a welldefined meaning.https://en.wikipedia.org/wiki/Colorless_green_ideas_sleep_furiously isThis sentence is not true: "This sentence is not true" is trueI think you missed some quotation marks there. The outer sentence is
because the inner sentence is nonsense gibberish.
true, but the inner is perfectly wellformed and syntactically
correct.
also syntactically well formed and semantic gibberish.
On 11/9/2024 10:03 AM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/9/2024 5:01 AM, joes wrote:
On 11/8/24 12:25 PM, olcott wrote:
Gödel showed otherwise.That formal systems that only apply truth preserving >>>>>>>>>>>>> operations
to expressions of their formal language that have been >>>>>>>>>>>>> stipulated to be true cannot possibly be undecidable is proven >>>>>>>>>>>>> to be over-your-head on the basis that you have no actual >>>>>>>>>>>>> reasoning as a rebuttal.
That is counter-factual within my precise specification.
That's untrue - you don't have a precise specification. And even if you
did, Gödel's theorem would still hold.
When truth is only derived by starting with
truth and applying truth preserving operations
then unprovable in PA becomes untrue in PA.
No. Unprovable will remain.
*Like I said you don't pay f-cking attention*
*Like I said you don't pay f-cking attention*
*Like I said you don't pay f-cking attention*
Unprovable(L,x) means Untrue(L,x)
Unprovable(L,~x) means Unfalse(L,x)
~True(L,x) ^ ~True(L, ~x) means ~Truth-Bearer(L,x)
Everyone is so sure that whatever I say must be wrong
that they don't pay any f-cking attention to what I say.
The above paragraph <is> infallibly correct.
Garbage. When you spout objectively wrong stuff, people don't need to
look at the details to know it's wrong. Wrong is wrong. Gödel's theorem >> is just as correct as 2 + 2 = 4 is.
[ .... ]
--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
On 11/9/2024 12:47 PM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/9/2024 11:58 AM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/9/2024 10:03 AM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
When truth is only derived by starting with
truth and applying truth preserving operations
then unprovable in PA becomes untrue in PA.
No. Unprovable will remain.
*Like I said you don't pay f-cking attention*
Stop swearing. I don't pay much attention to your provably false
utterances, no. Life is too short.
That you denigrate what I say without paying attention to what
I say <is> the definition of reckless disregard for the truth
that loses defamation cases.
Not at all. I denigrate your lies, where by lies I mean the emphatic
utterances of falsehood due to a lack of expertise in the subject matter.
See the beginning of this subthread.
You are not doing that. I am redefining the foundation
of the notion of a formal system and calling this a
lie can have your house confiscated for defamation.
You are the one with reckless disregard for the truth. You haven't even
bothered to read the introductory texts which would help you understand
what the truth is.
Hint: Gödel's theorem applies in any sufficiently powerful logical
system, and the bar for "sufficiently powerful" is not high.
Unless it is stipulated at the foundation of the notion of
formal systems that ~Provable(PA, g) simply means ~True(PA, g).
If you're going to redefine the word provable to mean something else,
you'll need some other word to mean what provable means to everybody
else.
I am correcting the somewhat ill-founded notion of provable
to only mean applying truth preserving operations to finite
string expressions of language.
Unprovable(L,x) means Untrue(L,x)
Unprovable(L,~x) means Unfalse(L,x)
~True(L,x) ^ ~True(L, ~x) means ~Truth-Bearer(L,x)
If you're going to change the standard meaning of standard words, you'll >>>> find communicating with other people somewhat strained and difficult.
ZFC did the same thing and that was the ONLY way
that Russell's Paradox was resolved.
No, they didn't do the same thing. They stayed within the bounds of
logic.
ZFC DID NOT STAY WITHIN THE BOUNDS OF NAIVE SET THEORY
And yes, they resolved a paradox. There is no paradox for your
"system" to resolve, even if it were logically coherent.
When ~Provable(PA,g) means ~True(PA,g) then
incompleteness cannot exist.
OK, That's a proof by contradiction that ~provable cannot mean ~true.
The assumption that ~Provable(PA, g) does not mean ~True(PA, g)
cannot correctly be the basis for any proof because it is only
an assumption.
We know, by Gödel's Theorem that incompleteness does exist. So the
initial proposition cannot hold, or it is in an inconsistent system.
Only on the basis of the assumption that
~Provable(PA, g) does not mean ~True(PA, g)
Get rid of that single assumption AND EVERYTHING CHANGES
[ .... ]
--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
On 11/9/2024 12:47 PM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/9/2024 11:58 AM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/9/2024 10:03 AM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/9/2024 5:01 AM, joes wrote:
On 11/8/24 12:25 PM, olcott wrote:
Gödel showed otherwise.That formal systems that only apply truth preserving >>>>>>>>>>>>>>>>> operations to expressions of their formal language that >>>>>>>>>>>>>>>>> have been stipulated to be true cannot possibly be >>>>>>>>>>>>>>>>> undecidable is proven to be over-your-head on the basis >>>>>>>>>>>>>>>>> that you have no actual reasoning as a rebuttal.
That is counter-factual within my precise specification.
That's untrue - you don't have a precise specification. And even >>>>>> if you
did, Gödel's theorem would still hold.
When truth is only derived by starting with
truth and applying truth preserving operations
then unprovable in PA becomes untrue in PA.
No. Unprovable will remain.
*Like I said you don't pay f-cking attention*
Stop swearing. I don't pay much attention to your provably false
utterances, no. Life is too short.
That you denigrate what I say without paying attention to what
I say <is> the definition of reckless disregard for the truth
that loses defamation cases.
Not at all. I denigrate your lies, where by lies I mean the emphatic
utterances of falsehood due to a lack of expertise in the subject matter.
See the beginning of this subthread.
You are not doing that. I am redefining the foundation
of the notion of a formal system and calling this a
lie can have your house confiscated for defamation.
You are the one with reckless disregard for the truth. You haven't even
bothered to read the introductory texts which would help you understand
what the truth is.
*I am redefining the foundation of the notion of a formal system*
*I am redefining the foundation of the notion of a formal system*
*I am redefining the foundation of the notion of a formal system*
*I am redefining the foundation of the notion of a formal system*
*I am redefining the foundation of the notion of a formal system*
*I am redefining the foundation of the notion of a formal system*
*I am redefining the foundation of the notion of a formal system*
*I am redefining the foundation of the notion of a formal system*
*I am redefining the foundation of the notion of a formal system*
*I am redefining the foundation of the notion of a formal system*
*I am redefining the foundation of the notion of a formal system*
*I am redefining the foundation of the notion of a formal system*
*I am redefining the foundation of the notion of a formal system*
*I am redefining the foundation of the notion of a formal system*
*I am redefining the foundation of the notion of a formal system*
*I am redefining the foundation of the notion of a formal system*
*I am redefining the foundation of the notion of a formal system*
I have no fear of you starting a defamation case against me. For a
start, you'd have to learn some German, and for another thing, I'd win on
the merits.
Hint: Gödel's theorem applies in any sufficiently powerful logical
system, and the bar for "sufficiently powerful" is not high.
Unless it is stipulated at the foundation of the notion of
formal systems that ~Provable(PA, g) simply means ~True(PA, g).
If you're going to redefine the word provable to mean something else,
you'll need some other word to mean what provable means to everybody
else.
I am correcting the somewhat ill-founded notion of provable
to only mean applying truth preserving operations to finite
string expressions of language.
I am correcting the somewhat ill-founded notion of provable
to only mean applying truth preserving operations to finite
string expressions of language.
I am correcting the somewhat ill-founded notion of provable
to only mean applying truth preserving operations to finite
string expressions of language.
I am correcting the somewhat ill-founded notion of provable
to only mean applying truth preserving operations to finite
string expressions of language.
I am correcting the somewhat ill-founded notion of provable
to only mean applying truth preserving operations to finite
string expressions of language.
Unprovable(L,x) means Untrue(L,x)
Unprovable(L,~x) means Unfalse(L,x)
~True(L,x) ^ ~True(L, ~x) means ~Truth-Bearer(L,x)
If you're going to change the standard meaning of standard words,
you'll
find communicating with other people somewhat strained and difficult.
ZFC did the same thing and that was the ONLY way
that Russell's Paradox was resolved.
No, they didn't do the same thing. They stayed within the bounds of
logic.
ZFC DID NOT STAY WITHIN THE BOUNDS OF NAIVE SET THEORY
ZFC DID NOT STAY WITHIN THE BOUNDS OF NAIVE SET THEORY
ZFC DID NOT STAY WITHIN THE BOUNDS OF NAIVE SET THEORY
ZFC DID NOT STAY WITHIN THE BOUNDS OF NAIVE SET THEORY
ZFC DID NOT STAY WITHIN THE BOUNDS OF NAIVE SET THEORY
And yes, they resolved a paradox. There is no paradox for your
"system" to resolve, even if it were logically coherent.
When ~Provable(PA,g) means ~True(PA,g) then
incompleteness cannot exist.
OK, That's a proof by contradiction that ~provable cannot mean ~true.
The assumption that ~Provable(PA, g) does not mean ~True(PA, g)
cannot correctly be the basis for any proof because it is only
an assumption.
We
know, by Gödel's Theorem that incompleteness does exist. So the initial >> proposition cannot hold, or it is in an inconsistent system.
Only on the basis of the assumption that
~Provable(PA, g) does not mean ~True(PA, g)
Get rid of that single assumption AND EVERYTHING CHANGES
[ .... ]
--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
On 11/9/2024 1:32 PM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
The assumption that ~Provable(PA, g) does not mean ~True(PA, g)
cannot correctly be the basis for any proof because it is only
an assumption.
It is an assumption which swifly leads to a contradiction, therefore must
be false.
You just said that the current foundation of logic leads to a
contradiction. Too many negations you got confused.
When we assume that only provable from the axioms
of PA derives True(PA, g) then (PA ⊢ g) merely means
~True(PA, g) THIS DOES NOT LEAD TO ANY CONTRADICTION.
But you don't understand the concept of proof by contradiction, and
you lack the basic humility to accept what experts say, so I don't
expect this to sink in.
We know, by Gödel's Theorem that incompleteness does exist. So the
initial proposition cannot hold, or it is in an inconsistent system.
Only on the basis of the assumption that
~Provable(PA, g) does not mean ~True(PA, g)
No, there is no such assumption. There are definitions of provable and
of true, and Gödel proved that these cannot be identical.
*He never proved that they cannot be identical*
The way that sound deductive inference is defined
to work is that they must be identical.
A conclusion IS ONLY true when applying truth
preserving operations to true premises.
It is very stupid of you to say that Gödel refuted that.
--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
On 11/9/2024 1:32 PM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
The assumption that ~Provable(PA, g) does not mean ~True(PA, g)
cannot correctly be the basis for any proof because it is only
an assumption.
It is an assumption which swifly leads to a contradiction, therefore must
be false.
You just said that the current foundation of logic leads to a
contradiction. Too many negations you got confused.
When we assume that only provable from the axioms
of PA derives True(PA, g) then (PA ⊢ g) merely means
~True(PA, g) THIS DOES NOT LEAD TO ANY CONTRADICTION.
But you don't understand the concept of proof by
contradiction, and you lack the basic humility to accept what experts
say, so I don't expect this to sink in.
We know, by Gödel's Theorem that incompleteness does exist. So the
initial proposition cannot hold, or it is in an inconsistent system.
Only on the basis of the assumption that
~Provable(PA, g) does not mean ~True(PA, g)
No, there is no such assumption. There are definitions of provable and
of true, and Gödel proved that these cannot be identical.
*He never proved that they cannot be identical*
The way that sound deductive inference is defined
to work is that they must be identical.
A conclusion IS ONLY true when applying truth
preserving operations to true premises.
It is very stupid of you to say that Gödel refuted that.
On 11/9/2024 2:53 PM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/9/2024 1:32 PM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
The way that sound deductive inference is defined
to work is that they must be identical.
Whatever "sound deductive inference" means. If you are right, then
"sound deductive inference" is incoherent garbage.
*Validity and Soundness*
A deductive argument is said to be valid if and only
if it takes a form that makes it impossible for the
premises to be true and the conclusion nevertheless
to be false. Otherwise, a deductive argument is said
to be invalid.
A deductive argument is sound if and only if it is
both valid, and all of its premises are actually
true. Otherwise, a deductive argument is unsound. https://iep.utm.edu/val-snd/
Thus your ignorance and not mine.
A conclusion IS ONLY true when applying truth
preserving operations to true premises.
I'm not sure what that adds to the argument.
It is already specified that a conclusion can only be
true when truth preserving operations are applied to
expressions of language known to be true.
That Gödel's proof didn't understand that this <is>
the actual foundation of mathematical logic is his
mistake.
Unprovable in PA has always meant untrue in PA when
viewed within the deductive inference foundation of
mathematical logic.
--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
On 11/9/2024 3:45 PM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/9/2024 2:53 PM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/9/2024 1:32 PM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
Is a very stupid thing to say."sound deductive inference" is incoherent garbage.
A conclusion IS ONLY true when applying truth
preserving operations to true premises.
I'm not sure what that adds to the argument.
It is already specified that a conclusion can only be
true when truth preserving operations are applied to
expressions of language known to be true.
That Gödel's proof didn't understand that this <is>
the actual foundation of mathematical logic is his
mistake.
You're lying by lack of expertise, again. Gödel understood mathematical
logic full well (indeed, played a significant part in its development),
He utterly failed to understand that his understanding
of provable in meta-math cannot mean true in PA unless
also provable in PA according to the deductive inference
foundation of all logic.
and he made no mistakes in his proof. Had he done so, they would have
been identified by other mathematicians by now.
That other people shared his lack of understanding
is no evidence that it is not a lack of understanding.
Unprovable in PA has always meant untrue in PA when
viewed within the deductive inference foundation of
mathematical logic.
Yet another lie by lack of expertise.
Truth is not any majority rule.
That everyone else got this wrong
is not my mistake.
Unprovable and untrue have been proven to be different things, whether
in the system of counting numbers or anything else containing it.
Generically epistemology always requires provability.
Mathematical knowledge is not allowed to diverge from
the way that knowledge itself generically works.
Whatever you might mean by "the deductive inference foundation of
mathematical logic" - is that another one of your "trademarks"?
Do you think that mathematical logic just popped
into existence fully formed out of no where?
All coherent knowledge fits into an inheritance hierarchy
knowledge ontology. A non fit means incoherence.
https://en.wikipedia.org/wiki/Ontology_(information_science)
--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
On 11/9/2024 12:47 PM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:You are not doing that. I am redefining the foundation of the notion of
On 11/9/2024 11:58 AM, Alan Mackenzie wrote:Not at all. I denigrate your lies, where by lies I mean the emphatic
olcott <polcott333@gmail.com> wrote:
On 11/9/2024 10:03 AM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/9/2024 5:01 AM, joes wrote:
On 11/8/24 12:25 PM, olcott wrote:
That you denigrate what I say without paying attention to what I sayStop swearing. I don't pay much attention to your provably false*Like I said you don't pay f-cking attention*That's untrue - you don't have a precise specification. And evenThat is counter-factual within my precise specification.Gödel showed otherwise.That formal systems that only apply truth preserving >>>>>>>>>>>>>>>>> operations to expressions of their formal language that >>>>>>>>>>>>>>>>> have been stipulated to be true cannot possibly be >>>>>>>>>>>>>>>>> undecidable is proven to be over-your-head on the basis >>>>>>>>>>>>>>>>> that you have no actual reasoning as a rebuttal.
if you did, Gödel's theorem would still hold.
When truth is only derived by starting with truth and applyingNo. Unprovable will remain.
truth preserving operations then unprovable in PA becomes untrue >>>>>>> in PA.
utterances, no. Life is too short.
<is> the definition of reckless disregard for the truth that loses
defamation cases.
utterances of falsehood due to a lack of expertise in the subject
matter.
See the beginning of this subthread.
a formal system and calling this a lie can have your house confiscated
for defamation.
You are the one with reckless disregard for the truth. You haven't
even bothered to read the introductory texts which would help you
understand what the truth is.
I have no fear of you starting a defamation case against me. For a
start, you'd have to learn some German, and for another thing, I'd win
on the merits.
Hint: Gödel's theorem applies in any sufficiently powerful logicalUnless it is stipulated at the foundation of the notion of formal
system, and the bar for "sufficiently powerful" is not high.
systems that ~Provable(PA, g) simply means ~True(PA, g).
If you're going to redefine the word provable to mean something else,I am correcting the somewhat ill-founded notion of provable to only mean applying truth preserving operations to finite string expressions of language.
you'll need some other word to mean what provable means to everybody
else.
It’s a very safe assumption, as it keeps both possibilities for theZFC DID NOT STAY WITHIN THE BOUNDS OF NAIVE SET THEORYNo, they didn't do the same thing. They stayed within the bounds ofZFC did the same thing and that was the ONLY way that Russell'sUnprovable(L,x) means Untrue(L,x)If you're going to change the standard meaning of standard words,
Unprovable(L,~x) means Unfalse(L,x)
~True(L,x) ^ ~True(L, ~x) means ~Truth-Bearer(L,x)
you'll find communicating with other people somewhat strained and
difficult.
Paradox was resolved.
logic.
And yes, they resolved a paradox. There is no paradox for yourThe assumption that ~Provable(PA, g) does not mean ~True(PA, g) cannot correctly be the basis for any proof because it is only an assumption.
"system" to resolve, even if it were logically coherent.
When ~Provable(PA,g) means ~True(PA,g) then incompleteness cannotOK, That's a proof by contradiction that ~provable cannot mean ~true.
exist.
We know, by Gödel's Theorem that incompleteness does exist. So theOnly on the basis of the assumption that ~Provable(PA, g) does not mean ~True(PA, g)
initial proposition cannot hold, or it is in an inconsistent system.
Get rid of that single assumption AND EVERYTHING CHANGES
On 11/9/2024 4:35 PM, joes wrote:
Am Sat, 09 Nov 2024 13:00:22 -0600 schrieb olcott:
On 11/9/2024 12:47 PM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:You are not doing that. I am redefining the foundation of the notion of
On 11/9/2024 11:58 AM, Alan Mackenzie wrote:Not at all. I denigrate your lies, where by lies I mean the emphatic >>>> utterances of falsehood due to a lack of expertise in the subject
olcott <polcott333@gmail.com> wrote:
On 11/9/2024 10:03 AM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/9/2024 5:01 AM, joes wrote:
On 11/8/24 12:25 PM, olcott wrote:
That you denigrate what I say without paying attention to what I say >>>>> <is> the definition of reckless disregard for the truth that losesStop swearing. I don't pay much attention to your provably false >>>>>> utterances, no. Life is too short.*Like I said you don't pay f-cking attention*That's untrue - you don't have a precise specification. And even >>>>>>>> if you did, Gödel's theorem would still hold.That is counter-factual within my precise specification.Gödel showed otherwise.That formal systems that only apply truth preserving >>>>>>>>>>>>>>>>>>> operations to expressions of their formal language that >>>>>>>>>>>>>>>>>>> have been stipulated to be true cannot possibly be >>>>>>>>>>>>>>>>>>> undecidable is proven to be over-your-head on the basis >>>>>>>>>>>>>>>>>>> that you have no actual reasoning as a rebuttal.
When truth is only derived by starting with truth and applying >>>>>>>>> truth preserving operations then unprovable in PA becomes untrue >>>>>>>>> in PA.No. Unprovable will remain.
defamation cases.
matter.
See the beginning of this subthread.
a formal system and calling this a lie can have your house confiscated
for defamation.
Go on, sue him, liar.
You are the one with reckless disregard for the truth. You haven't
even bothered to read the introductory texts which would help you
understand what the truth is.
I have no fear of you starting a defamation case against me. For a
start, you'd have to learn some German, and for another thing, I'd win >>>> on the merits.
Hint: Gödel's theorem applies in any sufficiently powerful logical >>>>>> system, and the bar for "sufficiently powerful" is not high.Unless it is stipulated at the foundation of the notion of formal
systems that ~Provable(PA, g) simply means ~True(PA, g).
That doesn’t make ~g provable.
If you're going to redefine the word provable to mean something else,I am correcting the somewhat ill-founded notion of provable to only mean >>> applying truth preserving operations to finite string expressions of
you'll need some other word to mean what provable means to everybody
else.
language.
What else do you think it meant?
https://en.wikipedia.org/wiki/Principle_of_explosion
Does not apply any truth preserving operations to
its premises.
If ~Provable in PA was understood to mean ~True
in PA then Gödel could not exist.
ZFC DID NOT STAY WITHIN THE BOUNDS OF NAIVE SET THEORYNo, they didn't do the same thing. They stayed within the bounds ofZFC did the same thing and that was the ONLY way that Russell'sUnprovable(L,x) means Untrue(L,x)If you're going to change the standard meaning of standard words,
Unprovable(L,~x) means Unfalse(L,x)
~True(L,x) ^ ~True(L, ~x) means ~Truth-Bearer(L,x)
you'll find communicating with other people somewhat strained and
difficult.
Paradox was resolved.
logic.
And yes, they resolved a paradox. There is no paradox for yourThe assumption that ~Provable(PA, g) does not mean ~True(PA, g) cannot
"system" to resolve, even if it were logically coherent.
When ~Provable(PA,g) means ~True(PA,g) then incompleteness cannotOK, That's a proof by contradiction that ~provable cannot mean ~true.
exist.
correctly be the basis for any proof because it is only an assumption.
It’s a very safe assumption, as it keeps both possibilities for the
truth value of g open.
It directly causes false conclusions by violating
the sound deductive inference model.
It is wasn't for stupid mistakes like this one
Nazi propaganda would have been put down as false
before it had a chance to take root in the USA
and swing the elections.
We know, by Gödel's Theorem that incompleteness does exist. So theOnly on the basis of the assumption that ~Provable(PA, g) does not mean
initial proposition cannot hold, or it is in an inconsistent system.
~True(PA, g)
Get rid of that single assumption AND EVERYTHING CHANGES
On 11/9/2024 10:03 AM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/9/2024 5:01 AM, joes wrote:
On 11/8/24 12:25 PM, olcott wrote:
Gödel showed otherwise.That formal systems that only apply truth preserving operations >>>>>>>>>>>>> to expressions of their formal language that have been >>>>>>>>>>>>> stipulated to be true cannot possibly be undecidable is proven >>>>>>>>>>>>> to be over-your-head on the basis that you have no actual >>>>>>>>>>>>> reasoning as a rebuttal.
That is counter-factual within my precise specification.
That's untrue - you don't have a precise specification. And even if you
did, Gödel's theorem would still hold.
When truth is only derived by starting with
truth and applying truth preserving operations
then unprovable in PA becomes untrue in PA.
No. Unprovable will remain.
*Like I said you don't pay f-cking attention*
On 11/9/2024 5:01 AM, joes wrote:
Am Fri, 08 Nov 2024 18:39:34 -0600 schrieb olcott:
On 11/8/2024 6:33 PM, Richard Damon wrote:Gödel showed otherwise.
On 11/8/24 6:36 PM, olcott wrote:
On 11/8/2024 3:59 PM, Richard Damon wrote:
On 11/8/24 4:17 PM, olcott wrote:
On 11/8/2024 12:31 PM, Richard Damon wrote:
On 11/8/24 1:08 PM, olcott wrote:
On 11/8/2024 12:02 PM, Richard Damon wrote:
On 11/8/24 12:25 PM, olcott wrote:
That formal systems that only apply truth preserving operations >>>>>>>>>>> to expressions of their formal language that have been
stipulated to be true cannot possibly be undecidable is proven >>>>>>>>>>> to be over-your-head on the basis that you have no actual >>>>>>>>>>> reasoning as a rebuttal.
That is counter-factual within my precise specification.
When truth is only derived by starting with
truth and applying truth preserving operations
then unprovable in PA becomes untrue in PA.
Everyone is so sure that whatever I say must be wrong
On 11/9/2024 4:28 PM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/9/2024 3:45 PM, Alan Mackenzie wrote:
Gödel understood mathematical logic full well (indeed, played a
significant part in its development),
He utterly failed to understand that his understanding
of provable in meta-math cannot mean true in PA unless
also provable in PA according to the deductive inference
foundation of all logic.
You're lying in your usual fashion, namely by lack of expertise. It is
entirely your lack of understanding. If Gödel's proof was not rigorously >> correct, his result would have been long discarded. It is correct.
Even if every other detail is 100% correct without
"true and unprovable" (the heart of incompleteness)
it utterly fails to make its incompleteness conclusion.
Perhaps you simply don't understand it at that level
thus will never have any idea that I proved I am correct.
--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
On 11/9/2024 11:58 AM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/9/2024 10:03 AM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/9/2024 5:01 AM, joes wrote:
On 11/8/24 12:25 PM, olcott wrote:
Gödel showed otherwise.That formal systems that only apply truth preserving operations >>>>>>>>>>>>>>> to expressions of their formal language that have been >>>>>>>>>>>>>>> stipulated to be true cannot possibly be undecidable is proven >>>>>>>>>>>>>>> to be over-your-head on the basis that you have no actual >>>>>>>>>>>>>>> reasoning as a rebuttal.
That is counter-factual within my precise specification.
That's untrue - you don't have a precise specification. And even if you >>>> did, Gödel's theorem would still hold.
When truth is only derived by starting with
truth and applying truth preserving operations
then unprovable in PA becomes untrue in PA.
No. Unprovable will remain.
*Like I said you don't pay f-cking attention*
Stop swearing. I don't pay much attention to your provably false
utterances, no. Life is too short.
That you denigrate what I say without paying attention to what
I say <is> the definition of reckless disregard for the truth
that loses defamation cases.
Hint: Gödel's theorem applies in any sufficiently powerful logical
system, and the bar for "sufficiently powerful" is not high.
Unless it is stipulated at the foundation of the notion of
formal systems that ~Provable(PA, g) simply means ~True(PA, g).
Unprovable(L,x) means Untrue(L,x)
Unprovable(L,~x) means Unfalse(L,x)
~True(L,x) ^ ~True(L, ~x) means ~Truth-Bearer(L,x)
If you're going to change the standard meaning of standard words, you'll
find communicating with other people somewhat strained and difficult.
ZFC did the same thing and that was the ONLY way
that Russell's Paradox was resolved.
When ~Provable(PA,g) means ~True(PA,g) then
incompleteness cannot exist.
On 11/10/2024 4:03 AM, Alan Mackenzie wrote:
In comp.theory olcott <polcott333@gmail.com> wrote:
On 11/9/2024 4:28 PM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/9/2024 3:45 PM, Alan Mackenzie wrote:
[ .... ]
Gödel understood mathematical logic full well (indeed, played a
significant part in its development),
He utterly failed to understand that his understanding
of provable in meta-math cannot mean true in PA unless
also provable in PA according to the deductive inference
foundation of all logic.
You're lying in your usual fashion, namely by lack of expertise. It
is entirely your lack of understanding. If Gödel's proof was not
rigorously correct, his result would have been long discarded. It
is correct.
Even if every other detail is 100% correct without
"true and unprovable" (the heart of incompleteness)
it utterly fails to make its incompleteness conclusion.
You are, of course, wrong here. You are too ignorant to make such a
judgment. I believe you've never even read through and verified a proof
of Gödel's theorem.
If you had a basis in reasoning to show that I was wrong
on this specific point you could provide it.
You have no basis in reasoning on this specific point all you have is presumption.
Perhaps you simply don't understand it at that level
thus will never have any idea that I proved I am correct.
More lies. You don't even understand what the word "proved" means.
Here is what Mathworld construes as proof ....
--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
On 11/9/2024 2:53 PM, Alan Mackenzie wrote:I see nothing about provability in there. I mean, if something is
olcott <polcott333@gmail.com> wrote:
On 11/9/2024 1:32 PM, Alan Mackenzie wrote:I did not say that, at least I didn't mean to. You've trimmed the
olcott <polcott333@gmail.com> wrote:
You just said that the current foundation of logic leads to aThe assumption that ~Provable(PA, g) does not mean ~True(PA, g)It is an assumption which swifly leads to a contradiction, therefore
cannot correctly be the basis for any proof because it is only an
assumption.
must be false.
contradiction. Too many negations you got confused.
context unusually severely, so it's difficult to see what I did say.
When we assume that only provable from the axioms of PA derivesI can't make out your weasel word "derives". There are true things in
True(PA, g) then (PA ⊢ g) merely means ~True(PA, g) THIS DOES NOT LEAD >>> TO ANY CONTRADICTION.
any system which can't be proved in that system. Unless that system is
inconsistent, or so restricted in scope that it can't do counting.
This is another example of lying by lack of expertise. You are simply*He never proved that they cannot be identical*No, there is no such assumption. There are definitions of provableWe know, by Gödel's Theorem that incompleteness does exist. So the >>>>>> initial proposition cannot hold, or it is in an inconsistentOnly on the basis of the assumption that ~Provable(PA, g) does not
system.
mean ~True(PA, g)
and of true, and Gödel proved that these cannot be identical.
wrong, there.
A deductive argument is sound if and only if it is both valid, and allThe way that sound deductive inference is defined to work is that theyWhatever "sound dedective inference" means. If you are right, then
must be identical.
"sound deductive inference" is incoherent garbage.
of its premises are actually true. Otherwise, a deductive argument is unsound.
No, it can also be true without a derivation even existing. ThisIt is already specified that a conclusion can only be true when truth preserving operations are applied to expressions of language known to be true.A conclusion IS ONLY true when applying truth preserving operations toI'm not sure what that adds to the argument.
true premises.
That Gödel's proof didn't understand that this <is> the actualHe did not think one could derive true expressions by applying
foundation of mathematical logic is his mistake.
Unprovable in PA has always meant untrue in PA when viewed within the deductive inference foundation of mathematical logic.No. Even then it leaves the negation of the unprovable statement
On 11/10/2024 4:03 AM, Alan Mackenzie wrote:
In comp.theory olcott <polcott333@gmail.com> wrote:
On 11/9/2024 4:28 PM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/9/2024 3:45 PM, Alan Mackenzie wrote:
[ .... ]
Gödel understood mathematical logic full well (indeed, played a
significant part in its development),
He utterly failed to understand that his understanding
of provable in meta-math cannot mean true in PA unless
also provable in PA according to the deductive inference
foundation of all logic.
You're lying in your usual fashion, namely by lack of expertise. It is >>>> entirely your lack of understanding. If Gödel's proof was not
rigorously
correct, his result would have been long discarded. It is correct.
Even if every other detail is 100% correct without
"true and unprovable" (the heart of incompleteness)
it utterly fails to make its incompleteness conclusion.
You are, of course, wrong here. You are too ignorant to make such a
judgment. I believe you've never even read through and verified a proof
of Gödel's theorem.
If you had a basis in reasoning to show that I was wrong
on this specific point you could provide it. You have no
basis in reasoning on this specific point all you have is
presumption.
Perhaps you simply don't understand it at that level
thus will never have any idea that I proved I am correct.
More lies. You don't even understand what the word "proved" means.
Here is what Mathworld construes as proof
A rigorous mathematical argument which unequivocally
demonstrates the truth of a given proposition. A
mathematical statement that has been proven is called
a theorem. https://mathworld.wolfram.com/Proof.html
the principle of explosion is the law according to which any statement
can be proven from a contradiction. https://en.wikipedia.org/wiki/Principle_of_explosion
Validity and Soundness
A deductive argument is said to be valid if and only
if it takes a form that makes it impossible for the
premises to be true and the conclusion nevertheless
to be false. Otherwise, a deductive argument is said
to be invalid.
A deductive argument is sound if and only if it is
both valid, and all of its premises are actually true.
Otherwise, a deductive argument is unsound.
https://iep.utm.edu/val-snd/
Here is the PL Olcott correction / clarification of all of
them. A proof begins with a set of expressions of language
known to be true (true premises) and derives a conclusion
that is a necessary consequence by applying truth preserving
operations to the true premises.
Mathworld
is correct yet fails to provide enough details.
The principle of explosion
is incorrect because its conclusion is not a necessary
consequence of applying truth preserving operations.
It fails to require semantic relevance.
Validity and Soundness
is incorrect because its conclusion is not a necessary
consequence of applying truth preserving operations.
It fails to require semantic relevance.
--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
On 11/10/2024 10:37 AM, Alan Mackenzie wrote:
In comp.theory olcott <polcott333@gmail.com> wrote:
On 11/10/2024 4:03 AM, Alan Mackenzie wrote:
In comp.theory olcott <polcott333@gmail.com> wrote:
On 11/9/2024 4:28 PM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/9/2024 3:45 PM, Alan Mackenzie wrote:
[ .... ]
Gödel understood mathematical logic full well (indeed, played a >>>>>>>> significant part in its development),
He utterly failed to understand that his understanding
of provable in meta-math cannot mean true in PA unless
also provable in PA according to the deductive inference
foundation of all logic.
You're lying in your usual fashion, namely by lack of expertise. It >>>>>> is entirely your lack of understanding. If Gödel's proof was not >>>>>> rigorously correct, his result would have been long discarded. It >>>>>> is correct.
Even if every other detail is 100% correct without
"true and unprovable" (the heart of incompleteness)
it utterly fails to make its incompleteness conclusion.
You are, of course, wrong here. You are too ignorant to make such a
judgment. I believe you've never even read through and verified a proof >>>> of Gödel's theorem.
If you had a basis in reasoning to show that I was wrong
on this specific point you could provide it.
I have read through and understood a proof of Gödel's theorem, and it was >> correct. Therefore you are wrong in what you assert. You have never
read such a proof, otherwise you would have said so. Therefore, on this
matter, you are ignorant, certainly when compared with me.
You have no basis in reasoning on this specific point all you have is
presumption.
It is you who is lacking any basis in what you say. I have already given
my bases for calling out your falsehoods.
Perhaps you simply don't understand it at that level
thus will never have any idea that I proved I am correct.
More lies. You don't even understand what the word "proved" means.
Here is what Mathworld construes as proof ....
I didn't say you couldn't search the web and find descriptions of what a
proof is. I said that you, you personally, don't understand those
descriptions.
I would furthermore propose you have never read and understood a
mathematical proof, and I also propose you have never constructed such a
proof yourself. If I am wrong here, feel free to counter these
propositions.
A thorough understanding of mathematical proof is a prerequisite for
talking meaningfully about things like Gödel's therem. You lack that
prerequisite, therefore all your false statements about it are lies by
lack of expertise.
In other words you can only dodge and thus not address my
specific point ....
.... and can only assert that you generally believe that I must somehow
be wrong ....
.... even if you yourself can't possibly point out exact where and how
this specific point is in any way incorrect:
Even if every other detail of Gödel's proof is 100% correct
when we require that true in PA requires a sequence of truth
preserving operations from the axioms of PA, then unprovable
is PA merely means untrue in PA and does not show that PA is
in any way incomplete.
You don't actually understand these things ....
.... you merely dogmatically accept that Gödel must be correct entirely
on the basis that so many people believe that he is correct.
--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
On 11/10/2024 1:13 PM, Richard Damon wrote:I still can’t see how this makes ~C provable.
On 11/10/24 10:11 AM, olcott wrote:
On 11/10/2024 4:03 AM, Alan Mackenzie wrote:
In comp.theory olcott <polcott333@gmail.com> wrote:
On 11/9/2024 4:28 PM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/9/2024 3:45 PM, Alan Mackenzie wrote:
Sorry, but until you actually and formally fully define your logicWhen C is a necessary consequence of the Haskell Curry elementary
system, you can't start using it.
theorems of L (Thus stipulated to be true in L) then and only then is C
is True in L.
This simple change does get rid of incompleteness because Incomplete(L)
is superseded and replaced by Incorrect(L,x).
On 11/10/2024 1:04 PM, Alan Mackenzie wrote:
I have addressed your point perfectly well. Gödel's theorem is correct,
therefore you are wrong. What part of that don't you understand?
YOU FAIL TO SHOW THE DETAILS OF HOW THIS DOES
NOT GET RID OF INCOMPLETENESS.
--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
On 11/10/2024 1:13 PM, Richard Damon wrote:
On 11/10/24 10:11 AM, olcott wrote:
On 11/10/2024 4:03 AM, Alan Mackenzie wrote:
In comp.theory olcott <polcott333@gmail.com> wrote:
On 11/9/2024 4:28 PM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/9/2024 3:45 PM, Alan Mackenzie wrote:
[ .... ]
Gödel understood mathematical logic full well (indeed, played a >>>>>>>> significant part in its development),
He utterly failed to understand that his understanding
of provable in meta-math cannot mean true in PA unless
also provable in PA according to the deductive inference
foundation of all logic.
You're lying in your usual fashion, namely by lack of expertise.Even if every other detail is 100% correct without
It is
entirely your lack of understanding. If Gödel's proof was not
rigorously
correct, his result would have been long discarded. It is correct. >>>>
"true and unprovable" (the heart of incompleteness)
it utterly fails to make its incompleteness conclusion.
You are, of course, wrong here. You are too ignorant to make such a
judgment. I believe you've never even read through and verified a
proof
of Gödel's theorem.
If you had a basis in reasoning to show that I was wrong
on this specific point you could provide it. You have no
basis in reasoning on this specific point all you have is
presumption.
If you gave some actual formal basis for your reasoning, then perhaps
a formal reply could be made.
Since your arguement starts with mis-interpreatations of what Godel's
proof does, you start off in error.
Perhaps you simply don't understand it at that level
thus will never have any idea that I proved I am correct.
More lies. You don't even understand what the word "proved" means.
Here is what Mathworld construes as proof
A rigorous mathematical argument which unequivocally
demonstrates the truth of a given proposition. A
mathematical statement that has been proven is called
a theorem. https://mathworld.wolfram.com/Proof.html
the principle of explosion is the law according to which any
statement can be proven from a contradiction.
https://en.wikipedia.org/wiki/Principle_of_explosion
Right, and I have shown your that proof, and you haven't shown what
statement in that proof is wrong, so you have accepted it.
Thus, YOU are the one disagreeing with yourself.
Validity and Soundness
A deductive argument is said to be valid if and only
if it takes a form that makes it impossible for the
premises to be true and the conclusion nevertheless
to be false. Otherwise, a deductive argument is said
to be invalid.
A deductive argument is sound if and only if it is
both valid, and all of its premises are actually true.
Otherwise, a deductive argument is unsound.
https://iep.utm.edu/val-snd/
Here is the PL Olcott correction / clarification of all of
them. A proof begins with a set of expressions of language
known to be true (true premises) and derives a conclusion
that is a necessary consequence by applying truth preserving
operations to the true premises.
But you aren't allowed to CHANGE those meanings.
Within the philosophy of logic assumptions
can be changed to see where t that lead.
Sorry, but until you actually and formally fully define your logic
system, you can't start using it.
We don't really have a symbols for truth preserving operations.
When C is a necessary consequence of the Haskell Curry
elementary theorems of L (Thus stipulated to be true in L)
then and only then is C is True in L. https://www.liarparadox.org/Haskell_Curry_45.pdf
(Haskell_Curry_Elementary_Theorems(L) □ C) ≡ True(L, C)
This simple change does get rid of incompleteness because
Incomplete(L) is superseded and replaced by Incorrect(L,x).
And, if you want to talk in your logic system, you can't say it
refutes arguments built in other logic system.
ZFC proves that naive set theory was incoherent.
Russell's paradox still exists in incoherent naive set theory.
On 11/10/2024 2:36 PM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/10/2024 1:04 PM, Alan Mackenzie wrote:
[ .... ]
I have addressed your point perfectly well. Gödel's theorem is correct, >>>> therefore you are wrong. What part of that don't you understand?
YOU FAIL TO SHOW THE DETAILS OF HOW THIS DOES
NOT GET RID OF INCOMPLETENESS.
The details are unimportant. Gödel's theorem is correct.
In other words you simply don't understand these
things well enough ....
.... to understand that when we change their basis the conclusion
changes.
You are a learned-by-rote guy that accepts what you
memorized as infallible gospel.
Your ideas contradict that theorem.
When we start with a different foundation then incompleteness
ceases to exist just like the different foundation of ZFC
eliminates Russell's Paradox.
Therefore your ideas are incorrect. Again, the precise details are
unimportant,
So you have no clue how ZFC eliminated Russell's Paradox.
The details are unimportant and you never heard of ZFC
or Russell's Paradox anyway.
--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
On 11/10/2024 3:52 AM, Mikko wrote:
On 2024-11-09 18:05:38 +0000, olcott said:
On 11/9/2024 11:58 AM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/9/2024 10:03 AM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/9/2024 5:01 AM, joes wrote:
On 11/8/24 12:25 PM, olcott wrote:
Gödel showed otherwise.That formal systems that only apply truth preserving >>>>>>>>>>>>>>>>> operations
to expressions of their formal language that have been >>>>>>>>>>>>>>>>> stipulated to be true cannot possibly be undecidable is >>>>>>>>>>>>>>>>> proven
to be over-your-head on the basis that you have no actual >>>>>>>>>>>>>>>>> reasoning as a rebuttal.
That is counter-factual within my precise specification.
That's untrue - you don't have a precise specification. And even >>>>>> if you
did, Gödel's theorem would still hold.
When truth is only derived by starting with
truth and applying truth preserving operations
then unprovable in PA becomes untrue in PA.
No. Unprovable will remain.
*Like I said you don't pay f-cking attention*
Stop swearing. I don't pay much attention to your provably false
utterances, no. Life is too short.
That you denigrate what I say without paying attention to what
I say <is> the definition of reckless disregard for the truth
that loses defamation cases.
Hint: Gödel's theorem applies in any sufficiently powerful logical
system, and the bar for "sufficiently powerful" is not high.
Unless it is stipulated at the foundation of the notion of
formal systems that ~Provable(PA, g) simply means ~True(PA, g).
Unprovable(L,x) means Untrue(L,x)
Unprovable(L,~x) means Unfalse(L,x)
~True(L,x) ^ ~True(L, ~x) means ~Truth-Bearer(L,x)
If you're going to change the standard meaning of standard words,
you'll
find communicating with other people somewhat strained and difficult.
ZFC did the same thing and that was the ONLY way
that Russell's Paradox was resolved.
When ~Provable(PA,g) means ~True(PA,g) then
incompleteness cannot exist.
But it doesn't. "Provable(PA,g)" means that there is a proof on g in PA
and "~Provable(PA,g)" means that there is not. These meanings are don't
involve your "True" in any way. You may define "True" as a synonym to
"Provable" but formal synonyms are not useful.
We can ALWAYS prove that any expression of language is true or
not on the basis of other expressions of language when we have a
coherent definition of True(L,x).
That Gödel relies on True(meta-math, g) to mean True(PA, g)
is a stupid mistake that enables Incomplete(PA) to exist.
On 11/10/2024 10:03 PM, Richard Damon wrote:
On 11/10/24 10:08 PM, olcott wrote:
On 11/10/2024 3:52 AM, Mikko wrote:
On 2024-11-09 18:05:38 +0000, olcott said:
On 11/9/2024 11:58 AM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/9/2024 10:03 AM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/9/2024 5:01 AM, joes wrote:
On 11/8/24 12:25 PM, olcott wrote:
g can be proven in meta-math.That Gödel relies on True(meta-math, g) to mean True(PA, g)Which just shows you don't understand how formal systems, and their
is a stupid mistake that enables Incomplete(PA) to exist.
meta-systems are constructed.
Are trollish head games really worth the possible cost of eternalAbsolutely.
damnation?
On 11/10/2024 3:52 AM, Mikko wrote:
On 2024-11-09 18:05:38 +0000, olcott said:
On 11/9/2024 11:58 AM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/9/2024 10:03 AM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/9/2024 5:01 AM, joes wrote:
On 11/8/24 12:25 PM, olcott wrote:
Gödel showed otherwise.That formal systems that only apply truth preserving operations
to expressions of their formal language that have been >>>>>>>>>>>>>>>>> stipulated to be true cannot possibly be undecidable is proven
to be over-your-head on the basis that you have no actual >>>>>>>>>>>>>>>>> reasoning as a rebuttal.
That is counter-factual within my precise specification.
That's untrue - you don't have a precise specification. And even if you
did, Gödel's theorem would still hold.
When truth is only derived by starting with
truth and applying truth preserving operations
then unprovable in PA becomes untrue in PA.
No. Unprovable will remain.
*Like I said you don't pay f-cking attention*
Stop swearing. I don't pay much attention to your provably false
utterances, no. Life is too short.
That you denigrate what I say without paying attention to what
I say <is> the definition of reckless disregard for the truth
that loses defamation cases.
Hint: Gödel's theorem applies in any sufficiently powerful logical
system, and the bar for "sufficiently powerful" is not high.
Unless it is stipulated at the foundation of the notion of
formal systems that ~Provable(PA, g) simply means ~True(PA, g).
Unprovable(L,x) means Untrue(L,x)
Unprovable(L,~x) means Unfalse(L,x)
~True(L,x) ^ ~True(L, ~x) means ~Truth-Bearer(L,x)
If you're going to change the standard meaning of standard words, you'll >>>> find communicating with other people somewhat strained and difficult.
ZFC did the same thing and that was the ONLY way
that Russell's Paradox was resolved.
When ~Provable(PA,g) means ~True(PA,g) then
incompleteness cannot exist.
But it doesn't. "Provable(PA,g)" means that there is a proof on g in PA
and "~Provable(PA,g)" means that there is not. These meanings are don't
involve your "True" in any way. You may define "True" as a synonym to
"Provable" but formal synonyms are not useful.
We can ALWAYS prove that any expression of language is true or
not on the basis of other expressions of language when we have a
coherent definition of True(L,x).
That Gödel relies on True(meta-math, g) to mean True(PA, g)
is a stupid mistake that enables Incomplete(PA) to exist.
On 11/10/2024 10:03 PM, Richard Damon wrote:
On 11/10/24 10:08 PM, olcott wrote:
On 11/10/2024 3:52 AM, Mikko wrote:
On 2024-11-09 18:05:38 +0000, olcott said:
On 11/9/2024 11:58 AM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/9/2024 10:03 AM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/9/2024 5:01 AM, joes wrote:
On 11/8/24 12:25 PM, olcott wrote:
Gödel showed otherwise.That formal systems that only apply truth preserving operations
to expressions of their formal language that have been >>>>>>>>>>>>>>>>>>> stipulated to be true cannot possibly be undecidable is proven
to be over-your-head on the basis that you have no actual >>>>>>>>>>>>>>>>>>> reasoning as a rebuttal.
That is counter-factual within my precise specification.
That's untrue - you don't have a precise specification. And even if you
did, Gödel's theorem would still hold.
When truth is only derived by starting with
truth and applying truth preserving operations
then unprovable in PA becomes untrue in PA.
No. Unprovable will remain.
*Like I said you don't pay f-cking attention*
Stop swearing. I don't pay much attention to your provably false >>>>>> utterances, no. Life is too short.
That you denigrate what I say without paying attention to what
I say <is> the definition of reckless disregard for the truth
that loses defamation cases.
Hint: Gödel's theorem applies in any sufficiently powerful logical >>>>>> system, and the bar for "sufficiently powerful" is not high.
Unless it is stipulated at the foundation of the notion of
formal systems that ~Provable(PA, g) simply means ~True(PA, g).
Unprovable(L,x) means Untrue(L,x)
Unprovable(L,~x) means Unfalse(L,x)
~True(L,x) ^ ~True(L, ~x) means ~Truth-Bearer(L,x)
If you're going to change the standard meaning of standard words, you'll >>>>>> find communicating with other people somewhat strained and difficult. >>>>>>
ZFC did the same thing and that was the ONLY way
that Russell's Paradox was resolved.
When ~Provable(PA,g) means ~True(PA,g) then
incompleteness cannot exist.
But it doesn't. "Provable(PA,g)" means that there is a proof on g in PA >>>> and "~Provable(PA,g)" means that there is not. These meanings are don't >>>> involve your "True" in any way. You may define "True" as a synonym to
"Provable" but formal synonyms are not useful.
We can ALWAYS prove that any expression of language is true or
not on the basis of other expressions of language when we have a
coherent definition of True(L,x).
No, we can't.
Proof(Olcott) means a sequence of truth preserving operations
that many not be finite.
On 11/10/2024 10:03 PM, Richard Damon wrote:
On 11/10/24 10:07 PM, olcott wrote:
On 11/10/2024 4:19 PM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/10/2024 2:36 PM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/10/2024 1:04 PM, Alan Mackenzie wrote:
[ .... ]
I have addressed your point perfectly well. Gödel's theorem is >>>>>>>> correct,YOU FAIL TO SHOW THE DETAILS OF HOW THIS DOES
therefore you are wrong. What part of that don't you understand? >>>>
NOT GET RID OF INCOMPLETENESS.
The details are unimportant. Gödel's theorem is correct.
In other words you simply don't understand these
things well enough ....
Not at all. It's you that doesn't understand them well enough to
make it
worthwhile trying to discuss things with you.
.... to understand that when we change their basis the conclusion
changes.
You're at too high a level of abstraction. When your new basis has
counting numbers, it's either inconsistent, or Gödel's theorem
applies to
it.
Finally we are getting somewhere.
You know what levels of abstraction are.
You are a learned-by-rote guy that accepts what you
memorized as infallible gospel.
You're an uneducated boor. So uneducated that you don't grasp that
learning by rote simply doesn't cut it at a university.
Your ideas contradict that theorem.
When we start with a different foundation then incompleteness
ceases to exist just like the different foundation of ZFC
eliminates Russell's Paradox.
No. You'd like it to, but it doesn't work that way.
[ .... ]
Therefore your ideas are incorrect. Again, the precise details are >>>>>> unimportant,
So you have no clue how ZFC eliminated Russell's Paradox.
The details are unimportant and you never heard of ZFC
or Russell's Paradox anyway.
Russell's paradox is a different thing from Gödel's theorem. The
latter
put to rest for ever the vainglorious falsehood that we could prove
everything that was true.
Ah so you don't understand HOW ZFC eliminated Russell's Paradox.
We can ALWAYS prove that any expression of language is true or not
on the basis of other expressions of language when we have a coherent
definition of True(L,x).
No, we can't.
We can sometimes prove it is true if we can find the sequence of steps
that establish it.
We can sometime prove it is false if we can find the sequence of steps
that refute it.
Since there are potentially an INFINITE number of possible proofs for
either of these until we find one of them, we don't know if the
statement IS provable or refutable.
Your problem is you think that knowledge and truth are the same, but
knowledge is only a subset of truth, and there are unknown truths, and
even unknowable truths in any reasonably complicated system.
Part of your issue is you seem to only think in very simple systems
where exhaustive searching might actually be viable.
That Gödel relies on True(meta-math, g) to mean True(PA, g)
is a stupid mistake that enables Incomplete(PA) to exist.
Which just shows you don't understand how formal systems, and their
meta-systems are constructed.
It does not matter how they are constructed the only
thing that matters is the functional end result.
*When we construe True(L,x) this way*
When g is a necessary consequence of the Haskell Curry
elementary theorems of PA (Thus stipulated to be true in PA)
then and only then is g is True in PA.
https://www.liarparadox.org/Haskell_Curry_45.pdf (Haskell_Curry_Elementary_Theorems(PA) □ g) ≡ True(PA, g)
If there is no sequence of truth preserving operations
in PA from its Haskell_Curry_Elementary_Theorems to g
then it can be construed that g is simply not true in PA.
Incorrect(PA,g) ≡ (True(PA, g) ∧ True(PA, ~g))
Your ignorance doesn't make the claim not true, just shows that you
are just stupid and a pathological liar.
That you say this without providing any supporting reasoning
indicates that you may not have an actual clue about these
thing and instead only have mere empty bluster.
I am not a liar and you are acting like a goofy nitwit.
On 11/10/2024 2:39 PM, joes wrote:
Am Sun, 10 Nov 2024 14:07:44 -0600 schrieb olcott:
On 11/10/2024 1:13 PM, Richard Damon wrote:I still can’t see how this makes ~C provable.
On 11/10/24 10:11 AM, olcott wrote:
On 11/10/2024 4:03 AM, Alan Mackenzie wrote:
In comp.theory olcott <polcott333@gmail.com> wrote:
On 11/9/2024 4:28 PM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/9/2024 3:45 PM, Alan Mackenzie wrote:
Sorry, but until you actually and formally fully define your logicWhen C is a necessary consequence of the Haskell Curry elementary
system, you can't start using it.
theorems of L (Thus stipulated to be true in L) then and only then is C
is True in L.
This simple change does get rid of incompleteness because Incomplete(L)
is superseded and replaced by Incorrect(L,x).
If C is not provable it is merely rejected as incorrect
not used as any basis to determine that L is incomplete.
For many reasons: "A sequence of truth preserving operations"
is a much better term than the term "provable".
On 11/11/2024 4:26 AM, Mikko wrote:
On 2024-11-11 03:08:36 +0000, olcott said:
On 11/10/2024 3:52 AM, Mikko wrote:
On 2024-11-09 18:05:38 +0000, olcott said:
On 11/9/2024 11:58 AM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/9/2024 10:03 AM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/9/2024 5:01 AM, joes wrote:
On 11/8/24 12:25 PM, olcott wrote:
Gödel showed otherwise.That formal systems that only apply truth preserving >>>>>>>>>>>>>>>>>>> operations
to expressions of their formal language that have been >>>>>>>>>>>>>>>>>>> stipulated to be true cannot possibly be undecidable >>>>>>>>>>>>>>>>>>> is proven
to be over-your-head on the basis that you have no >>>>>>>>>>>>>>>>>>> actual
reasoning as a rebuttal.
That is counter-factual within my precise specification.
That's untrue - you don't have a precise specification. And
even if you
did, Gödel's theorem would still hold.
When truth is only derived by starting with
truth and applying truth preserving operations
then unprovable in PA becomes untrue in PA.
No. Unprovable will remain.
*Like I said you don't pay f-cking attention*
Stop swearing. I don't pay much attention to your provably false >>>>>> utterances, no. Life is too short.
That you denigrate what I say without paying attention to what
I say <is> the definition of reckless disregard for the truth
that loses defamation cases.
Hint: Gödel's theorem applies in any sufficiently powerful logical >>>>>> system, and the bar for "sufficiently powerful" is not high.
Unless it is stipulated at the foundation of the notion of
formal systems that ~Provable(PA, g) simply means ~True(PA, g).
Unprovable(L,x) means Untrue(L,x)
Unprovable(L,~x) means Unfalse(L,x)
~True(L,x) ^ ~True(L, ~x) means ~Truth-Bearer(L,x)
If you're going to change the standard meaning of standard words,
you'll
find communicating with other people somewhat strained and difficult. >>>>>>
ZFC did the same thing and that was the ONLY way
that Russell's Paradox was resolved.
When ~Provable(PA,g) means ~True(PA,g) then
incompleteness cannot exist.
But it doesn't. "Provable(PA,g)" means that there is a proof on g in PA >>>> and "~Provable(PA,g)" means that there is not. These meanings are don't >>>> involve your "True" in any way. You may define "True" as a synonym to
"Provable" but formal synonyms are not useful.
We can ALWAYS prove that any expression of language is true or
not on the basis of other expressions of language when we have a
coherent definition of True(L,x).
Not relevant.
It <is> relevant in that it does refute the Tarski
Undefinability theorem that <is> isomorphic to incompleteness.
The meaning of "Provable(PA,g)" does not depend on
the definition of "True(L,x)". "Provable(PA,g)" is false because
there is no proof of g in PA. For the same reason "Provable(PA,~g)"
is false.
There is no proof of Tarski's x in his Theory only
because x is incoherent in his theory. https://liarparadox.org/Tarski_275_276.pdf
Let {T} be such a theory. Then the elementary
statements which belong to {T} we shall call the
elementary theorems of {T}; we also say that
these elementary statements are true for {T}.
Thus, given {T}, an elementary theorem is an
elementary statement which is true.
https://www.liarparadox.org/Haskell_Curry_45.pdf
Haskell Curry is referring to a set of expressions that are
stipulated to be true in T.
We define True(L, x) to mean x is a necessary consequence of
the Haskell Curry elementary theorems of L. (Haskell_Curry_Elementary_Theorems(L) □ x) ≡ True(L, x)
x = "What time is it?"
True(English, x) == false
True(English, ~x) == false
∴ Not_a_Truth_Bearer(English, x)
Under math rules we would declare that English is incomplete
because neither x nor ~x is provable in English.
There are actually infinitely many sentences of PA that could be used
instead of g to show incompleteness but one is enoubh.
That Gödel relies on True(meta-math, g) to mean True(PA, g)
is a stupid mistake that enables Incomplete(PA) to exist.
Gödel proved Provable(meta-math, "~Provable(PA,g) ∧ ~Provable(PA,g)").
That is the same thing as proving:
This sentence is not true: "This sentence is not true" is true.
On 11/6/2024 10:45 AM, Alan Mackenzie wrote:
Andy Walker <anw@cuboid.co.uk> wrote:
On 04/11/2024 14:05, Mikko wrote:
Then show how two statements about distinct topics can disagree.I disagree. [:-)]That is not a disagreement.[...] The statement itself does not changeDisagree. There is a clear advantage in distinguishing those >>>>>>> who make [honest] mistakes from those who wilfully mislead.
when someone states it so there is no clear advantage in
saying that the statement was not a lie until someone stated
it.
You've had the free, introductory five-minute argument; the
half-hour argument has to be paid for. [:-)]
[Perhaps more helpfully, "distinct" is your invention. One same >>> statement can be either true or false, a mistake or a lie, depending on
the context (time. place and motivation) within which it is uttered.
Plenty of examples both in everyday life and in science, inc maths. Eg, >>> "It's raining!", "The angles of a triangle sum to 180 degrees.", "The
Sun goes round the Earth.". Each of those is true in some contexts, false >>> and a mistake in others, false and a lie in yet others. English has clear >>> distinctions between these, which it is useful to maintain; it is not
useful to describe them as "lies" in the absence of any context, eg when >>> the statement has not yet been uttered.]
There is another sense in which something could be a lie. If, for
example, I emphatically asserted some view about the minutiae of medical
surgery, in opposition to the standard view accepted by practicing
surgeons, no matter how sincere I might be in that belief, I would be
lying. Lying by ignorance.
That is a lie unless you qualify your statement with X is a
lie(unintentional false statement). It is more truthful to
say that statement X is rejected as untrue by a consensus of
medical opinion.
This allows for the possibility that the consensus is not
infallible. No one here allows for the possibility that the
current received view is not infallible. Textbooks on the
theory of computation are NOT the INFALLIBLE word of God.
Peter Olcott is likewise ignorant about mathematical logic. So in
that sense, the false things he continually asserts _are_ lies.
*It is not at all that I am ignorant of mathematical logic* It is that
I am not a mindless robot that is programmed by textbook opinions.
--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
On 11/6/2024 10:45 AM, Alan Mackenzie wrote:
Andy Walker <anw@cuboid.co.uk> wrote:
On 04/11/2024 14:05, Mikko wrote:
Then show how two statements about distinct topics can disagree.I disagree. [:-)]That is not a disagreement.[...] The statement itself does not changeDisagree. There is a clear advantage in distinguishing those
when someone states it so there is no clear advantage in
saying that the statement was not a lie until someone stated
it.
who make [honest] mistakes from those who wilfully mislead.
You've had the free, introductory five-minute argument; the
half-hour argument has to be paid for. [:-)]
[Perhaps more helpfully, "distinct" is your invention. One same
statement can be either true or false, a mistake or a lie, depending on
the context (time. place and motivation) within which it is uttered.
Plenty of examples both in everyday life and in science, inc maths. Eg, >>> "It's raining!", "The angles of a triangle sum to 180 degrees.", "The
Sun goes round the Earth.". Each of those is true in some contexts,
false
and a mistake in others, false and a lie in yet others. English has
clear
distinctions between these, which it is useful to maintain; it is not
useful to describe them as "lies" in the absence of any context, eg when >>> the statement has not yet been uttered.]
There is another sense in which something could be a lie. If, for
example, I empatically asserted some view about the minutiae of medical
surgery, in opposition to the standard view accepted by practicing
surgeons, no matter how sincere I might be in that belief, I would be
lying. Lying by ignorance.
That is a lie unless you qualify your statement with X is a
lie(unintentional false statement). It is more truthful to
say that statement X is rejected as untrue by a consensus of
medical opinion.
This allows for the possibility that the consensus is not
infallible. No one here allows for the possibility that the
current received view is not infallible. Textbooks on the
theory of computation are NOT the INFALLIBLE word of God.
Peter Olcott is likewise ignorant about mathematical logic. So in that
sense, the false things he continually asserts _are_ lies.
*It is not at all that I am ignorant of mathematical logic*
It is that I am not a mindless robot that is programmed by
textbook opinions.
Just like ZFC corrected the error of naive set theory
alternative views on mathematical logic do resolve their
Russell's Paradox like issues.
(Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)))
When True(L,x) is only a sequence of truth preserving operations
applied to x in L and False(L, x) is only a sequence of truth
preserving operations applied to ~x in L then Incomplete(L)
becomes Not_Truth_Bearer(L,x).
This is not any lack of understanding of mathematical logic.
It is my refusing to be a mindless robot and accept mathematical
logic as it is currently defined as inherently infallible.
--
Andy Walker, Nottingham.
Andy's music pages: www.cuboid.me.uk/andy/Music
Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Peerson >>
On 11/6/2024 2:34 PM, Alan Mackenzie wrote:
In comp.theory olcott <polcott333@gmail.com> wrote:
On 11/6/2024 10:45 AM, Alan Mackenzie wrote:
There is another sense in which something could be a lie. If, for
example, I emphatically asserted some view about the minutiae of
medical surgery, in opposition to the standard view accepted by
practicing surgeons, no matter how sincere I might be in that
belief, I would be lying. Lying by ignorance.
That is a lie unless you qualify your statement with X is a
lie(unintentional false statement). It is more truthful to
say that statement X is rejected as untrue by a consensus of
medical opinion.
No, as so often, you've missed the nuances. The essence of the
scenario is making emphatic statements in a topic which requires
expertise, but that expertise is missing. Such as me laying down the
law about surgery or you doing the same in mathematical logic.
It is not at all my lack of expertise on mathematical logic
it is your ignorance of philosophy of logic as shown by you
lack of understanding of the difference between "a priori"
and "a posteriori" knowledge.
Surgical procedures and mathematical logic are in fundamentally
different classes of knowledge.
This allows for the possibility that the consensus is not
infallible. No one here allows for the possibility that the
current received view is not infallible. Textbooks on the
theory of computation are NOT the INFALLIBLE word of God.
Gods have got nothing to do with it. 2 + 2 = 4, the fact that the
world is a ball, not flat, Gödel's theorem, and the halting problem,
have all been demonstrated beyond any doubt whatsoever.
Regarding the last two they would have said the same thing about
Russell's Paradox and what is now known as naive set theory at the
time.
That you can't begin to imagine that mathematical logic might
not be infallible is definitely an error on your part ....
.... as proven by your failure to point put any error in the following:
(Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)))
--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
On 11/8/2024 5:58 AM, Alan Mackenzie wrote:
In comp.theory olcott <polcott333@gmail.com> wrote:
On 11/6/2024 2:34 PM, Alan Mackenzie wrote:
In comp.theory olcott <polcott333@gmail.com> wrote:
On 11/6/2024 10:45 AM, Alan Mackenzie wrote:
[ .... ]
There is another sense in which something could be a lie. If, for >>>>>> example, I emphatically asserted some view about the minutiae of
medical surgery, in opposition to the standard view accepted by
practicing surgeons, no matter how sincere I might be in that
belief, I would be lying. Lying by ignorance.
That is a lie unless you qualify your statement with X is a
lie(unintentional false statement). It is more truthful to
say that statement X is rejected as untrue by a consensus of
medical opinion.
No, as so often, you've missed the nuances. The essence of the
scenario is making emphatic statements in a topic which requires
expertise, but that expertise is missing. Such as me laying down the
law about surgery or you doing the same in mathematical logic.
It is not at all my lack of expertise on mathematical logic
it is your ignorance of philosophy of logic as shown by you
lack of understanding of the difference between "a priori"
and "a posteriori" knowledge.
Garbage.
Surgical procedures and mathematical logic are in fundamentally
different classes of knowledge.
But the necessity of expertise is present in both, equally. Emphatically
to assert falsehoods when expertise is lacking is a form of lying. That
is what you do.
This allows for the possibility that the consensus is not
infallible. No one here allows for the possibility that the
current received view is not infallible. Textbooks on the
theory of computation are NOT the INFALLIBLE word of God.
Gods have got nothing to do with it. 2 + 2 = 4, the fact that the
world is a ball, not flat, Gödel's theorem, and the halting problem,
have all been demonstrated beyond any doubt whatsoever.
Regarding the last two they would have said the same thing about
Russell's Paradox and what is now known as naive set theory at the
time.
There's no "would have said" regarding Russell's paradox. Nobody would
have asserted the correctness of naive set theory, a part of mathematics
then at the forefront of research and still in flux. We've moved beyond
that point in the last hundred years.
And you are continually stating that theorems like 2 + 2 = 4 are false.
That is a lie. I never said anything like that and you know it.
Here is what I actually said:
When the operations are limited to applying truth preserving
operations to expressions of language that are stipulated to
be true then
True(L,x) ≡ (L ⊢ x) and False(L, x) ≡ (L ⊢ ~x)
Then
(Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)))
becomes
(¬TruthBearer(L,x) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x))) Incompleteness utterly ceases to exist
--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
On 11/8/2024 9:05 AM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/8/2024 5:58 AM, Alan Mackenzie wrote:
And you are continually stating that theorems like 2 + 2 = 4 are false.
That is a lie. I never said anything like that and you know it.
Now who's lying? You have frequently denied the truth of proven
mathematical facts like 2 + 2 = 4.
Never and you are a damned (going to actual Hell) liar for
saying so.
As I have continually made clear in
my posts "like 2 + 2 = 4" includes the halting theorem, Gödel's theorem,
and Tarski's theorem.
Your misconceptions are not my errors.
You cannot possibly prove that they are infallible
that best that you can show is that you believe they
are infallible.
Here is what I actually said:
When the operations are limited to applying truth preserving
operations to expressions of language that are stipulated to
be true then
True(L,x) ≡ (L ⊢ x) and False(L, x) ≡ (L ⊢ ~x)
Then
(Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)))
becomes
(¬TruthBearer(L,x) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x))) >>> Incompleteness utterly ceases to exist
Incompleteness is an essential property of logic systems
Rejecting what I say out-of-hand on the basis that you don't
believe what I say is far far less than no rebuttal at all.
What I said about is a semantic tautology just like
2 + 3 = 5. Formal systems are only incomplete when
the term "incomplete" is a euphemism for the inability
of formal systems to correctly determine the truth
value of non-truth-bearers.
which can do anything at all. If what you assert is true (which I
doubt), then your system would be incapable of doing anything useful.
--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
On 11/8/2024 10:02 AM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/8/2024 9:05 AM, Alan Mackenzie wrote:
Now who's lying? You have frequently denied the truth of proven
mathematical facts like 2 + 2 = 4.
Never and you are a damned (going to actual Hell) liar for
saying so.
Hahahaha! There is no actual Hell.
Let me repeat: you have frequently denied the truth of proven
mathematical facts like 2 + 2 = 4.
As I have continually made clear in
my posts "like 2 + 2 = 4" includes the halting theorem, Gödel's theorem, >>>> and Tarski's theorem.
Your misconceptions are not my errors.
It is you who has misconceptions, evident to all in this newsgroup who
have studied the subject.
My "mistakes" are merely the presumption that the current
received view of these things is infallible.
You cannot possibly prove that they are infallible
that best that you can show is that you believe they
are infallible.
Here is where your lack of expertise shows itself. All the above
theorems have been proven beyond any doubt.
Within their faulty foundations.
In the same way that naive set theory was a faulty foundation.
It was not initially called naive set theory. It was only called
that when someone noticed its error.
In that respect they are all like 2 + 2 = 4. But you're right in a
sense. I couldn't personally prove these things any more; but I know
where to go to find the proofs. And I don't "believe they are
infallible"; I've studied, understood, and checked proofs that they
are true.
OK good some honesty.
Incompleteness is an essential property of logic systems
Rejecting what I say out-of-hand on the basis that you don't
believe what I say is far far less than no rebuttal at all.
As I said, it's not a matter of "belief". It's a matter of certain
knowledge stemming from having studied for and having a degree in maths.
You understand what the received view is.
My view is inconsistent with the received view therefore
(when one assumes that the received view is infallible)
I must be wrong.
I reject what you say because it's objectively wrong. Just as if you
said 2 + 2 = 5.
What I said about is a semantic tautology just like
2 + 3 = 5. Formal systems are only incomplete when
the term "incomplete" is a euphemism for the inability
of formal systems to correctly determine the truth
value of non-truth-bearers.
No. You lack the expertise.
I know how the current systems work and I disagree
that they are correct. This is not any lack of expertise.
As you already admitted you don't understand these
things well enough to even see what I am saying.
--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
On 11/8/2024 9:05 AM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/8/2024 5:58 AM, Alan Mackenzie wrote:
In comp.theory olcott <polcott333@gmail.com> wrote:
On 11/6/2024 2:34 PM, Alan Mackenzie wrote:
In comp.theory olcott <polcott333@gmail.com> wrote:
On 11/6/2024 10:45 AM, Alan Mackenzie wrote:
[ .... ]
There is another sense in which something could be a lie. If, for >>>>>>>> example, I emphatically asserted some view about the minutiae of >>>>>>>> medical surgery, in opposition to the standard view accepted by >>>>>>>> practicing surgeons, no matter how sincere I might be in that
belief, I would be lying. Lying by ignorance.
That is a lie unless you qualify your statement with X is a
lie(unintentional false statement). It is more truthful to
say that statement X is rejected as untrue by a consensus of
medical opinion.
No, as so often, you've missed the nuances. The essence of the
scenario is making emphatic statements in a topic which requires
expertise, but that expertise is missing. Such as me laying down the >>>>>> law about surgery or you doing the same in mathematical logic.
It is not at all my lack of expertise on mathematical logic
it is your ignorance of philosophy of logic as shown by you
lack of understanding of the difference between "a priori"
and "a posteriori" knowledge.
Garbage.
Surgical procedures and mathematical logic are in fundamentally
different classes of knowledge.
But the necessity of expertise is present in both, equally.
Emphatically
to assert falsehoods when expertise is lacking is a form of lying.
That
is what you do.
This allows for the possibility that the consensus is not
infallible. No one here allows for the possibility that the
current received view is not infallible. Textbooks on the
theory of computation are NOT the INFALLIBLE word of God.
Gods have got nothing to do with it. 2 + 2 = 4, the fact that the >>>>>> world is a ball, not flat, Gödel's theorem, and the halting problem, >>>>>> have all been demonstrated beyond any doubt whatsoever.
Regarding the last two they would have said the same thing about
Russell's Paradox and what is now known as naive set theory at the
time.
There's no "would have said" regarding Russell's paradox. Nobody would >>>> have asserted the correctness of naive set theory, a part of
mathematics
then at the forefront of research and still in flux. We've moved
beyond
that point in the last hundred years.
And you are continually stating that theorems like 2 + 2 = 4 are false.
That is a lie. I never said anything like that and you know it.
Now who's lying? You have frequently denied the truth of proven
mathematical facts like 2 + 2 = 4.
Never and you are a damned (going to actual Hell) liar for
saying so.
As I have continually made clear in
my posts "like 2 + 2 = 4" includes the halting theorem, Gödel's theorem,
and Tarski's theorem.
Your misconceptions are not my errors.
You cannot possibly prove that they are infallible
that best that you can show is that you believe they
are infallible.
Here is what I actually said:
When the operations are limited to applying truth preserving
operations to expressions of language that are stipulated to
be true then
True(L,x) ≡ (L ⊢ x) and False(L, x) ≡ (L ⊢ ~x)
Then
(Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)))
becomes
(¬TruthBearer(L,x) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x))) >>> Incompleteness utterly ceases to exist
Incompleteness is an essential property of logic systems
Rejecting what I say out-of-hand on the basis that you don't
believe what I say is far far less than no rebuttal at all.
What I said about is a semantic tautology just like
2 + 3 = 5. Formal systems are only incomplete when
the term "incomplete" is a euphemism for the inability
of formal systems to correctly determine the truth
value of non-truth-bearers.
which can do
anything at all. If what you assert is true (which I doubt), then your
system would be incapable of doing anything useful.
--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
That formal systems that only apply truth preserving
operations to expressions of their formal language
that have been stipulated to be true cannot possibly
be undecidable is proven to be over-your-head on the
basis that you have no actual reasoning as a rebuttal.
--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
That formal systems that only apply truth preserving
operations to expressions of their formal language
that have been stipulated to be true cannot possibly
be undecidable is proven to be over-your-head on the
basis that you have no actual reasoning as a rebuttal.
On 11/8/2024 12:02 PM, Richard Damon wrote:
On 11/8/24 12:25 PM, olcott wrote:
That formal systems that only apply truth preserving
operations to expressions of their formal language
that have been stipulated to be true cannot possibly
be undecidable is proven to be over-your-head on the
basis that you have no actual reasoning as a rebuttal.
No, all you have done is shown that you don't undertstand what you are
talking about.
Godel PROVED that the FORMAL SYSTEM that his proof started in, is
unable to PROVE that the statement G, being "that no Natural Number g,
that satifies a particularly designed Primitive Recursive
Relationship" is true, but also shows (using the Meta-Mathematics that
derived the PRR for the original Formal System) that no such number
can exist.
The equivocation of switching formal systems from PA to meta-math.
None of what I say is all that difficult unless one's
primary purpose is to be disagreeable.
--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
On 11/8/2024 9:05 AM, Alan Mackenzie wrote:They are proven. Show the error.
olcott <polcott333@gmail.com> wrote:
On 11/8/2024 5:58 AM, Alan Mackenzie wrote:
In comp.theory olcott <polcott333@gmail.com> wrote:
On 11/6/2024 2:34 PM, Alan Mackenzie wrote:
In comp.theory olcott <polcott333@gmail.com> wrote:
On 11/6/2024 10:45 AM, Alan Mackenzie wrote:
As I have continually made clear in my posts "like 2 + 2 = 4" includesYour misconceptions are not my errors.
the halting theorem, Gödel's theorem, and Tarski's theorem.
You cannot possibly prove that they are infallible that best that you
can show is that you believe they are infallible.
What is the truth value of non-truth-bearers then?What I said about is a semantic tautology just like 2 + 3 = 5. FormalWhen the operations are limited to applying truth preservingIncompleteness is an essential property of logic systems
operations to expressions of language that are stipulated to be true
then True(L,x) ≡ (L ⊢ x) and False(L, x) ≡ (L ⊢ ~x)
Then (Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x))) becomes
(¬TruthBearer(L,x) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x))) >>> Incompleteness utterly ceases to exist
systems are only incomplete when the term "incomplete" is a dysphemism
for the inability of formal systems to correctly determine the truth
value of non-truth-bearers.
--which can do anything at all. If what you assert is true (which I
doubt), then your system would be incapable of doing anything useful.
On 11/8/2024 12:32 PM, Alan Mackenzie wrote:
Much of what you say is wrong. That you strongly assert false things
outside your understanding is a form of lying. For what it's worth, I
find you highly disagreeable, and your contempt for truth and knowledge
truly despicable.
That you want to disagree with this .... on the basis of Ad Hominem
attacks makes you look like a nitwit.
--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
On 11/8/2024 12:31 PM, Richard Damon wrote:
On 11/8/24 1:08 PM, olcott wrote:
On 11/8/2024 12:02 PM, Richard Damon wrote:
On 11/8/24 12:25 PM, olcott wrote:
That formal systems that only apply truth preserving
operations to expressions of their formal language
that have been stipulated to be true cannot possibly
be undecidable is proven to be over-your-head on the
basis that you have no actual reasoning as a rebuttal.
No, all you have done is shown that you don't undertstand what you
are talking about.
Godel PROVED that the FORMAL SYSTEM that his proof started in, is
unable to PROVE that the statement G, being "that no Natural Number
g, that satifies a particularly designed Primitive Recursive
Relationship" is true, but also shows (using the Meta-Mathematics
that derived the PRR for the original Formal System) that no such
number can exist.
The equivocation of switching formal systems from PA to meta-math.
No, it just shows you don't understand how meta-systems work.
IT SHOWS THAT I KNOW IT IS STUPID TO
CONSTRUE TRUE IN META-MATH AS TRUE IN PA.
THAT YOU DON'T UNDERSTAND THIS IS STUPID IS YOUR ERROR.
The Formal System is PA. that defines the basic axioms that are to be
used to establish the truth of the statement and where to attempt the
proof of it.
The Meta-Math, is an EXTENSION to PA, where we add a number of
additional axioms, none that contradict any of the axions of PA, but
in particular, assign each axiom and needed proven statement in PA to
a prime number. These provide the additional semantics in the Meta-
Math to understand the new meaning that a number could have, and with
that semantics, using just the mathematics of PA, the PRR is derived
that with the semantics of the MM becomes a proof-checker.
Note, the Meta-Math is carefully constructed so that there is a
correlation of truth, such that anything true in PA is true in MM, and
anything statement shown in MM to be true, that doesn't use the
additional terms defined, is also true in PA.
There is no equivocation in that, as nothing changed meaning, only
some things that didn't have a semantic meaning (like a number) now does.
If you want to try to show an actual error or equivocation, go ahead
and try, but so far, all you have done is shown that you don't even
seem to understand what a Formal System is, since you keep on wanting
to "re- invent them" but just repeat the basic definition of them.
On 11/8/2024 3:59 PM, Richard Damon wrote:
On 11/8/24 4:17 PM, olcott wrote:
On 11/8/2024 12:31 PM, Richard Damon wrote:
On 11/8/24 1:08 PM, olcott wrote:
On 11/8/2024 12:02 PM, Richard Damon wrote:
On 11/8/24 12:25 PM, olcott wrote:
That formal systems that only apply truth preserving
operations to expressions of their formal language
that have been stipulated to be true cannot possibly
be undecidable is proven to be over-your-head on the
basis that you have no actual reasoning as a rebuttal.
No, all you have done is shown that you don't undertstand what you >>>>>> are talking about.
Godel PROVED that the FORMAL SYSTEM that his proof started in, is
unable to PROVE that the statement G, being "that no Natural
Number g, that satifies a particularly designed Primitive
Recursive Relationship" is true, but also shows (using the Meta-
Mathematics that derived the PRR for the original Formal System)
that no such number can exist.
The equivocation of switching formal systems from PA to meta-math.
No, it just shows you don't understand how meta-systems work.
IT SHOWS THAT I KNOW IT IS STUPID TO
CONSTRUE TRUE IN META-MATH AS TRUE IN PA.
THAT YOU DON'T UNDERSTAND THIS IS STUPID IS YOUR ERROR.
But, as I pointed out, the way Meta-Math is derived from PA,
Meta-math <IS NOT> PA.
Meta-math <IS NOT> PA.
Meta-math <IS NOT> PA.
Meta-math <IS NOT> PA.
True in meta-math <IS NOT> True in PA.
True in meta-math <IS NOT> True in PA.
True in meta-math <IS NOT> True in PA.
True in meta-math <IS NOT> True in PA.
This sentence is not true: "This sentence is not true"
is only true because the inner sentence is bullshit gibberish.
On 11/8/2024 6:33 PM, Richard Damon wrote:
On 11/8/24 6:36 PM, olcott wrote:
On 11/8/2024 3:59 PM, Richard Damon wrote:
On 11/8/24 4:17 PM, olcott wrote:
On 11/8/2024 12:31 PM, Richard Damon wrote:
On 11/8/24 1:08 PM, olcott wrote:
On 11/8/2024 12:02 PM, Richard Damon wrote:
On 11/8/24 12:25 PM, olcott wrote:
That formal systems that only apply truth preserving
operations to expressions of their formal language
that have been stipulated to be true cannot possibly
be undecidable is proven to be over-your-head on the
basis that you have no actual reasoning as a rebuttal.
No, all you have done is shown that you don't undertstand what >>>>>>>> you are talking about.
Godel PROVED that the FORMAL SYSTEM that his proof started in, >>>>>>>> is unable to PROVE that the statement G, being "that no Natural >>>>>>>> Number g, that satifies a particularly designed Primitive
Recursive Relationship" is true, but also shows (using the Meta- >>>>>>>> Mathematics that derived the PRR for the original Formal System) >>>>>>>> that no such number can exist.
The equivocation of switching formal systems from PA to meta-math. >>>>>>>
No, it just shows you don't understand how meta-systems work.
IT SHOWS THAT I KNOW IT IS STUPID TO
CONSTRUE TRUE IN META-MATH AS TRUE IN PA.
THAT YOU DON'T UNDERSTAND THIS IS STUPID IS YOUR ERROR.
But, as I pointed out, the way Meta-Math is derived from PA,
Meta-math <IS NOT> PA.
Meta-math <IS NOT> PA.
Meta-math <IS NOT> PA.
Meta-math <IS NOT> PA.
True in meta-math <IS NOT> True in PA.
True in meta-math <IS NOT> True in PA.
True in meta-math <IS NOT> True in PA.
True in meta-math <IS NOT> True in PA.
This sentence is not true: "This sentence is not true"
is only true because the inner sentence is bullshit gibberish.
But MM has exactly the same axioms and rules as PA, so anything
established by that set of axioms and rules in MM is established in PA
too.
There are additional axioms in MM, but the rules are built specifically
One single level of indirect reference CHANGES EVERYTHING.
PA speaks PA. Meta-math speaks ABOUT PA.
The liar paradox is nonsense gibberish except when applied
to itself, then it becomes true.
On 11/8/2024 6:58 PM, Richard Damon wrote:
On 11/8/24 7:39 PM, olcott wrote:
On 11/8/2024 6:33 PM, Richard Damon wrote:
On 11/8/24 6:36 PM, olcott wrote:One single level of indirect reference CHANGES EVERYTHING.
On 11/8/2024 3:59 PM, Richard Damon wrote:
On 11/8/24 4:17 PM, olcott wrote:
On 11/8/2024 12:31 PM, Richard Damon wrote:
On 11/8/24 1:08 PM, olcott wrote:
On 11/8/2024 12:02 PM, Richard Damon wrote:
On 11/8/24 12:25 PM, olcott wrote:
That formal systems that only apply truth preserving
operations to expressions of their formal language
that have been stipulated to be true cannot possibly
be undecidable is proven to be over-your-head on the
basis that you have no actual reasoning as a rebuttal.
No, all you have done is shown that you don't undertstand what >>>>>>>>>> you are talking about.
Godel PROVED that the FORMAL SYSTEM that his proof started in, >>>>>>>>>> is unable to PROVE that the statement G, being "that no
Natural Number g, that satifies a particularly designed
Primitive Recursive Relationship" is true, but also shows
(using the Meta- Mathematics that derived the PRR for the
original Formal System) that no such number can exist.
The equivocation of switching formal systems from PA to meta-math. >>>>>>>>>
No, it just shows you don't understand how meta-systems work.
IT SHOWS THAT I KNOW IT IS STUPID TO
CONSTRUE TRUE IN META-MATH AS TRUE IN PA.
THAT YOU DON'T UNDERSTAND THIS IS STUPID IS YOUR ERROR.
But, as I pointed out, the way Meta-Math is derived from PA,
Meta-math <IS NOT> PA.
Meta-math <IS NOT> PA.
Meta-math <IS NOT> PA.
Meta-math <IS NOT> PA.
True in meta-math <IS NOT> True in PA.
True in meta-math <IS NOT> True in PA.
True in meta-math <IS NOT> True in PA.
True in meta-math <IS NOT> True in PA.
This sentence is not true: "This sentence is not true"
is only true because the inner sentence is bullshit gibberish.
But MM has exactly the same axioms and rules as PA, so anything
established by that set of axioms and rules in MM is established in
PA too.
There are additional axioms in MM, but the rules are built specifically >>>
PA speaks PA. Meta-math speaks ABOUT PA.
The liar paradox is nonsense gibberish except when applied
to itself, then it becomes true.
No, Meta-Math speaks PA, because is includes ALL the axioms and rules
of PA, so it can speak PA.
You just don't understand what a meta-system is.
In C we can have a pointer to a character string
and a pointer to a pointer to a character string.
The pointer to pointer is one level of indirect
reference away form the pointer to the character string.
I know exactly what a meta-system is. It is a system that
refers to the underlying system by one level of indirect
reference. PA talks PA meta-math talks ABOUT PA.
On 11/11/2024 4:33 AM, Mikko wrote:
On 2024-11-11 04:41:24 +0000, olcott said:
On 11/10/2024 10:03 PM, Richard Damon wrote:
On 11/10/24 10:08 PM, olcott wrote:
On 11/10/2024 3:52 AM, Mikko wrote:
On 2024-11-09 18:05:38 +0000, olcott said:
On 11/9/2024 11:58 AM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/9/2024 10:03 AM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/9/2024 5:01 AM, joes wrote:
On 11/8/24 12:25 PM, olcott wrote:
Gödel showed otherwise.That formal systems that only apply truth preserving operations
to expressions of their formal language that have been >>>>>>>>>>>>>>>>>>>>> stipulated to be true cannot possibly be undecidable is proven
to be over-your-head on the basis that you have no actual >>>>>>>>>>>>>>>>>>>>> reasoning as a rebuttal.
That is counter-factual within my precise specification.
That's untrue - you don't have a precise specification. And even if you
did, Gödel's theorem would still hold.
When truth is only derived by starting with
truth and applying truth preserving operations
then unprovable in PA becomes untrue in PA.
No. Unprovable will remain.
*Like I said you don't pay f-cking attention*
Stop swearing. I don't pay much attention to your provably false >>>>>>>> utterances, no. Life is too short.
That you denigrate what I say without paying attention to what
I say <is> the definition of reckless disregard for the truth
that loses defamation cases.
Hint: Gödel's theorem applies in any sufficiently powerful logical >>>>>>>> system, and the bar for "sufficiently powerful" is not high.
Unless it is stipulated at the foundation of the notion of
formal systems that ~Provable(PA, g) simply means ~True(PA, g).
Unprovable(L,x) means Untrue(L,x)
Unprovable(L,~x) means Unfalse(L,x)
~True(L,x) ^ ~True(L, ~x) means ~Truth-Bearer(L,x)
If you're going to change the standard meaning of standard words, you'll
find communicating with other people somewhat strained and difficult. >>>>>>>>
ZFC did the same thing and that was the ONLY way
that Russell's Paradox was resolved.
When ~Provable(PA,g) means ~True(PA,g) then
incompleteness cannot exist.
But it doesn't. "Provable(PA,g)" means that there is a proof on g in PA >>>>>> and "~Provable(PA,g)" means that there is not. These meanings are don't >>>>>> involve your "True" in any way. You may define "True" as a synonym to >>>>>> "Provable" but formal synonyms are not useful.
We can ALWAYS prove that any expression of language is true or
not on the basis of other expressions of language when we have a
coherent definition of True(L,x).
No, we can't.
Proof(Olcott) means a sequence of truth preserving operations
that many not be finite.
With a hyperfinite sequnce it is possible to prove a false claim.
It will always be possible to merely prove a false claim.
What ceases to be possible is proving that a false claim is true.
The most obvious truth preserving operation is the identity operation.
Its result is the same as its premise, so the truth valure of the
result must be the same as the truth value of the premise. So we
can form a hyperfinite sequence
1 = 1, 1 = 1, 1 = 1, ... , 1 = 2, 1 = 2, 1 = 2
where ... denotes infinitely manu intermedate steps. The first equation
is true, every other equation is as ture as the one before it and the
last equation is false.
On 11/11/2024 4:26 AM, Mikko wrote:
On 2024-11-11 03:08:36 +0000, olcott said:
On 11/10/2024 3:52 AM, Mikko wrote:
On 2024-11-09 18:05:38 +0000, olcott said:
On 11/9/2024 11:58 AM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/9/2024 10:03 AM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/9/2024 5:01 AM, joes wrote:
On 11/8/24 12:25 PM, olcott wrote:
Gödel showed otherwise.That formal systems that only apply truth preserving operations
to expressions of their formal language that have been >>>>>>>>>>>>>>>>>>> stipulated to be true cannot possibly be undecidable is proven
to be over-your-head on the basis that you have no actual >>>>>>>>>>>>>>>>>>> reasoning as a rebuttal.
That is counter-factual within my precise specification.
That's untrue - you don't have a precise specification. And even if you
did, Gödel's theorem would still hold.
When truth is only derived by starting with
truth and applying truth preserving operations
then unprovable in PA becomes untrue in PA.
No. Unprovable will remain.
*Like I said you don't pay f-cking attention*
Stop swearing. I don't pay much attention to your provably false >>>>>> utterances, no. Life is too short.
That you denigrate what I say without paying attention to what
I say <is> the definition of reckless disregard for the truth
that loses defamation cases.
Hint: Gödel's theorem applies in any sufficiently powerful logical >>>>>> system, and the bar for "sufficiently powerful" is not high.
Unless it is stipulated at the foundation of the notion of
formal systems that ~Provable(PA, g) simply means ~True(PA, g).
Unprovable(L,x) means Untrue(L,x)
Unprovable(L,~x) means Unfalse(L,x)
~True(L,x) ^ ~True(L, ~x) means ~Truth-Bearer(L,x)
If you're going to change the standard meaning of standard words, you'll >>>>>> find communicating with other people somewhat strained and difficult. >>>>>>
ZFC did the same thing and that was the ONLY way
that Russell's Paradox was resolved.
When ~Provable(PA,g) means ~True(PA,g) then
incompleteness cannot exist.
But it doesn't. "Provable(PA,g)" means that there is a proof on g in PA >>>> and "~Provable(PA,g)" means that there is not. These meanings are don't >>>> involve your "True" in any way. You may define "True" as a synonym to
"Provable" but formal synonyms are not useful.
We can ALWAYS prove that any expression of language is true or
not on the basis of other expressions of language when we have a
coherent definition of True(L,x).
Not relevant.
It <is> relevant in that it does refute the Tarski
Undefinability theorem that <is> isomorphic to incompleteness.
On 11/10/2024 4:19 PM, Alan Mackenzie wrote:
Russell's paradox is a different thing from Gödel's theorem. The latter
put to rest for ever the vainglorious falsehood that we could prove
everything that was true.
Ah so you don't understand HOW ZFC eliminated Russell's Paradox.
We can ALWAYS prove that any expression of language is true or not
on the basis of other expressions of language when we have a coherent definition of True(L,x).
That Gödel relies on True(meta-math, g) to mean True(PA, g)
is a stupid mistake that enables Incomplete(PA) to exist.
--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
On 11/10/2024 2:36 PM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/10/2024 1:04 PM, Alan Mackenzie wrote:
[ .... ]
I have addressed your point perfectly well. Gödel's theorem is
correct,
therefore you are wrong. What part of that don't you understand?
YOU FAIL TO SHOW THE DETAILS OF HOW THIS DOES
NOT GET RID OF INCOMPLETENESS.
The details are unimportant. Gödel's theorem is correct. Your ideas
contradict that theorem. Therefore your ideas are incorrect. Again, the >> precise details are unimportant, and you wouldn't understand them
anyway. Your ideas are as coherent as 2 + 2 = 5.
Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x))
When the above foundational definition ceases to exist then
Gödel's proof cannot prove incompleteness.
*You just don't understand this at its foundational level*
Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x))That only defines the term „incomplete”. The non-derivable sentences continue to exist.
When the above foundational definition ceases to exist then Gödel's
proof cannot prove incompleteness.
On 11/10/2024 2:36 PM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/10/2024 1:04 PM, Alan Mackenzie wrote:
[ .... ]
I have addressed your point perfectly well. Gödel's theorem is correct, >>>> therefore you are wrong. What part of that don't you understand?
YOU FAIL TO SHOW THE DETAILS OF HOW THIS DOES
NOT GET RID OF INCOMPLETENESS.
The details are unimportant. Gödel's theorem is correct. Your ideas
contradict that theorem. Therefore your ideas are incorrect. Again, the
precise details are unimportant, and you wouldn't understand them
anyway. Your ideas are as coherent as 2 + 2 = 5.
Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x))
When the above foundational definition ceases to exist then
Gödel's proof cannot prove incompleteness.
On 11/10/2024 2:36 PM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/10/2024 1:04 PM, Alan Mackenzie wrote:
[ .... ]
I have addressed your point perfectly well. Gödel's theorem is correct, >>>> therefore you are wrong. What part of that don't you understand?
YOU FAIL TO SHOW THE DETAILS OF HOW THIS DOES
NOT GET RID OF INCOMPLETENESS.
The details are unimportant. Gödel's theorem is correct. Your ideas
contradict that theorem. Therefore your ideas are incorrect. Again, the
precise details are unimportant, and you wouldn't understand them
anyway. Your ideas are as coherent as 2 + 2 = 5.
Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x))
When the above foundational definition ceases to exist then
Gödel's proof cannot prove incompleteness.
*You just don't understand this at its foundational level*
--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
On 11/11/2024 9:06 AM, Richard Damon wrote:
On 11/10/24 5:01 PM, olcott wrote:
On 11/10/2024 2:39 PM, joes wrote:
Am Sun, 10 Nov 2024 14:07:44 -0600 schrieb olcott:
On 11/10/2024 1:13 PM, Richard Damon wrote:I still can’t see how this makes ~C provable.
On 11/10/24 10:11 AM, olcott wrote:
On 11/10/2024 4:03 AM, Alan Mackenzie wrote:
In comp.theory olcott <polcott333@gmail.com> wrote:
On 11/9/2024 4:28 PM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/9/2024 3:45 PM, Alan Mackenzie wrote:
Sorry, but until you actually and formally fully define your logic >>>>>> system, you can't start using it.When C is a necessary consequence of the Haskell Curry elementary
theorems of L (Thus stipulated to be true in L) then and only then
is C
is True in L.
This simple change does get rid of incompleteness because
Incomplete(L)
is superseded and replaced by Incorrect(L,x).
If C is not provable it is merely rejected as incorrect
not used as any basis to determine that L is incomplete.
For many reasons: "A sequence of truth preserving operations"
is a much better term than the term "provable".
But since there exist statements that are True but not Provable.
except by your incorrect definition of Provable, your logic is just
broken.
There cannot possibly be any expressions of language that
are true in L that are not determined to be true on the
basis of applying a sequence of truth preserving operations
in L to Haskell_Curry_Elementary_Theorems in L.
https://www.liarparadox.org/Haskell_Curry_45.pdf
Everything that is true on the basis of its meaning
expressed in language is shown to be true this exact
same way.
On 11/13/2024 5:57 AM, Alan Mackenzie wrote:What does incompleteness mean then?
olcott <polcott333@gmail.com> wrote:
On 11/10/2024 2:36 PM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/10/2024 1:04 PM, Alan Mackenzie wrote:
The details are unimportant. Gödel's theorem is correct. Your ideas >>>> contradict that theorem. Therefore your ideas are incorrect. Again,I have addressed your point perfectly well. Gödel's theorem isYOU FAIL TO SHOW THE DETAILS OF HOW THIS DOES NOT GET RID OF
correct,
therefore you are wrong. What part of that don't you understand?
INCOMPLETENESS.
the precise details are unimportant, and you wouldn't understand them
anyway. Your ideas are as coherent as 2 + 2 = 5.
Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)) When the above
foundational definition ceases to exist then Gödel's proof cannot
prove incompleteness.
What on Earth do you mean by a definition "ceasing to exist"? Do youWhen the definition of Incompleteness:
mean you shut your eyes and pretend you can't see it?
Incompleteness exists as a concept, whether you like it or not.
Gödel's theorem is proven, whether you like it or not (evidently the
latter).
Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x))
becomes
¬TruthBearer(L,x) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x))
Then meeting the criteria for incompleteness means something else
entirely and incompleteness can no longer be proven.
Neither did Alan claim that you did.As for your attempts to pretend that unprovable statements are the sameI never said that ~True(L,x) == False(L,x).
as false statements,
I have been saying the direct opposite of your claim forThen if G is false, ~G must be true, but you want it to also be false.
years now. False(L, x) == True(L, ~x)
--Mark Twain got it right when he asked "How many legs does a dog have if
you call a tail a leg?". To which the answer is "Four: calling a tail
a leg doesn't make it one.".
On 11/13/2024 5:57 AM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x))
When the above foundational definition ceases to exist then
Gödel's proof cannot prove incompleteness.
*You just don't understand this at its foundational level*
You make me laugh, sometimes (at you, not with you).
What on Earth do you mean by a definition "ceasing to exist"? Do you
mean you shut your eyes and pretend you can't see it?
It is very easy if your weren't stuck in rebuttal mode
not giving a rat's ass for truth you would already know.
A set as a member of itself ceases to exist in ZFC, thus
making Russell's Paradox cease to exist in ZFC.
Incompleteness exists as a concept, whether you like it or not. Gödel's
theorem is proven, whether you like it or not (evidently the latter).
When the definition of Incompleteness:
Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x))
becomes
¬TruthBearer(L,x) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x))
Then meeting the criteria for incompleteness means something
else entirely and incompleteness can no longer be proven.
After 2000 years most of the greatest experts in the world
still believe that "This sentence is not true" is undecidable
rather than incorrect.
As for your attempts to pretend that unprovable statements are the
same as false statements,
I never said anything like that.
You are so stuck on rebuttal that you can't even keep track on the
exact words that I actually said.
I never said that ~True(L,x) == False(L,x). That is an egregious
error on your part. I have been saying the direct opposite of your
claim for years now. False(L, x) == True(L, ~x)
There cannot possibly be any expressions of language that
are true in L that are not determined to be true on the
basis of applying a sequence of truth preserving operations
in L to Haskell_Curry_Elementary_Theorems in L.
https://www.liarparadox.org/Haskell_Curry_45.pdf
Everything that is true on the basis of its meaning
expressed in language is shown to be true this exact
same way, within this same language.
Logicians take the prior work of other humans as inherently
infallible. Philosophers of logic examine alternative views
that may be more coherent.
Mark Twain got it right when he asked "How many legs
does a dog have if you call a tail a leg?". To which the answer is
"Four: calling a tail a leg doesn't make it one.".
--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
On 11/13/2024 4:45 AM, Mikko wrote:
On 2024-11-12 23:17:20 +0000, olcott said:
On 11/10/2024 2:36 PM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/10/2024 1:04 PM, Alan Mackenzie wrote:
[ .... ]
I have addressed your point perfectly well. Gödel's theorem is
correct,
therefore you are wrong. What part of that don't you understand?
YOU FAIL TO SHOW THE DETAILS OF HOW THIS DOES
NOT GET RID OF INCOMPLETENESS.
The details are unimportant. Gödel's theorem is correct. Your ideas >>>> contradict that theorem. Therefore your ideas are incorrect.
Again, the
precise details are unimportant, and you wouldn't understand them
anyway. Your ideas are as coherent as 2 + 2 = 5.
Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x))
That's correct (although T is usually used instead of L).
Per this definition the first order group theory and the first order
Peano arithmetic are incomplete.
Every language that can by any means express self-contradiction
incorrectly shows that its formal system is incomplete.
It becomes baseless.When the above foundational definition ceases to exist then
Gödel's proof cannot prove incompleteness.
I doesn't cease to exist.
On 11/13/2024 4:45 AM, Mikko wrote:
On 2024-11-12 23:17:20 +0000, olcott said:
On 11/10/2024 2:36 PM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/10/2024 1:04 PM, Alan Mackenzie wrote:
[ .... ]
I have addressed your point perfectly well. Gödel's theorem is correct,
therefore you are wrong. What part of that don't you understand?
YOU FAIL TO SHOW THE DETAILS OF HOW THIS DOES
NOT GET RID OF INCOMPLETENESS.
The details are unimportant. Gödel's theorem is correct. Your ideas >>>> contradict that theorem. Therefore your ideas are incorrect. Again, the
precise details are unimportant, and you wouldn't understand them
anyway. Your ideas are as coherent as 2 + 2 = 5.
Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x))
That's correct (although T is usually used instead of L).
Per this definition the first order group theory and the first order
Peano arithmetic are incomplete.
Every language that can by any means express self-contradiction
incorrectly shows that its formal system is incomplete.
As for your attempts to pretend that unprovable statements are the same
as false statements, Mark Twain got it right when he asked "How many legs does a dog have if you call a tail a leg?". To which the answer is
"Four: calling a tail a leg doesn't make it one.".
On 11/13/24 11:50 AM, olcott wrote:
On 11/13/2024 10:33 AM, joes wrote:
Am Wed, 13 Nov 2024 09:11:13 -0600 schrieb olcott:
On 11/13/2024 5:57 AM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/10/2024 2:36 PM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/10/2024 1:04 PM, Alan Mackenzie wrote:
The details are unimportant. Gödel's theorem is correct. Your ideasI have addressed your point perfectly well. Gödel's theorem is >>>>>>>>> correct,INCOMPLETENESS.
therefore you are wrong. What part of that don't you understand? >>>>>>>> YOU FAIL TO SHOW THE DETAILS OF HOW THIS DOES NOT GET RID OF
contradict that theorem. Therefore your ideas are incorrect. Again, >>>>>>> the precise details are unimportant, and you wouldn't understand them >>>>>>> anyway. Your ideas are as coherent as 2 + 2 = 5.
Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)) When the above
foundational definition ceases to exist then Gödel's proof cannot >>>>>> prove incompleteness.
What on Earth do you mean by a definition "ceasing to exist"? Do you >>>>> mean you shut your eyes and pretend you can't see it?When the definition of Incompleteness:
Incompleteness exists as a concept, whether you like it or not.
Gödel's theorem is proven, whether you like it or not (evidently the >>>>> latter).
Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x))
becomes
¬TruthBearer(L,x) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)) >>>> Then meeting the criteria for incompleteness means something else
entirely and incompleteness can no longer be proven.
What does incompleteness mean then?
Incompleteness ceases to exist the same way that Russell's
Paradox ceases to exist in ZFC.
Not until your create your logic system like Z & F did to make ZFC.
On 2024-11-14 01:09:18 +0000, Richard Damon said:
On 11/13/24 11:50 AM, olcott wrote:
Incompleteness ceases to exist the same way that Russell's
Paradox ceases to exist in ZFC.
Not until your create your logic system like Z & F did to make ZFC.
It wouldn't be that simple. Zermelo and Fraenkel accepted ordinary logic
but Olcott wants to reject that so he would need to start with building
a new logical foundation.
On 11/14/2024 2:52 AM, Mikko wrote:
On 2024-11-14 01:09:18 +0000, Richard Damon said:
On 11/13/24 11:50 AM, olcott wrote:
On 11/13/2024 10:33 AM, joes wrote:
Am Wed, 13 Nov 2024 09:11:13 -0600 schrieb olcott:
On 11/13/2024 5:57 AM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/10/2024 2:36 PM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/10/2024 1:04 PM, Alan Mackenzie wrote:
The details are unimportant. Gödel's theorem is correct. Your >>>>>>>>> ideasI have addressed your point perfectly well. Gödel's theorem is >>>>>>>>>>> correct,YOU FAIL TO SHOW THE DETAILS OF HOW THIS DOES NOT GET RID OF >>>>>>>>>> INCOMPLETENESS.
therefore you are wrong. What part of that don't you
understand?
contradict that theorem. Therefore your ideas are incorrect. >>>>>>>>> Again,
the precise details are unimportant, and you wouldn't
understand them
anyway. Your ideas are as coherent as 2 + 2 = 5.
Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)) When the
above
foundational definition ceases to exist then Gödel's proof cannot >>>>>>>> prove incompleteness.
What on Earth do you mean by a definition "ceasing to exist"? Do >>>>>>> youWhen the definition of Incompleteness:
mean you shut your eyes and pretend you can't see it?
Incompleteness exists as a concept, whether you like it or not.
Gödel's theorem is proven, whether you like it or not (evidently the >>>>>>> latter).
Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)) >>>>>> becomes
¬TruthBearer(L,x) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x))
Then meeting the criteria for incompleteness means something else
entirely and incompleteness can no longer be proven.
What does incompleteness mean then?
Incompleteness ceases to exist the same way that Russell's
Paradox ceases to exist in ZFC.
Not until your create your logic system like Z & F did to make ZFC.
It wouldn't be that simple. Zermelo and Fraenkel accepted ordinary logic
but Olcott wants to reject that so he would need to start with building
a new logical foundation.
Their foundation was not ordinary logic. They began with
the incoherent foundation of naive set theory and fixed it.
It is pretty dumb that you tried to get away with saying
that a set containing itself was a part of ordinary logic.
On 11/14/2024 5:53 PM, Richard Damon wrote:
On 11/14/24 6:40 PM, olcott wrote:
On 11/14/2024 2:39 AM, Mikko wrote:
On 2024-11-13 23:01:50 +0000, olcott said:
On 11/13/2024 4:45 AM, Mikko wrote:
On 2024-11-12 23:17:20 +0000, olcott said:
On 11/10/2024 2:36 PM, Alan Mackenzie wrote:That's correct (although T is usually used instead of L).
olcott <polcott333@gmail.com> wrote:
On 11/10/2024 1:04 PM, Alan Mackenzie wrote:
[ .... ]
I have addressed your point perfectly well. Gödel's theorem >>>>>>>>>> is correct,YOU FAIL TO SHOW THE DETAILS OF HOW THIS DOES
therefore you are wrong. What part of that don't you understand? >>>>>>>>
NOT GET RID OF INCOMPLETENESS.
The details are unimportant. Gödel's theorem is correct. Your >>>>>>>> ideas
contradict that theorem. Therefore your ideas are incorrect. >>>>>>>> Again, the
precise details are unimportant, and you wouldn't understand them >>>>>>>> anyway. Your ideas are as coherent as 2 + 2 = 5.
Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)) >>>>>>
Per this definition the first order group theory and the first order >>>>>> Peano arithmetic are incomplete.
Every language that can by any means express self-contradiction
incorrectly shows that its formal system is incomplete.
That "incorrectly shows" is non-sense. A language does not show,
incorrectly or otherwise. A proof shows but not incorrectly. But
for a proof you need a theory, i.e. more than just a language.
That a theory can't prove something is usually not provable in the
theory itself but usually needs be proven in another theory, one
that can be interpreted as a metatheory.
*So in other words you just don't get it*
When you start with truth and only apply truth preserving
operations then you necessarily end up with truth.
Right, but that truth might not be PROVABLE (by a finite proof that
establishes Knowledge) as Truth is allowed to be established by
infinite chains.
All of analytic truth is specified as relations between
expressions of language. When these relations do not exist
neither does the truth of these expressions.
I am looking at this on the basis of how truth itself
actually works. You are looking at this on the basis
of memorized dogma.
On 11/14/2024 8:42 PM, Richard Damon wrote:
On 11/14/24 9:26 PM, olcott wrote:
On 11/14/2024 5:53 PM, Richard Damon wrote:
On 11/14/24 6:40 PM, olcott wrote:
On 11/14/2024 2:39 AM, Mikko wrote:
On 2024-11-13 23:01:50 +0000, olcott said:
On 11/13/2024 4:45 AM, Mikko wrote:
On 2024-11-12 23:17:20 +0000, olcott said:
On 11/10/2024 2:36 PM, Alan Mackenzie wrote:That's correct (although T is usually used instead of L).
olcott <polcott333@gmail.com> wrote:
On 11/10/2024 1:04 PM, Alan Mackenzie wrote:
[ .... ]
I have addressed your point perfectly well. Gödel's theorem >>>>>>>>>>>> is correct,
therefore you are wrong. What part of that don't you >>>>>>>>>>>> understand?
YOU FAIL TO SHOW THE DETAILS OF HOW THIS DOES
NOT GET RID OF INCOMPLETENESS.
The details are unimportant. Gödel's theorem is correct. >>>>>>>>>> Your ideas
contradict that theorem. Therefore your ideas are incorrect. >>>>>>>>>> Again, the
precise details are unimportant, and you wouldn't understand them >>>>>>>>>> anyway. Your ideas are as coherent as 2 + 2 = 5.
Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)) >>>>>>>>
Per this definition the first order group theory and the first >>>>>>>> order
Peano arithmetic are incomplete.
Every language that can by any means express self-contradiction
incorrectly shows that its formal system is incomplete.
That "incorrectly shows" is non-sense. A language does not show,
incorrectly or otherwise. A proof shows but not incorrectly. But
for a proof you need a theory, i.e. more than just a language.
That a theory can't prove something is usually not provable in the >>>>>> theory itself but usually needs be proven in another theory, one
that can be interpreted as a metatheory.
*So in other words you just don't get it*
When you start with truth and only apply truth preserving
operations then you necessarily end up with truth.
Right, but that truth might not be PROVABLE (by a finite proof that
establishes Knowledge) as Truth is allowed to be established by
infinite chains.
All of analytic truth is specified as relations between
expressions of language. When these relations do not exist
neither does the truth of these expressions.
But in FORMAL LOGIC, that analytic Truth is specified as the axioms of
the system, and the approved logical operations for the system.
You confuse "Formal Logic" with "Philosophy" due to your ignorance of
them.
I am looking at this on the basis of how truth itself
actually works. You are looking at this on the basis
of memorized dogma.
No, because you logic is based on LIES, because you are trying to
redefine fundamental terms within the system, as opposed to doiing the
work to make a system the way you want, likely because you are just to
ignorant to do the work,
Logic never has been free to override and supersede how
truth itself fundamentally works.
Logic confused itself by not breaking things down to
their barest essence. There is no such thing as any
analytic expression of language that is true having
nothing that shows it is true.
If Goldbach conjecture is true then there is some
finite or infinite sequence of truth preserving
operations that shows this, otherwise it is not true.
On 11/14/2024 2:52 AM, Mikko wrote:
On 2024-11-14 01:09:18 +0000, Richard Damon said:
On 11/13/24 11:50 AM, olcott wrote:
On 11/13/2024 10:33 AM, joes wrote:
Am Wed, 13 Nov 2024 09:11:13 -0600 schrieb olcott:
On 11/13/2024 5:57 AM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/10/2024 2:36 PM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/10/2024 1:04 PM, Alan Mackenzie wrote:
The details are unimportant. Gödel's theorem is correct. Your ideasI have addressed your point perfectly well. Gödel's theorem is >>>>>>>>>>> correct,
therefore you are wrong. What part of that don't you understand? >>>>>>>>>> YOU FAIL TO SHOW THE DETAILS OF HOW THIS DOES NOT GET RID OF >>>>>>>>>> INCOMPLETENESS.
contradict that theorem. Therefore your ideas are incorrect. Again,
the precise details are unimportant, and you wouldn't understand them >>>>>>>>> anyway. Your ideas are as coherent as 2 + 2 = 5.
Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)) When the above
foundational definition ceases to exist then Gödel's proof cannot >>>>>>>> prove incompleteness.
What on Earth do you mean by a definition "ceasing to exist"? Do you >>>>>>> mean you shut your eyes and pretend you can't see it?When the definition of Incompleteness:
Incompleteness exists as a concept, whether you like it or not.
Gödel's theorem is proven, whether you like it or not (evidently the >>>>>>> latter).
Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)) >>>>>> becomes
¬TruthBearer(L,x) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x))
Then meeting the criteria for incompleteness means something else
entirely and incompleteness can no longer be proven.
What does incompleteness mean then?
Incompleteness ceases to exist the same way that Russell's
Paradox ceases to exist in ZFC.
Not until your create your logic system like Z & F did to make ZFC.
It wouldn't be that simple. Zermelo and Fraenkel accepted ordinary logic
but Olcott wants to reject that so he would need to start with building
a new logical foundation.
Their foundation was not ordinary logic. They began with
the incoherent foundation of naive set theory and fixed it.
It is pretty dumb that you tried to get away with saying
that a set containing itself was a part of ordinary logic.
On 11/14/2024 2:39 AM, Mikko wrote:
On 2024-11-13 23:01:50 +0000, olcott said:
On 11/13/2024 4:45 AM, Mikko wrote:
On 2024-11-12 23:17:20 +0000, olcott said:
On 11/10/2024 2:36 PM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/10/2024 1:04 PM, Alan Mackenzie wrote:
[ .... ]
I have addressed your point perfectly well. Gödel's theorem is correct,YOU FAIL TO SHOW THE DETAILS OF HOW THIS DOES
therefore you are wrong. What part of that don't you understand? >>>>>>
NOT GET RID OF INCOMPLETENESS.
The details are unimportant. Gödel's theorem is correct. Your ideas >>>>>> contradict that theorem. Therefore your ideas are incorrect. Again, the
precise details are unimportant, and you wouldn't understand them
anyway. Your ideas are as coherent as 2 + 2 = 5.
Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x))
That's correct (although T is usually used instead of L).
Per this definition the first order group theory and the first order
Peano arithmetic are incomplete.
Every language that can by any means express self-contradiction
incorrectly shows that its formal system is incomplete.
That "incorrectly shows" is non-sense. A language does not show,
incorrectly or otherwise. A proof shows but not incorrectly. But
for a proof you need a theory, i.e. more than just a language.
That a theory can't prove something is usually not provable in the
theory itself but usually needs be proven in another theory, one
that can be interpreted as a metatheory.
*So in other words you just don't get it*
When you start with truth and only apply truth preserving
operations then you necessarily end up with truth.
On 11/14/2024 8:42 PM, Richard Damon wrote:
On 11/14/24 9:26 PM, olcott wrote:
On 11/14/2024 5:53 PM, Richard Damon wrote:
On 11/14/24 6:40 PM, olcott wrote:
On 11/14/2024 2:39 AM, Mikko wrote:
On 2024-11-13 23:01:50 +0000, olcott said:
On 11/13/2024 4:45 AM, Mikko wrote:
On 2024-11-12 23:17:20 +0000, olcott said:
On 11/10/2024 2:36 PM, Alan Mackenzie wrote:
olcott <polcott333@gmail.com> wrote:
On 11/10/2024 1:04 PM, Alan Mackenzie wrote:
That's correct (although T is usually used instead of L).Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)) >>>>>>>>I have addressed your point perfectly well. Gödel's theorem >>>>>>>>>>>> is correct,
therefore you are wrong. What part of that don't you >>>>>>>>>>>> understand?
YOU FAIL TO SHOW THE DETAILS OF HOW THIS DOES NOT GET RID OF >>>>>>>>>>> INCOMPLETENESS.
The details are unimportant. Gödel's theorem is correct. Your >>>>>>>>>> ideas contradict that theorem. Therefore your ideas are
incorrect. Again, the precise details are unimportant, and you >>>>>>>>>> wouldn't understand them anyway. Your ideas are as coherent as >>>>>>>>>> 2 + 2 = 5.
Per this definition the first order group theory and the first >>>>>>>> order Peano arithmetic are incomplete.
Every language that can by any means express self-contradiction
incorrectly shows that its formal system is incomplete.
That "incorrectly shows" is non-sense. A language does not show,
incorrectly or otherwise. A proof shows but not incorrectly. But
for a proof you need a theory, i.e. more than just a language.
That a theory can't prove something is usually not provable in the >>>>>> theory itself but usually needs be proven in another theory, one
that can be interpreted as a metatheory.
When you start with truth and only apply truth preserving operations >>>>> then you necessarily end up with truth.
But in FORMAL LOGIC, that analytic Truth is specified as the axioms ofRight, but that truth might not be PROVABLE (by a finite proof thatAll of analytic truth is specified as relations between expressions of
establishes Knowledge) as Truth is allowed to be established by
infinite chains.
language. When these relations do not exist neither does the truth of
these expressions.
the system, and the approved logical operations for the system.
You confuse "Formal Logic" with "Philosophy" due to your ignorance of
them.
Logic never has been free to override and supersede how truth itself fundamentally works.I am looking at this on the basis of how truth itself actually works.No, because you logic is based on LIES, because you are trying to
You are looking at this on the basis of memorized dogma.
redefine fundamental terms within the system, as opposed to doiing the
work to make a system the way you want, likely because you are just to
ignorant to do the work,
If Goldbach conjecture is true then there is some finite or infiniteYes. In the case of an infinite sequence we can not prove it.
sequence of truth preserving operations that shows this, otherwise it is
not true.
On 11/14/2024 8:42 PM, Richard Damon wrote:
On 11/14/24 9:26 PM, olcott wrote:
On 11/14/2024 5:53 PM, Richard Damon wrote:
On 11/14/24 6:40 PM, olcott wrote:
On 11/14/2024 2:39 AM, Mikko wrote:
On 2024-11-13 23:01:50 +0000, olcott said:
On 11/13/2024 4:45 AM, Mikko wrote:
On 2024-11-12 23:17:20 +0000, olcott said:
On 11/10/2024 2:36 PM, Alan Mackenzie wrote:That's correct (although T is usually used instead of L).
olcott <polcott333@gmail.com> wrote:
On 11/10/2024 1:04 PM, Alan Mackenzie wrote:
[ .... ]
I have addressed your point perfectly well. Gödel's theorem >>>>>>>>>>>> is correct,
therefore you are wrong. What part of that don't you >>>>>>>>>>>> understand?
YOU FAIL TO SHOW THE DETAILS OF HOW THIS DOES
NOT GET RID OF INCOMPLETENESS.
The details are unimportant. Gödel's theorem is correct. >>>>>>>>>> Your ideas
contradict that theorem. Therefore your ideas are incorrect. >>>>>>>>>> Again, the
precise details are unimportant, and you wouldn't understand them >>>>>>>>>> anyway. Your ideas are as coherent as 2 + 2 = 5.
Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)) >>>>>>>>
Per this definition the first order group theory and the first >>>>>>>> order
Peano arithmetic are incomplete.
Every language that can by any means express self-contradiction
incorrectly shows that its formal system is incomplete.
That "incorrectly shows" is non-sense. A language does not show,
incorrectly or otherwise. A proof shows but not incorrectly. But
for a proof you need a theory, i.e. more than just a language.
That a theory can't prove something is usually not provable in the >>>>>> theory itself but usually needs be proven in another theory, one
that can be interpreted as a metatheory.
*So in other words you just don't get it*
When you start with truth and only apply truth preserving
operations then you necessarily end up with truth.
Right, but that truth might not be PROVABLE (by a finite proof that
establishes Knowledge) as Truth is allowed to be established by
infinite chains.
All of analytic truth is specified as relations between
expressions of language. When these relations do not exist
neither does the truth of these expressions.
But in FORMAL LOGIC, that analytic Truth is specified as the axioms of
the system, and the approved logical operations for the system.
You confuse "Formal Logic" with "Philosophy" due to your ignorance of
them.
I am looking at this on the basis of how truth itself
actually works. You are looking at this on the basis
of memorized dogma.
No, because you logic is based on LIES, because you are trying to
redefine fundamental terms within the system, as opposed to doiing the
work to make a system the way you want, likely because you are just to
ignorant to do the work,
Logic never has been free to override and supersede how
truth itself fundamentally works.
Logic confused itself by not breaking things down to
their barest essence. There is no such thing as any
analytic expression of language that is true having
nothing that shows it is true.
If Goldbach conjecture is true then there is some
finite or infinite sequence of truth preserving
operations that shows this, otherwise it is not true.
On 11/15/2024 3:03 AM, Mikko wrote:
On 2024-11-14 23:40:19 +0000, olcott said:
When you start with truth and only apply truth preserving
operations then you necessarily end up with truth.
And if you don't you prove nothing.
That is the basic model of all correct proofs. https://en.wikipedia.org/wiki/Syllogism#Basic_structure
On 11/15/2024 3:03 AM, Mikko wrote:
On 2024-11-14 23:40:19 +0000, olcott said:
When you start with truth and only apply truth preserving
operations then you necessarily end up with truth.
And if you don't you prove nothing.
That is the basic model of all correct proofs. https://en.wikipedia.org/wiki/Syllogism#Basic_structure
On 11/16/2024 3:11 AM, Mikko wrote:
On 2024-11-15 23:49:17 +0000, olcott said:
On 11/15/2024 3:03 AM, Mikko wrote:
On 2024-11-14 23:40:19 +0000, olcott said:
When you start with truth and only apply truth preserving
operations then you necessarily end up with truth.
And if you don't you prove nothing.
That is the basic model of all correct proofs.
https://en.wikipedia.org/wiki/Syllogism#Basic_structure
No, it is not. There are truth preserving transformations that do
not follow that pattern.
There are zero truth preserving operations that are not truth
preserving operations. The principle of explosion is not a
truth preserving operation. The full semantics of natural
can be extended to only apply truth preserving operations
to its own statement of basic fact.
For example, the reduction rule: if A,
B, and C are formulas, the recution rule permits that from
A ∨ B and ¬A ∨ C can be inferred B ∨ C.
That is the way the contradiction is supposed to work
A ∧ ¬A cancel each other out leaving B ∨ C.
A ∧ ¬A ∴ Trump is the Christ is proven (is nuts)
On 11/16/2024 8:32 AM, Richard Damon wrote:
On 11/16/24 9:21 AM, olcott wrote:
On 11/16/2024 3:11 AM, Mikko wrote:
On 2024-11-15 23:49:17 +0000, olcott said:
On 11/15/2024 3:03 AM, Mikko wrote:
On 2024-11-14 23:40:19 +0000, olcott said:
When you start with truth and only apply truth preserving
operations then you necessarily end up with truth.
And if you don't you prove nothing.
That is the basic model of all correct proofs.
https://en.wikipedia.org/wiki/Syllogism#Basic_structure
No, it is not. There are truth preserving transformations that do
not follow that pattern.
There are zero truth preserving operations that are not truth
preserving operations. The principle of explosion is not a
truth preserving operation. The full semantics of natural
can be extended to only apply truth preserving operations
to its own statement of basic fact.
But Syllogism is not the only form of "Truth Preserving Operations".
It is the foundation of necessarily correct reasoning.
True(L,x) ≡ Haskell_Curry_Elementary_Theorems(L) □ x
x is a necessary consequence of the expressions of the
language of L that have been stipulated to be true.
False(L,x) ≡ Haskell_Curry_Elementary_Theorems(L) □ ~x
~x is a necessary consequence of the expressions of the
language of L that have been stipulated to be true.
The above provides the basis for LLM AI systems to
distinguish facts from fictions.
That the provability operator has been replaced
with the necessity operator seems to require semantic
relevance. This prevents logic from diverging from
correct reasoning in many different ways such as
the principle of explosion.
IF that is all you accept, then be prepared for a very limited logic
system.
For example, the reduction rule: if A,
B, and C are formulas, the recution rule permits that from
A ∨ B and ¬A ∨ C can be inferred B ∨ C.
That is the way the contradiction is supposed to work
A ∧ ¬A cancel each other out leaving B ∨ C.
A ∧ ¬A ∴ Trump is the Christ is proven (is nuts)
Just shows you are the one that is NUTS.
Your problem is it seems you only understand the most elementary of
logic, but presume everyone one else is just using that most
elementary of logic.
Yes, With the most restricted set of rules, you can't get to
incompleteness, but that is because you can't create the system with
the power needed for the proof.
The problem is that having the fullness of the logic of Natual Numbers
is enough to cross the line, so your "Complete" Logic system can't
have that, but you just are too stupid to undetstand that limit,
because you don't know how any of your tools actually work.
When True(L,x) and False(L,x) are defined as above then
Truth_Bearer(L,x) ≡ (True(L,x) ∨ False(L,x))
eliminating the notion of Incomplete(L) previously defined by
Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x))
On 11/16/2024 3:11 AM, Mikko wrote:
On 2024-11-15 23:49:17 +0000, olcott said:
On 11/15/2024 3:03 AM, Mikko wrote:
On 2024-11-14 23:40:19 +0000, olcott said:
When you start with truth and only apply truth preserving
operations then you necessarily end up with truth.
And if you don't you prove nothing.
That is the basic model of all correct proofs.
https://en.wikipedia.org/wiki/Syllogism#Basic_structure
No, it is not. There are truth preserving transformations that do
not follow that pattern.
There are zero truth preserving operations that are not truth
preserving operations.
The principle of explosion is not a truth preserving operation.
For example, the reduction rule: if A, B, and C are formulas,
the recution rule permits that from A ∨ B and ¬A ∨ C can be
inferred B ∨ C.
That is the way the contradiction is supposed to work
A ∧ ¬A cancel each other out leaving B ∨ C.
A ∧ ¬A ∴ Trump is the Christ is proven (is nuts)
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