On 10/23/24 9:15 AM, olcott wrote:

On 10/23/2024 2:28 AM, Mikko wrote:

On 2024-10-22 14:02:01 +0000, olcott said:

On 10/22/2024 2:13 AM, Mikko wrote:

On 2024-10-21 13:52:28 +0000, olcott said:

On 10/21/2024 3:41 AM, Mikko wrote:

On 2024-10-20 15:32:45 +0000, olcott said:

The actual barest essence for formal systems and computations
is finite string transformation rules applied to finite strings.

Before you can start from that you need a formal theory that
can be interpreted as a theory of finite strings.

Not at all. The only theory needed are the operations
that can be performed on finite strings:
concatenation, substring, relational operator ...

You may try with an informal foundation but you need to make sure
that it is sufficicently well defined and that is easier with a
formal theory.

The minimal complete theory that I can think of computes
the sum of pairs of ASCII digit strings.

That is easily extended to Peano arithmetic.

As a bottom layer you need some sort of logic. There must be
unambifuous
rules about syntax and inference.

I already wrote this in C a long time ago.
It simply computes the sum the same way
that a first grader would compute the sum.

I have no idea how the first grade arithmetic
algorithm could be extended to PA.

Basically you define that the successor of X is X + 1. The only
primitive function of Peano arithmetic is the successor. Addition
and multiplication are recursively defined from the successor
function. Equality is often included in the underlying logic but
can be defined recursively from the successor function and the
order relation is defined similarly.

Anyway, the details are not important, only that it can be done.

First grade arithmetic can define a successor function
by merely applying first grade arithmetic to the pair
of ASCII digits strings of [0-1]+ and "1". https://en.wikipedia.org/wiki/Peano_axioms

But the problem is that "First Grade Arithmetic" doesn't PROVE that
fact, but ASSUMES it.

Just like your logic ASSUMES your resuls, but doesn't prove it, just in
your case it isn't true.

The first incompleteness theorem states that no consistent system of
axioms whose theorems can be listed by an effective procedure (i.e. an algorithm) is capable of proving all truths about the arithmetic of
natural numbers. For any such consistent formal system, there will
always be statements about natural numbers that are true, but that are unprovable within the system. https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems

When we boil this down to its first-grade arithmetic foundation
this would seem to mean that there are some cases where the
sum of a pair of ASCII digit strings cannot be computed.

Nope, because some truths of arithmetic aren't just "summing".

Godel's Primitive Recursive Relationship is just a 'simple' arithmetical operation, and the existance of numbers that satisfies it are truths
about that arithemtic.

He shows that there can not be a Natural number that satisfies that relationship, and that there can not be a proof in that system of axioms
to prove that fact, as in the meta-theory that the relationship was
developed in (but using only operations available in that theory) that
if such a number existed, (and only if such a number existed) it would
BE a proof of the statement, but that number would also make the
statement false.

Since you can not prove a false statement in a consistent system of
axioms, there can not be a Natural Number that satisfies it, and since
that also means a proof can not ezist (since the existance of the proof
creates a number that would satisfy that relationship).

Note, all of this is based on the ideas you are trying to form, but with understanding and looking at there consequence.

CHECKING a proof is a computable operation, as Godel Proved.

Encoding that into the mathematics of Natural Numbers is a possible
operation, as Godel Proved.

Forming that statement, is possible, as Godel proved, and thus, he shows
that he can make a statement that MUST be True, but also can not be
proven in the system.

Sorry, your logic just falls appart when put against the facts. Your
problem is you just don't understand how to formalize you statement's so
your logic isn't precise enough to do what you want, and that leads you
down false alleys.

--- SoupGate-Win32 v1.05
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On 2024-10-23 13:15:00 +0000, olcott said:

On 10/23/2024 2:28 AM, Mikko wrote:

On 2024-10-22 14:02:01 +0000, olcott said:

On 10/22/2024 2:13 AM, Mikko wrote:

On 2024-10-21 13:52:28 +0000, olcott said:

On 10/21/2024 3:41 AM, Mikko wrote:

On 2024-10-20 15:32:45 +0000, olcott said:

The actual barest essence for formal systems and computations
is finite string transformation rules applied to finite strings.

Before you can start from that you need a formal theory that
can be interpreted as a theory of finite strings.

Not at all. The only theory needed are the operations
that can be performed on finite strings:
concatenation, substring, relational operator ...

You may try with an informal foundation but you need to make sure
that it is sufficicently well defined and that is easier with a
formal theory.

The minimal complete theory that I can think of computes
the sum of pairs of ASCII digit strings.

That is easily extended to Peano arithmetic.

As a bottom layer you need some sort of logic. There must be unambifuous >>>> rules about syntax and inference.

I already wrote this in C a long time ago.
It simply computes the sum the same way
that a first grader would compute the sum.

I have no idea how the first grade arithmetic
algorithm could be extended to PA.

Basically you define that the successor of X is X + 1. The only
primitive function of Peano arithmetic is the successor. Addition
and multiplication are recursively defined from the successor
function. Equality is often included in the underlying logic but
can be defined recursively from the successor function and the
order relation is defined similarly.

Anyway, the details are not important, only that it can be done.

First grade arithmetic can define a successor function
by merely applying first grade arithmetic to the pair
of ASCII digits strings of [0-1]+ and "1". https://en.wikipedia.org/wiki/Peano_axioms

The first incompleteness theorem states that no consistent system of
axioms whose theorems can be listed by an effective procedure (i.e. an algorithm) is capable of proving all truths about the arithmetic of
natural numbers. For any such consistent formal system, there will
always be statements about natural numbers that are true, but that are unprovable within the system. https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems

When we boil this down to its first-grade arithmetic foundation
this would seem to mean that there are some cases where the
sum of a pair of ASCII digit strings cannot be computed.

No, it does not. Incompleteness theorem does not apply to artihmetic
that only has addition but not multiplication.

The incompleteness theorem is about theories that have quantifiers.
A specific arithmetic expression (i.e, with no variables of any kind)
always has a well defined value.

--
Mikko

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On 10/24/24 10:28 AM, olcott wrote:

On 10/24/2024 8:51 AM, Mikko wrote:

On 2024-10-23 13:15:00 +0000, olcott said:

On 10/23/2024 2:28 AM, Mikko wrote:

On 2024-10-22 14:02:01 +0000, olcott said:

On 10/22/2024 2:13 AM, Mikko wrote:

On 2024-10-21 13:52:28 +0000, olcott said:

On 10/21/2024 3:41 AM, Mikko wrote:

On 2024-10-20 15:32:45 +0000, olcott said:

The actual barest essence for formal systems and computations >>>>>>>>> is finite string transformation rules applied to finite strings. >>>>>>>>

Before you can start from that you need a formal theory that
can be interpreted as a theory of finite strings.

Not at all. The only theory needed are the operations
that can be performed on finite strings:
concatenation, substring, relational operator ...

You may try with an informal foundation but you need to make sure
that it is sufficicently well defined and that is easier with a
formal theory.

The minimal complete theory that I can think of computes
the sum of pairs of ASCII digit strings.

That is easily extended to Peano arithmetic.

As a bottom layer you need some sort of logic. There must be
unambifuous
rules about syntax and inference.

I already wrote this in C a long time ago.
It simply computes the sum the same way
that a first grader would compute the sum.

I have no idea how the first grade arithmetic
algorithm could be extended to PA.

Basically you define that the successor of X is X + 1. The only
primitive function of Peano arithmetic is the successor. Addition
and multiplication are recursively defined from the successor
function. Equality is often included in the underlying logic but
can be defined recursively from the successor function and the
order relation is defined similarly.

Anyway, the details are not important, only that it can be done.

First grade arithmetic can define a successor function
by merely applying first grade arithmetic to the pair
of ASCII digits strings of [0-1]+ and "1".
https://en.wikipedia.org/wiki/Peano_axioms

The first incompleteness theorem states that no consistent system of
axioms whose theorems can be listed by an effective procedure (i.e.
an algorithm) is capable of proving all truths about the arithmetic
of natural numbers. For any such consistent formal system, there will
always be statements about natural numbers that are true, but that
are unprovable within the system.
https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems

When we boil this down to its first-grade arithmetic foundation
this would seem to mean that there are some cases where the
sum of a pair of ASCII digit strings cannot be computed.

No, it does not. Incompleteness theorem does not apply to artihmetic
that only has addition but not multiplication.

The incompleteness theorem is about theories that have quantifiers.
A specific arithmetic expression (i.e, with no variables of any kind)
always has a well defined value.

So lets goes the next step and add multiplication to the algorithm:
(just like first grade arithmetic we perform multiplication
on arbitrary length ASCII digit strings just like someone would
do with pencil and paper).

Incompleteness cannot be defined. until we add variables and
quantification: There exists an X such that X * 11 = 132.
Every detail of every step until we get G is unprovable in F.

Yes, Incompleteness requires a certain degree of suffistication in the operations allowed, but that is all part of the "properties of the
Natural Numbers".

There is a critical boundary, beyound which if a logic system supports
it, it must be incomplete. Simple system can be complete.

I wouldn't be surprized if part of that line is the ability to create
concepts over a certain size of infinity.

Note, that while the Natural Numbers themselves are a countable
infinity, there are operations on them (like computing the number of
possible mappings of N to N) that can create uncountable infinities, and
that might be part of what makes the system incomplete.

That or the ability to make countable infinite sets, that you need to
confirm that All or None of the memebers have a property.

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On 2024-10-24 14:28:35 +0000, olcott said:

On 10/24/2024 8:51 AM, Mikko wrote:

On 2024-10-23 13:15:00 +0000, olcott said:

On 10/23/2024 2:28 AM, Mikko wrote:

On 2024-10-22 14:02:01 +0000, olcott said:

On 10/22/2024 2:13 AM, Mikko wrote:

On 2024-10-21 13:52:28 +0000, olcott said:

On 10/21/2024 3:41 AM, Mikko wrote:

On 2024-10-20 15:32:45 +0000, olcott said:

The actual barest essence for formal systems and computations >>>>>>>>> is finite string transformation rules applied to finite strings. >>>>>>>>

Before you can start from that you need a formal theory that
can be interpreted as a theory of finite strings.

Not at all. The only theory needed are the operations
that can be performed on finite strings:
concatenation, substring, relational operator ...

You may try with an informal foundation but you need to make sure
that it is sufficicently well defined and that is easier with a
formal theory.

The minimal complete theory that I can think of computes
the sum of pairs of ASCII digit strings.

That is easily extended to Peano arithmetic.

As a bottom layer you need some sort of logic. There must be unambifuous >>>>>> rules about syntax and inference.

I already wrote this in C a long time ago.
It simply computes the sum the same way
that a first grader would compute the sum.

I have no idea how the first grade arithmetic
algorithm could be extended to PA.

Basically you define that the successor of X is X + 1. The only
primitive function of Peano arithmetic is the successor. Addition
and multiplication are recursively defined from the successor
function. Equality is often included in the underlying logic but
can be defined recursively from the successor function and the
order relation is defined similarly.

Anyway, the details are not important, only that it can be done.

First grade arithmetic can define a successor function
by merely applying first grade arithmetic to the pair
of ASCII digits strings of [0-1]+ and "1".
https://en.wikipedia.org/wiki/Peano_axioms

The first incompleteness theorem states that no consistent system of
axioms whose theorems can be listed by an effective procedure (i.e. an
algorithm) is capable of proving all truths about the arithmetic of
natural numbers. For any such consistent formal system, there will
always be statements about natural numbers that are true, but that are
unprovable within the system.
https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems

When we boil this down to its first-grade arithmetic foundation
this would seem to mean that there are some cases where the
sum of a pair of ASCII digit strings cannot be computed.

No, it does not. Incompleteness theorem does not apply to artihmetic
that only has addition but not multiplication.

The incompleteness theorem is about theories that have quantifiers.
A specific arithmetic expression (i.e, with no variables of any kind)
always has a well defined value.

So lets goes the next step and add multiplication to the algorithm:
(just like first grade arithmetic we perform multiplication
on arbitrary length ASCII digit strings just like someone would
do with pencil and paper).

Incompleteness cannot be defined. until we add variables and
quantification: There exists an X such that X * 11 = 132.
Every detail of every step until we get G is unprovable in F.

Incompleteness is easier to define if you also add the power operator
to the arithmetic. Otherwise the expressions of provability and
incompleteness are more complicated. They become much simpler if
instead of arithmetic the fundamental theory is a theory of finite
strings. As you already observed, arithmetic is easy to do with
finite strings. The opposite is possible but much more complicated.

--
Mikko

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On 10/24/24 9:07 PM, olcott wrote:

On 10/24/2024 6:23 PM, Richard Damon wrote:

On 10/24/24 10:28 AM, olcott wrote:

On 10/24/2024 8:51 AM, Mikko wrote:

On 2024-10-23 13:15:00 +0000, olcott said:

On 10/23/2024 2:28 AM, Mikko wrote:

On 2024-10-22 14:02:01 +0000, olcott said:

On 10/22/2024 2:13 AM, Mikko wrote:

On 2024-10-21 13:52:28 +0000, olcott said:

On 10/21/2024 3:41 AM, Mikko wrote:

On 2024-10-20 15:32:45 +0000, olcott said:

The actual barest essence for formal systems and computations >>>>>>>>>>> is finite string transformation rules applied to finite strings. >>>>>>>>>>

Before you can start from that you need a formal theory that >>>>>>>>>> can be interpreted as a theory of finite strings.

Not at all. The only theory needed are the operations
that can be performed on finite strings:
concatenation, substring, relational operator ...

You may try with an informal foundation but you need to make sure >>>>>>>> that it is sufficicently well defined and that is easier with a >>>>>>>> formal theory.

The minimal complete theory that I can think of computes
the sum of pairs of ASCII digit strings.

That is easily extended to Peano arithmetic.

As a bottom layer you need some sort of logic. There must be
unambifuous
rules about syntax and inference.

I already wrote this in C a long time ago.
It simply computes the sum the same way
that a first grader would compute the sum.

I have no idea how the first grade arithmetic
algorithm could be extended to PA.

Basically you define that the successor of X is X + 1. The only
primitive function of Peano arithmetic is the successor. Addition
and multiplication are recursively defined from the successor
function. Equality is often included in the underlying logic but
can be defined recursively from the successor function and the
order relation is defined similarly.

Anyway, the details are not important, only that it can be done.

First grade arithmetic can define a successor function
by merely applying first grade arithmetic to the pair
of ASCII digits strings of [0-1]+ and "1".
https://en.wikipedia.org/wiki/Peano_axioms

The first incompleteness theorem states that no consistent system
of axioms whose theorems can be listed by an effective procedure
(i.e. an algorithm) is capable of proving all truths about the
arithmetic of natural numbers. For any such consistent formal
system, there will always be statements about natural numbers that
are true, but that are unprovable within the system.
https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems >>>>>
When we boil this down to its first-grade arithmetic foundation
this would seem to mean that there are some cases where the
sum of a pair of ASCII digit strings cannot be computed.

No, it does not. Incompleteness theorem does not apply to artihmetic
that only has addition but not multiplication.

The incompleteness theorem is about theories that have quantifiers.
A specific arithmetic expression (i.e, with no variables of any kind)
always has a well defined value.

So lets goes the next step and add multiplication to the algorithm:
(just like first grade arithmetic we perform multiplication
on arbitrary length ASCII digit strings just like someone would
do with pencil and paper).

Incompleteness cannot be defined. until we add variables and
quantification: There exists an X such that X * 11 = 132.
Every detail of every step until we get G is unprovable in F.

Yes, Incompleteness requires a certain degree of suffistication in the
operations allowed, but that is all part of the "properties of the
Natural Numbers".

There is a critical boundary, beyound which if a logic system supports
it, it must be incomplete. Simple system can be complete.

The inability to prove that incoherent expressions
are true such as the Tarski Undefinability theorem
is only because they are freaking incoherent.

But the expressions are only "incoherent" to stupid people like you.

People who know what they are talking about understand what those
expression mean.

All you are doing is proving that you don't undertstand what you are
talking about, and that you think lying is a correct form of logical proof.

This shows in the fact that your "arguements" seem to always be based on
using known logical fallacies.

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On 10/25/24 9:31 AM, olcott wrote:

On 10/25/2024 3:01 AM, Mikko wrote:

On 2024-10-24 14:28:35 +0000, olcott said:

On 10/24/2024 8:51 AM, Mikko wrote:

On 2024-10-23 13:15:00 +0000, olcott said:

On 10/23/2024 2:28 AM, Mikko wrote:

On 2024-10-22 14:02:01 +0000, olcott said:

On 10/22/2024 2:13 AM, Mikko wrote:

On 2024-10-21 13:52:28 +0000, olcott said:

On 10/21/2024 3:41 AM, Mikko wrote:

On 2024-10-20 15:32:45 +0000, olcott said:

The actual barest essence for formal systems and computations >>>>>>>>>>> is finite string transformation rules applied to finite strings. >>>>>>>>>>

Before you can start from that you need a formal theory that >>>>>>>>>> can be interpreted as a theory of finite strings.

Not at all. The only theory needed are the operations
that can be performed on finite strings:
concatenation, substring, relational operator ...

You may try with an informal foundation but you need to make sure >>>>>>>> that it is sufficicently well defined and that is easier with a >>>>>>>> formal theory.

The minimal complete theory that I can think of computes
the sum of pairs of ASCII digit strings.

That is easily extended to Peano arithmetic.

As a bottom layer you need some sort of logic. There must be
unambifuous
rules about syntax and inference.

I already wrote this in C a long time ago.
It simply computes the sum the same way
that a first grader would compute the sum.

I have no idea how the first grade arithmetic
algorithm could be extended to PA.

Basically you define that the successor of X is X + 1. The only
primitive function of Peano arithmetic is the successor. Addition
and multiplication are recursively defined from the successor
function. Equality is often included in the underlying logic but
can be defined recursively from the successor function and the
order relation is defined similarly.

Anyway, the details are not important, only that it can be done.

First grade arithmetic can define a successor function
by merely applying first grade arithmetic to the pair
of ASCII digits strings of [0-1]+ and "1".
https://en.wikipedia.org/wiki/Peano_axioms

The first incompleteness theorem states that no consistent system
of axioms whose theorems can be listed by an effective procedure
(i.e. an algorithm) is capable of proving all truths about the
arithmetic of natural numbers. For any such consistent formal
system, there will always be statements about natural numbers that
are true, but that are unprovable within the system.
https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems >>>>>
When we boil this down to its first-grade arithmetic foundation
this would seem to mean that there are some cases where the
sum of a pair of ASCII digit strings cannot be computed.

No, it does not. Incompleteness theorem does not apply to artihmetic
that only has addition but not multiplication.

The incompleteness theorem is about theories that have quantifiers.
A specific arithmetic expression (i.e, with no variables of any kind)
always has a well defined value.

So lets goes the next step and add multiplication to the algorithm:
(just like first grade arithmetic we perform multiplication
on arbitrary length ASCII digit strings just like someone would
do with pencil and paper).

Incompleteness cannot be defined. until we add variables and
quantification: There exists an X such that X * 11 = 132.
Every detail of every step until we get G is unprovable in F.

Incompleteness is easier to define if you also add the power operator
to the arithmetic. Otherwise the expressions of provability and
incompleteness are more complicated. They become much simpler if
instead of arithmetic the fundamental theory is a theory of finite
strings. As you already observed, arithmetic is easy to do with
finite strings. The opposite is possible but much more complicated.

The power operator can be built from repeated operations of
the multiply operator. Will a terabyte be enough to store
the Gödel numbers?

Likely depends on how big of a system you are making F.

Some of the numbers would definitely fit.

The numbers do get big very quickly, so a terabyte might not be big
enough if F is the fullness of PA.

That could be an interesting computational challenge to estimate the
size of the various Godel numbers.

Of course, just because it needs more memory than might be available,
doesn't make it wrong, it just shows the difference between what can be
done with strictly finite models, and what can be done in mathematics
which allows for the infinite model, just you need to show things with a
finite number steps from a finite set of axioms and a finite set of
operations.

Remember, it has been shown that with the right finite set of rules and
axioms, we can generate infinite sets.

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On 10/25/24 9:43 AM, olcott wrote:

On 10/25/2024 7:27 AM, Richard Damon wrote:

On 10/24/24 9:07 PM, olcott wrote:

On 10/24/2024 6:23 PM, Richard Damon wrote:

On 10/24/24 10:28 AM, olcott wrote:

On 10/24/2024 8:51 AM, Mikko wrote:

On 2024-10-23 13:15:00 +0000, olcott said:

On 10/23/2024 2:28 AM, Mikko wrote:

On 2024-10-22 14:02:01 +0000, olcott said:

On 10/22/2024 2:13 AM, Mikko wrote:

On 2024-10-21 13:52:28 +0000, olcott said:

On 10/21/2024 3:41 AM, Mikko wrote:

On 2024-10-20 15:32:45 +0000, olcott said:

The actual barest essence for formal systems and computations >>>>>>>>>>>>> is finite string transformation rules applied to finite >>>>>>>>>>>>> strings.

Before you can start from that you need a formal theory that >>>>>>>>>>>> can be interpreted as a theory of finite strings.

Not at all. The only theory needed are the operations
that can be performed on finite strings:
concatenation, substring, relational operator ...

You may try with an informal foundation but you need to make sure >>>>>>>>>> that it is sufficicently well defined and that is easier with a >>>>>>>>>> formal theory.

The minimal complete theory that I can think of computes >>>>>>>>>>> the sum of pairs of ASCII digit strings.

That is easily extended to Peano arithmetic.

As a bottom layer you need some sort of logic. There must be >>>>>>>>>> unambifuous
rules about syntax and inference.

I already wrote this in C a long time ago.
It simply computes the sum the same way
that a first grader would compute the sum.

I have no idea how the first grade arithmetic
algorithm could be extended to PA.

Basically you define that the successor of X is X + 1. The only >>>>>>>> primitive function of Peano arithmetic is the successor. Addition >>>>>>>> and multiplication are recursively defined from the successor
function. Equality is often included in the underlying logic but >>>>>>>> can be defined recursively from the successor function and the >>>>>>>> order relation is defined similarly.

Anyway, the details are not important, only that it can be done. >>>>>>>>

First grade arithmetic can define a successor function
by merely applying first grade arithmetic to the pair
of ASCII digits strings of [0-1]+ and "1".
https://en.wikipedia.org/wiki/Peano_axioms

The first incompleteness theorem states that no consistent system >>>>>>> of axioms whose theorems can be listed by an effective procedure >>>>>>> (i.e. an algorithm) is capable of proving all truths about the
arithmetic of natural numbers. For any such consistent formal
system, there will always be statements about natural numbers
that are true, but that are unprovable within the system.
https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems >>>>>>>
When we boil this down to its first-grade arithmetic foundation
this would seem to mean that there are some cases where the
sum of a pair of ASCII digit strings cannot be computed.

No, it does not. Incompleteness theorem does not apply to artihmetic >>>>>> that only has addition but not multiplication.

The incompleteness theorem is about theories that have quantifiers. >>>>>> A specific arithmetic expression (i.e, with no variables of any kind) >>>>>> always has a well defined value.

So lets goes the next step and add multiplication to the algorithm:
(just like first grade arithmetic we perform multiplication
on arbitrary length ASCII digit strings just like someone would
do with pencil and paper).

Incompleteness cannot be defined. until we add variables and
quantification: There exists an X such that X * 11 = 132.
Every detail of every step until we get G is unprovable in F.

Yes, Incompleteness requires a certain degree of suffistication in
the operations allowed, but that is all part of the "properties of
the Natural Numbers".

There is a critical boundary, beyound which if a logic system
supports it, it must be incomplete. Simple system can be complete.

The inability to prove that incoherent expressions
are true such as the Tarski Undefinability theorem
is only because they are freaking incoherent.

But the expressions are only "incoherent" to stupid people like you.

Is this sentence {true, false, truth_bearer}
"This sentence is not true."

But that is a non-sequitor, as it isn't the sentence actually used in
any of the proofs.

Any reply unsupported by correct reasoning will
be construed as baseless. Most of what you say
has no basis what-so-ever in correct reasoning.

Really, you mean by your concept of "Correct Reasoning" that thinks that
a progrma that halts can be correctly desceribed as non-halting?

A term that you have been unable to actually formally define, in part
because you just don't understand the language of logic.

Sorry, you are just proving your stupidity.

You claim the proofs are incorrect because they use incoherent
statements, but you can't even correctly quote them, because you do not understand what they say.

The "incoherence" is all in your head, because YOU are the ignorant one.

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On 2024-10-25 13:31:16 +0000, olcott said:

On 10/25/2024 3:01 AM, Mikko wrote:

On 2024-10-24 14:28:35 +0000, olcott said:

On 10/24/2024 8:51 AM, Mikko wrote:

On 2024-10-23 13:15:00 +0000, olcott said:

On 10/23/2024 2:28 AM, Mikko wrote:

On 2024-10-22 14:02:01 +0000, olcott said:

On 10/22/2024 2:13 AM, Mikko wrote:

On 2024-10-21 13:52:28 +0000, olcott said:

On 10/21/2024 3:41 AM, Mikko wrote:

On 2024-10-20 15:32:45 +0000, olcott said:

The actual barest essence for formal systems and computations >>>>>>>>>>> is finite string transformation rules applied to finite strings. >>>>>>>>>>

Before you can start from that you need a formal theory that >>>>>>>>>> can be interpreted as a theory of finite strings.

Not at all. The only theory needed are the operations
that can be performed on finite strings:
concatenation, substring, relational operator ...

You may try with an informal foundation but you need to make sure >>>>>>>> that it is sufficicently well defined and that is easier with a >>>>>>>> formal theory.

The minimal complete theory that I can think of computes
the sum of pairs of ASCII digit strings.

That is easily extended to Peano arithmetic.

As a bottom layer you need some sort of logic. There must be unambifuous
rules about syntax and inference.

I already wrote this in C a long time ago.
It simply computes the sum the same way
that a first grader would compute the sum.

I have no idea how the first grade arithmetic
algorithm could be extended to PA.

Basically you define that the successor of X is X + 1. The only
primitive function of Peano arithmetic is the successor. Addition
and multiplication are recursively defined from the successor
function. Equality is often included in the underlying logic but
can be defined recursively from the successor function and the
order relation is defined similarly.

Anyway, the details are not important, only that it can be done.

First grade arithmetic can define a successor function
by merely applying first grade arithmetic to the pair
of ASCII digits strings of [0-1]+ and "1".
https://en.wikipedia.org/wiki/Peano_axioms

The first incompleteness theorem states that no consistent system of >>>>> axioms whose theorems can be listed by an effective procedure (i.e. an >>>>> algorithm) is capable of proving all truths about the arithmetic of
natural numbers. For any such consistent formal system, there will
always be statements about natural numbers that are true, but that are >>>>> unprovable within the system.
https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems >>>>>
When we boil this down to its first-grade arithmetic foundation
this would seem to mean that there are some cases where the
sum of a pair of ASCII digit strings cannot be computed.

No, it does not. Incompleteness theorem does not apply to artihmetic
that only has addition but not multiplication.

The incompleteness theorem is about theories that have quantifiers.
A specific arithmetic expression (i.e, with no variables of any kind)
always has a well defined value.

So lets goes the next step and add multiplication to the algorithm:
(just like first grade arithmetic we perform multiplication
on arbitrary length ASCII digit strings just like someone would
do with pencil and paper).

Incompleteness cannot be defined. until we add variables and
quantification: There exists an X such that X * 11 = 132.
Every detail of every step until we get G is unprovable in F.

Incompleteness is easier to define if you also add the power operator
to the arithmetic. Otherwise the expressions of provability and
incompleteness are more complicated. They become much simpler if
instead of arithmetic the fundamental theory is a theory of finite
strings. As you already observed, arithmetic is easy to do with
finite strings. The opposite is possible but much more complicated.

The power operator can be built from repeated operations of
the multiply operator.

It is possible but to say that x is the z'th power of y is overly
complicated with a first order formula using just addition and
multiplication.

--
Mikko

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On 10/26/24 9:17 AM, olcott wrote:

On 10/26/2024 3:02 AM, Mikko wrote:

On 2024-10-25 13:31:16 +0000, olcott said:

On 10/25/2024 3:01 AM, Mikko wrote:

On 2024-10-24 14:28:35 +0000, olcott said:

On 10/24/2024 8:51 AM, Mikko wrote:

On 2024-10-23 13:15:00 +0000, olcott said:

On 10/23/2024 2:28 AM, Mikko wrote:

On 2024-10-22 14:02:01 +0000, olcott said:

On 10/22/2024 2:13 AM, Mikko wrote:

On 2024-10-21 13:52:28 +0000, olcott said:

On 10/21/2024 3:41 AM, Mikko wrote:

On 2024-10-20 15:32:45 +0000, olcott said:

The actual barest essence for formal systems and computations >>>>>>>>>>>>> is finite string transformation rules applied to finite >>>>>>>>>>>>> strings.

Before you can start from that you need a formal theory that >>>>>>>>>>>> can be interpreted as a theory of finite strings.

Not at all. The only theory needed are the operations
that can be performed on finite strings:
concatenation, substring, relational operator ...

You may try with an informal foundation but you need to make sure >>>>>>>>>> that it is sufficicently well defined and that is easier with a >>>>>>>>>> formal theory.

The minimal complete theory that I can think of computes >>>>>>>>>>> the sum of pairs of ASCII digit strings.

That is easily extended to Peano arithmetic.

As a bottom layer you need some sort of logic. There must be >>>>>>>>>> unambifuous
rules about syntax and inference.

I already wrote this in C a long time ago.
It simply computes the sum the same way
that a first grader would compute the sum.

I have no idea how the first grade arithmetic
algorithm could be extended to PA.

Basically you define that the successor of X is X + 1. The only >>>>>>>> primitive function of Peano arithmetic is the successor. Addition >>>>>>>> and multiplication are recursively defined from the successor
function. Equality is often included in the underlying logic but >>>>>>>> can be defined recursively from the successor function and the >>>>>>>> order relation is defined similarly.

Anyway, the details are not important, only that it can be done. >>>>>>>>

First grade arithmetic can define a successor function
by merely applying first grade arithmetic to the pair
of ASCII digits strings of [0-1]+ and "1".
https://en.wikipedia.org/wiki/Peano_axioms

The first incompleteness theorem states that no consistent system >>>>>>> of axioms whose theorems can be listed by an effective procedure >>>>>>> (i.e. an algorithm) is capable of proving all truths about the
arithmetic of natural numbers. For any such consistent formal
system, there will always be statements about natural numbers
that are true, but that are unprovable within the system.
https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems >>>>>>>
When we boil this down to its first-grade arithmetic foundation
this would seem to mean that there are some cases where the
sum of a pair of ASCII digit strings cannot be computed.

No, it does not. Incompleteness theorem does not apply to artihmetic >>>>>> that only has addition but not multiplication.

The incompleteness theorem is about theories that have quantifiers. >>>>>> A specific arithmetic expression (i.e, with no variables of any kind) >>>>>> always has a well defined value.

So lets goes the next step and add multiplication to the algorithm:
(just like first grade arithmetic we perform multiplication
on arbitrary length ASCII digit strings just like someone would
do with pencil and paper).

Incompleteness cannot be defined. until we add variables and
quantification: There exists an X such that X * 11 = 132.
Every detail of every step until we get G is unprovable in F.

Incompleteness is easier to define if you also add the power operator
to the arithmetic. Otherwise the expressions of provability and
incompleteness are more complicated. They become much simpler if
instead of arithmetic the fundamental theory is a theory of finite
strings. As you already observed, arithmetic is easy to do with
finite strings. The opposite is possible but much more complicated.

The power operator can be built from repeated operations of
the multiply operator.

It is possible but to say that x is the z'th power of y is overly
complicated with a first order formula using just addition and
multiplication.

Just imagine c functions that have enough memory to compute
sums and products of ASCII strings of digits using the same
method that people do.
char* sum(char* x, char *y);
char* product(char* x, char *y);
char* difference(char* x, char *y); // never returns < 0

char* power_of(char* base, char * power);
I am not going to bother with the rest of the steps.
We simply multiply  base times itself power times.

So, you are just admitting that the question just went over your head.

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On 2024-10-26 13:17:52 +0000, olcott said:

On 10/26/2024 3:02 AM, Mikko wrote:

On 2024-10-25 13:31:16 +0000, olcott said:

On 10/25/2024 3:01 AM, Mikko wrote:

On 2024-10-24 14:28:35 +0000, olcott said:

On 10/24/2024 8:51 AM, Mikko wrote:

On 2024-10-23 13:15:00 +0000, olcott said:

On 10/23/2024 2:28 AM, Mikko wrote:

On 2024-10-22 14:02:01 +0000, olcott said:

On 10/22/2024 2:13 AM, Mikko wrote:

On 2024-10-21 13:52:28 +0000, olcott said:

On 10/21/2024 3:41 AM, Mikko wrote:

On 2024-10-20 15:32:45 +0000, olcott said:

The actual barest essence for formal systems and computations >>>>>>>>>>>>> is finite string transformation rules applied to finite strings. >>>>>>>>>>>>

Before you can start from that you need a formal theory that >>>>>>>>>>>> can be interpreted as a theory of finite strings.

Not at all. The only theory needed are the operations
that can be performed on finite strings:
concatenation, substring, relational operator ...

You may try with an informal foundation but you need to make sure >>>>>>>>>> that it is sufficicently well defined and that is easier with a >>>>>>>>>> formal theory.

The minimal complete theory that I can think of computes >>>>>>>>>>> the sum of pairs of ASCII digit strings.

That is easily extended to Peano arithmetic.

As a bottom layer you need some sort of logic. There must be unambifuous
rules about syntax and inference.

I already wrote this in C a long time ago.
It simply computes the sum the same way
that a first grader would compute the sum.

I have no idea how the first grade arithmetic
algorithm could be extended to PA.

Basically you define that the successor of X is X + 1. The only >>>>>>>> primitive function of Peano arithmetic is the successor. Addition >>>>>>>> and multiplication are recursively defined from the successor
function. Equality is often included in the underlying logic but >>>>>>>> can be defined recursively from the successor function and the >>>>>>>> order relation is defined similarly.

Anyway, the details are not important, only that it can be done. >>>>>>>>

First grade arithmetic can define a successor function
by merely applying first grade arithmetic to the pair
of ASCII digits strings of [0-1]+ and "1".
https://en.wikipedia.org/wiki/Peano_axioms

The first incompleteness theorem states that no consistent system of >>>>>>> axioms whose theorems can be listed by an effective procedure (i.e. an >>>>>>> algorithm) is capable of proving all truths about the arithmetic of >>>>>>> natural numbers. For any such consistent formal system, there will >>>>>>> always be statements about natural numbers that are true, but that are >>>>>>> unprovable within the system.
https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems >>>>>>>
When we boil this down to its first-grade arithmetic foundation
this would seem to mean that there are some cases where the
sum of a pair of ASCII digit strings cannot be computed.

No, it does not. Incompleteness theorem does not apply to artihmetic >>>>>> that only has addition but not multiplication.

The incompleteness theorem is about theories that have quantifiers. >>>>>> A specific arithmetic expression (i.e, with no variables of any kind) >>>>>> always has a well defined value.

So lets goes the next step and add multiplication to the algorithm:
(just like first grade arithmetic we perform multiplication
on arbitrary length ASCII digit strings just like someone would
do with pencil and paper).

Incompleteness cannot be defined. until we add variables and
quantification: There exists an X such that X * 11 = 132.
Every detail of every step until we get G is unprovable in F.

Incompleteness is easier to define if you also add the power operator
to the arithmetic. Otherwise the expressions of provability and
incompleteness are more complicated. They become much simpler if
instead of arithmetic the fundamental theory is a theory of finite
strings. As you already observed, arithmetic is easy to do with
finite strings. The opposite is possible but much more complicated.

The power operator can be built from repeated operations of
the multiply operator.

It is possible but to say that x is the z'th power of y is overly
complicated with a first order formula using just addition and
multiplication.

Just imagine c functions that have enough memory to compute
sums and products of ASCII strings of digits using the same
method that people do.

Why just imagein? That is fairly easy to make. In some other lanugages
(e.g. Python, Javascript) it is alread in the library or as a built-in
feature.

--
Mikko

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On 10/27/24 10:21 AM, olcott wrote:

On 10/27/2024 3:37 AM, Mikko wrote:

On 2024-10-26 13:17:52 +0000, olcott said:

Just imagine c functions that have enough memory to compute
sums and products of ASCII strings of digits using the same
method that people do.

Why just imagein? That is fairly easy to make. In some other lanugages
(e.g. Python, Javascript) it is alread in the library or as a built-in
feature.

OK next I want to see the actual Godel numbers and the
arithmetic steps used to derive them.

Then do it. His proof detailed exactly how ot do it, he just didn't
limit himself to Peano, as he wanted to prove it for ANY system that
meet the minimum requirements.

Of course, the first requirement is to fully understand the Axioms of
the system you want to use, and how that system derives its needed rules
of arithmetic from those, at EVERY step in fine detail.

So, why don't you first make a listing of that, then do the encoding step.

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On 2024-10-27 14:21:25 +0000, olcott said:

On 10/27/2024 3:37 AM, Mikko wrote:

On 2024-10-26 13:17:52 +0000, olcott said:

Just imagine c functions that have enough memory to compute
sums and products of ASCII strings of digits using the same
method that people do.

Why just imagein? That is fairly easy to make. In some other lanugages
(e.g. Python, Javascript) it is alread in the library or as a built-in
feature.

OK next I want to see the actual Godel numbers and the
arithmetic steps used to derive them.

They can be found in any textbook of logic that discusses undecidability.
If you need to ask about details tell us which book you are using.

--
Mikko

--- SoupGate-Win32 v1.05
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On 10/30/24 8:16 AM, olcott wrote:

On 10/30/2024 5:02 AM, Mikko wrote:

On 2024-10-27 14:21:25 +0000, olcott said:

On 10/27/2024 3:37 AM, Mikko wrote:

On 2024-10-26 13:17:52 +0000, olcott said:

Just imagine c functions that have enough memory to compute
sums and products of ASCII strings of digits using the same
method that people do.

Why just imagein? That is fairly easy to make. In some other lanugages >>>> (e.g. Python, Javascript) it is alread in the library or as a built-in >>>> feature.

OK next I want to see the actual Godel numbers and the
arithmetic steps used to derive them.

They can be found in any textbook of logic that discusses undecidability.
If you need to ask about details tell us which book you are using.

Every single digit of the entire natural numbers
not any symbolic name for such a number.

It might be the case that one number fills 100 books
of 1000 pages each.

But, before you can get those, you need to define the Meta-System you
are building them in, as that changes the numbers.

So, if you want to see a sample set of the numbers, do the work to get them.

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On 2024-10-30 12:16:02 +0000, olcott said:

On 10/30/2024 5:02 AM, Mikko wrote:

On 2024-10-27 14:21:25 +0000, olcott said:

On 10/27/2024 3:37 AM, Mikko wrote:

On 2024-10-26 13:17:52 +0000, olcott said:

Just imagine c functions that have enough memory to compute
sums and products of ASCII strings of digits using the same
method that people do.

Why just imagein? That is fairly easy to make. In some other lanugages >>>> (e.g. Python, Javascript) it is alread in the library or as a built-in >>>> feature.

OK next I want to see the actual Godel numbers and the
arithmetic steps used to derive them.

They can be found in any textbook of logic that discusses undecidability.
If you need to ask about details tell us which book you are using.

Every single digit of the entire natural numbers
not any symbolic name for such a number.

Just evaluate the expressions shown in the books.

It might be the case that one number fills 100 books
of 1000 pages each.

You fill find out when you evaluate the expressions. If you use Gödel's original numbering you will need larger numbers than strictly necessary.
If you first encode symbols with a finite set of characters you can
encode everything with finite set of characters. Then you can encode
those character strings as integers. The number of digits can be determined from the length of the character strings. Besides, computations are much
faster than with Gödel's powers of primes.

--
Mikko

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Am Thu, 31 Oct 2024 07:19:18 -0500 schrieb olcott:

On 10/31/2024 5:34 AM, Mikko wrote:

On 2024-10-30 12:16:02 +0000, olcott said:

On 10/30/2024 5:02 AM, Mikko wrote:

On 2024-10-27 14:21:25 +0000, olcott said:

On 10/27/2024 3:37 AM, Mikko wrote:

On 2024-10-26 13:17:52 +0000, olcott said:

Just imagine c functions that have enough memory to compute sums >>>>>>> and products of ASCII strings of digits using the same method that >>>>>>> people do.

Why just imagein? That is fairly easy to make. In some other
lanugages (e.g. Python, Javascript) it is alread in the library or >>>>>> as a built-in feature.

OK next I want to see the actual Godel numbers and the arithmetic
steps used to derive them.

They can be found in any textbook of logic that discusses
undecidability.
If you need to ask about details tell us which book you are using.

Every single digit of the entire natural numbers not any symbolic name
for such a number.

Just evaluate the expressions shown in the books.

To me they are all nonsense gibberish. How one can convert a proof about arithmetic into a proof about provability seems to be flatly false.

The key is selfreference. There is a number that encodes the sentence
"the sentence with the number [the number that this sentence encodes to]
is not provable".

It might be the case that one number fills 100 books of 1000 pages
each.

You fill find out when you evaluate the expressions. If you use Gödel's
original numbering you will need larger numbers than strictly
necessary. If you first encode symbols with a finite set of characters
you can encode everything with finite set of characters.

A book a trillion light years deep?

Is finite.

Then you can encode those character strings as integers. The number of
digits can be determined from the length of the character strings.
Besides, computations are much faster than with Gödel's powers of
primes.

--
Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.

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On 10/31/24 8:19 AM, olcott wrote:

On 10/31/2024 5:34 AM, Mikko wrote:

On 2024-10-30 12:16:02 +0000, olcott said:

On 10/30/2024 5:02 AM, Mikko wrote:

On 2024-10-27 14:21:25 +0000, olcott said:

On 10/27/2024 3:37 AM, Mikko wrote:

On 2024-10-26 13:17:52 +0000, olcott said:

Just imagine c functions that have enough memory to compute
sums and products of ASCII strings of digits using the same
method that people do.

Why just imagein? That is fairly easy to make. In some other
lanugages
(e.g. Python, Javascript) it is alread in the library or as a
built-in
feature.

OK next I want to see the actual Godel numbers and the
arithmetic steps used to derive them.

They can be found in any textbook of logic that discusses
undecidability.
If you need to ask about details tell us which book you are using.

Every single digit of the entire natural numbers
not any symbolic name for such a number.

Just evaluate the expressions shown in the books.

To me they are all nonsense gibberish. How one
can convert a proof about arithmetic into a
proof about provability seems to be flatly false.

And to assert that just because something seems "gibberish" to you means
it is false, just proves that you don't undetstand how logic works.

It might be the case that one number fills 100 books
of 1000 pages each.

You fill find out when you evaluate the expressions. If you use Gödel's
original numbering you will need larger numbers than strictly necessary.
If you first encode symbols with a finite set of characters you can
encode everything with finite set of characters.

A book a trillion light years deep?

Maybe, but that isn't important to the proof.

Then you can encode
those character strings as integers. The number of digits can be
determined
from the length of the character strings. Besides, computations are much
faster than with Gödel's powers of primes.

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On 2024-10-31 12:19:18 +0000, olcott said:

On 10/31/2024 5:34 AM, Mikko wrote:

On 2024-10-30 12:16:02 +0000, olcott said:

On 10/30/2024 5:02 AM, Mikko wrote:

On 2024-10-27 14:21:25 +0000, olcott said:

On 10/27/2024 3:37 AM, Mikko wrote:

On 2024-10-26 13:17:52 +0000, olcott said:

Just imagine c functions that have enough memory to compute
sums and products of ASCII strings of digits using the same
method that people do.

Why just imagein? That is fairly easy to make. In some other lanugages >>>>>> (e.g. Python, Javascript) it is alread in the library or as a built-in >>>>>> feature.

OK next I want to see the actual Godel numbers and the
arithmetic steps used to derive them.

They can be found in any textbook of logic that discusses undecidability. >>>> If you need to ask about details tell us which book you are using.

Every single digit of the entire natural numbers
not any symbolic name for such a number.

Just evaluate the expressions shown in the books.

To me they are all nonsense gibberish.

The books define everything needed in order to understand the encoding
rules.

Encoding nonsense gibberish as a string of digits is trivial.

How one
can convert a proof about arithmetic into a
proof about provability seems to be flatly false.

You needn't. The proof about provability is given in the books so
you needn't any comversion.

It might be the case that one number fills 100 books
of 1000 pages each.

You fill find out when you evaluate the expressions. If you use Gödel's
original numbering you will need larger numbers than strictly necessary.
If you first encode symbols with a finite set of characters you can
encode everything with finite set of characters.

A book a trillion light years deep?

The number of digits in a Gödel number can be computed with less effort
than the Gödel number itself. Still easier to compute a rough estimate.

--
Mikko

--- SoupGate-Win32 v1.05
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On 2024-10-31 14:18:40 +0000, olcott said:

On 10/31/2024 8:58 AM, joes wrote:

Am Thu, 31 Oct 2024 07:19:18 -0500 schrieb olcott:

On 10/31/2024 5:34 AM, Mikko wrote:

On 2024-10-30 12:16:02 +0000, olcott said:

On 10/30/2024 5:02 AM, Mikko wrote:

On 2024-10-27 14:21:25 +0000, olcott said:

On 10/27/2024 3:37 AM, Mikko wrote:

On 2024-10-26 13:17:52 +0000, olcott said:

Just imagine c functions that have enough memory to compute sums >>>>>>>>> and products of ASCII strings of digits using the same method that >>>>>>>>> people do.

Why just imagein? That is fairly easy to make. In some other
lanugages (e.g. Python, Javascript) it is alread in the library or >>>>>>>> as a built-in feature.

OK next I want to see the actual Godel numbers and the arithmetic >>>>>>> steps used to derive them.

They can be found in any textbook of logic that discusses
undecidability.
If you need to ask about details tell us which book you are using.

Every single digit of the entire natural numbers not any symbolic name >>>>> for such a number.

Just evaluate the expressions shown in the books.

To me they are all nonsense gibberish. How one can convert a proof about >>> arithmetic into a proof about provability seems to be flatly false.

The key is selfreference. There is a number that encodes the sentence
"the sentence with the number [the number that this sentence encodes to]
is not provable".

Can you please hit return before you reply?
Your reply is always buried too close to what you are replying to.

We simply reject pathological self-reference lie
ZFC did and the issue ends.

You cannot reject any number from atrithmetic. If you do the result is
not arithmetic anymore.

--
Mikko

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On 11/1/24 7:50 AM, olcott wrote:

On 11/1/2024 3:44 AM, Mikko wrote:

On 2024-10-31 12:19:18 +0000, olcott said:

On 10/31/2024 5:34 AM, Mikko wrote:

On 2024-10-30 12:16:02 +0000, olcott said:

On 10/30/2024 5:02 AM, Mikko wrote:

On 2024-10-27 14:21:25 +0000, olcott said:

On 10/27/2024 3:37 AM, Mikko wrote:

On 2024-10-26 13:17:52 +0000, olcott said:

Just imagine c functions that have enough memory to compute
sums and products of ASCII strings of digits using the same
method that people do.

Why just imagein? That is fairly easy to make. In some other
lanugages
(e.g. Python, Javascript) it is alread in the library or as a
built-in
feature.

OK next I want to see the actual Godel numbers and the
arithmetic steps used to derive them.

They can be found in any textbook of logic that discusses
undecidability.
If you need to ask about details tell us which book you are using. >>>>>>

Every single digit of the entire natural numbers
not any symbolic name for such a number.

Just evaluate the expressions shown in the books.

To me they are all nonsense gibberish.

The books define everything needed in order to understand the encoding
rules.

Encoding nonsense gibberish as a string of digits is trivial.

How one
can convert a proof about arithmetic into a
proof about provability seems to be flatly false.

You needn't. The proof about provability is given in the books so
you needn't any comversion.

So you are saying that the Gödel sentence has nothing
to do with

BEGIN:(Gödel 1931:39-41)
  ...We are therefore confronted with a proposition which
  asserts its own unprovability.
END:(Gödel 1931:39-41)

Sort of. The Godel sentence only allows us to derive such a statement,
and only in the meta-logic that was used to develop the relationship
that the Godel sentence uses.

Making arithmetic say anything about provability
seems like making an angel food cake out of lug nuts,
cannot possible be done.

But the testing of a proof HAS BEEN shown to be computable, and thus
subject to the laws of arithmetic.

You are just showing your ignorance.

It might be the case that one number fills 100 books
of 1000 pages each.

You fill find out when you evaluate the expressions. If you use Gödel's >>>> original numbering you will need larger numbers than strictly
necessary.
If you first encode symbols with a finite set of characters you can
encode everything with finite set of characters.

A book a trillion light years deep?

The number of digits in a Gödel number can be computed with less effort
than the Gödel number itself. Still easier to compute a rough estimate.

So you have no idea how to compute the Gödel numbers.

He didn't say that.

The problem with computing a Godel Number, is you first need to choose
the System you want to express them about, and then the Meta-System that allocates the primes to the various elements of the system.

For the typical systems used, it could well be that the work needed to
actually produce said numbers exceeds the ability for a person to do in
their lifetime using a supercomputer.

But that doesn't mean that he didn't show they could be produced.

Godel's proof was never about "feasability" but possibility.

After all, we don't need the numbers to understand the proof that used
those numbers is correct.

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On 2024-11-01 11:50:24 +0000, olcott said:

On 11/1/2024 3:44 AM, Mikko wrote:

On 2024-10-31 12:19:18 +0000, olcott said:

On 10/31/2024 5:34 AM, Mikko wrote:

On 2024-10-30 12:16:02 +0000, olcott said:

On 10/30/2024 5:02 AM, Mikko wrote:

On 2024-10-27 14:21:25 +0000, olcott said:

On 10/27/2024 3:37 AM, Mikko wrote:

On 2024-10-26 13:17:52 +0000, olcott said:

Just imagine c functions that have enough memory to compute
sums and products of ASCII strings of digits using the same
method that people do.

Why just imagein? That is fairly easy to make. In some other lanugages >>>>>>>> (e.g. Python, Javascript) it is alread in the library or as a built-in >>>>>>>> feature.

OK next I want to see the actual Godel numbers and the
arithmetic steps used to derive them.

They can be found in any textbook of logic that discusses undecidability.
If you need to ask about details tell us which book you are using. >>>>>>

Every single digit of the entire natural numbers
not any symbolic name for such a number.

Just evaluate the expressions shown in the books.

To me they are all nonsense gibberish.

The books define everything needed in order to understand the encoding
rules.

Encoding nonsense gibberish as a string of digits is trivial.

How one
can convert a proof about arithmetic into a
proof about provability seems to be flatly false.

You needn't. The proof about provability is given in the books so
you needn't any comversion.

So you are saying that the Gödel sentence has nothing
to do with

BEGIN:(Gödel 1931:39-41)
...We are therefore confronted with a proposition which
asserts its own unprovability.
END:(Gödel 1931:39-41)

Nothing is too strong but the connection is not arithmetic.
That "asserts its own unprovability" refers to a non-arithmetic
interpretation of an arithmetic formula.

Making arithmetic say anything about provability
seems like making an angel food cake out of lug nuts,
cannot possible be done.

Numbers have features and formulas have features. Therefore it is
possible to compare features of formulas to features of numbers.

It might be the case that one number fills 100 books
of 1000 pages each.

You fill find out when you evaluate the expressions. If you use Gödel's >>>> original numbering you will need larger numbers than strictly necessary. >>>> If you first encode symbols with a finite set of characters you can
encode everything with finite set of characters.

A book a trillion light years deep?

The number of digits in a Gödel number can be computed with less effort
than the Gödel number itself. Still easier to compute a rough estimate.

So you have no idea how to compute the Gödel numbers.

As I aleady told, I have an idea how to encode formulas with smaller
numbers than the numbers Gödel used.

--
Mikko

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On 2024-11-01 11:53:00 +0000, olcott said:

On 11/1/2024 3:47 AM, Mikko wrote:

On 2024-10-31 14:18:40 +0000, olcott said:

On 10/31/2024 8:58 AM, joes wrote:

Am Thu, 31 Oct 2024 07:19:18 -0500 schrieb olcott:

On 10/31/2024 5:34 AM, Mikko wrote:

On 2024-10-30 12:16:02 +0000, olcott said:

On 10/30/2024 5:02 AM, Mikko wrote:

On 2024-10-27 14:21:25 +0000, olcott said:

On 10/27/2024 3:37 AM, Mikko wrote:

On 2024-10-26 13:17:52 +0000, olcott said:

Just imagine c functions that have enough memory to compute sums >>>>>>>>>>> and products of ASCII strings of digits using the same method that >>>>>>>>>>> people do.

Why just imagein? That is fairly easy to make. In some other >>>>>>>>>> lanugages (e.g. Python, Javascript) it is alread in the library or >>>>>>>>>> as a built-in feature.

OK next I want to see the actual Godel numbers and the arithmetic >>>>>>>>> steps used to derive them.

They can be found in any textbook of logic that discusses
undecidability.
If you need to ask about details tell us which book you are using. >>>>>>> Every single digit of the entire natural numbers not any symbolic name >>>>>>> for such a number.

Just evaluate the expressions shown in the books.

To me they are all nonsense gibberish. How one can convert a proof about >>>>> arithmetic into a proof about provability seems to be flatly false.

The key is selfreference. There is a number that encodes the sentence
"the sentence with the number [the number that this sentence encodes to] >>>> is not provable".

Can you please hit return before you reply?
Your reply is always buried too close to what you are replying to.

We simply reject pathological self-reference lie
ZFC did and the issue ends.

You cannot reject any number from atrithmetic. If you do the result is
not arithmetic anymore.

I claims that his whole proof is nonsense until you
provide 1200% concrete proof otherwise.

Crackpots claim all all sorts of things. There is no way to change that
so there is no point to try.

All of arithmetic is inherently computable and
any non-arithmetic operation on a number is a type
mismatch error.

There are arithmetic functions and predicates that are not Turing
computable. For example, Busy Beaver.

--
Mikko

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On 11/2/24 7:05 AM, olcott wrote:

On 11/2/2024 3:29 AM, Mikko wrote:

On 2024-11-01 11:53:00 +0000, olcott said:

On 11/1/2024 3:47 AM, Mikko wrote:

On 2024-10-31 14:18:40 +0000, olcott said:

On 10/31/2024 8:58 AM, joes wrote:

Am Thu, 31 Oct 2024 07:19:18 -0500 schrieb olcott:

On 10/31/2024 5:34 AM, Mikko wrote:

On 2024-10-30 12:16:02 +0000, olcott said:

On 10/30/2024 5:02 AM, Mikko wrote:

On 2024-10-27 14:21:25 +0000, olcott said:

On 10/27/2024 3:37 AM, Mikko wrote:

On 2024-10-26 13:17:52 +0000, olcott said:

Just imagine c functions that have enough memory to compute >>>>>>>>>>>>> sums
and products of ASCII strings of digits using the same >>>>>>>>>>>>> method that
people do.

Why just imagein? That is fairly easy to make. In some other >>>>>>>>>>>> lanugages (e.g. Python, Javascript) it is alread in the >>>>>>>>>>>> library or
as a built-in feature.

OK next I want to see the actual Godel numbers and the
arithmetic
steps used to derive them.

They can be found in any textbook of logic that discusses
undecidability.
If you need to ask about details tell us which book you are >>>>>>>>>> using.

Every single digit of the entire natural numbers not any
symbolic name
for such a number.

Just evaluate the expressions shown in the books.

To me they are all nonsense gibberish. How one can convert a
proof about
arithmetic into a proof about provability seems to be flatly false. >>>>>

The key is selfreference. There is a number that encodes the sentence >>>>>> "the sentence with the number [the number that this sentence
encodes to]
is not provable".

Can you please hit return before you reply?
Your reply is always buried too close to what you are replying to.

We simply reject pathological self-reference lie
ZFC did and the issue ends.

You cannot reject any number from atrithmetic. If you do the result is >>>> not arithmetic anymore.

I claims that his whole proof is nonsense until you
provide 1200% concrete proof otherwise.

Crackpots claim all all sorts of things. There is no way to change that
so there is no point to try.

All of arithmetic is inherently computable and
any non-arithmetic operation on a number is a type
mismatch error.

There are arithmetic functions and predicates that are not Turing
computable. For example, Busy Beaver.

Not computable because of self-reference is a different class
than not computable for other reasons. The Goldbach conjecture
seems not computable only because it seems to require an infinite
number of steps.

No, because the ability to form self-reference is totally inherent in
the nature of Turing Complete computation systems.

The essential fact that infinity is not finite is the core reason for
both non-computable natures.

Note, we don't know if Goldbach is non-computable, and in fact, if we
did, we would have the answer, and thus it can not be proven to be non-computable, as the proof of truth of Goldback being non-computable
would be a proof that it was true, as if the Goldback conjecture is
false, there is a compuation that proves that.

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On 2024-11-02 11:05:20 +0000, olcott said:

On 11/2/2024 3:29 AM, Mikko wrote:

On 2024-11-01 11:53:00 +0000, olcott said:

On 11/1/2024 3:47 AM, Mikko wrote:

On 2024-10-31 14:18:40 +0000, olcott said:

On 10/31/2024 8:58 AM, joes wrote:

Am Thu, 31 Oct 2024 07:19:18 -0500 schrieb olcott:

On 10/31/2024 5:34 AM, Mikko wrote:

On 2024-10-30 12:16:02 +0000, olcott said:

On 10/30/2024 5:02 AM, Mikko wrote:

On 2024-10-27 14:21:25 +0000, olcott said:

On 10/27/2024 3:37 AM, Mikko wrote:

On 2024-10-26 13:17:52 +0000, olcott said:

Just imagine c functions that have enough memory to compute sums >>>>>>>>>>>>> and products of ASCII strings of digits using the same method that
people do.

Why just imagein? That is fairly easy to make. In some other >>>>>>>>>>>> lanugages (e.g. Python, Javascript) it is alread in the library or >>>>>>>>>>>> as a built-in feature.

OK next I want to see the actual Godel numbers and the arithmetic >>>>>>>>>>> steps used to derive them.

They can be found in any textbook of logic that discusses
undecidability.
If you need to ask about details tell us which book you are using. >>>>>>>>> Every single digit of the entire natural numbers not any symbolic name

for such a number.

Just evaluate the expressions shown in the books.

To me they are all nonsense gibberish. How one can convert a proof about
arithmetic into a proof about provability seems to be flatly false. >>>>>

The key is selfreference. There is a number that encodes the sentence >>>>>> "the sentence with the number [the number that this sentence encodes to] >>>>>> is not provable".

Can you please hit return before you reply?
Your reply is always buried too close to what you are replying to.

We simply reject pathological self-reference lie
ZFC did and the issue ends.

You cannot reject any number from atrithmetic. If you do the result is >>>> not arithmetic anymore.

I claims that his whole proof is nonsense until you
provide 1200% concrete proof otherwise.

Crackpots claim all all sorts of things. There is no way to change that
so there is no point to try.

All of arithmetic is inherently computable and
any non-arithmetic operation on a number is a type
mismatch error.

There are arithmetic functions and predicates that are not Turing
computable. For example, Busy Beaver.

Not computable because of self-reference is a different class
than not computable for other reasons.

There is no self reference in Busy Beaver. Anyway, not Turing computable
is not Turing computable, whatever the reason.

The Goldbach conjecture seems not computable only because it seems to
require an infinite number of steps.

It seems so but that is not really known.

--
Mikko

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On 2024-11-02 11:09:06 +0000, olcott said:

On 11/2/2024 3:37 AM, Mikko wrote:

On 2024-11-01 11:50:24 +0000, olcott said:

On 11/1/2024 3:44 AM, Mikko wrote:

On 2024-10-31 12:19:18 +0000, olcott said:

On 10/31/2024 5:34 AM, Mikko wrote:

On 2024-10-30 12:16:02 +0000, olcott said:

On 10/30/2024 5:02 AM, Mikko wrote:

On 2024-10-27 14:21:25 +0000, olcott said:

On 10/27/2024 3:37 AM, Mikko wrote:

On 2024-10-26 13:17:52 +0000, olcott said:

Just imagine c functions that have enough memory to compute >>>>>>>>>>> sums and products of ASCII strings of digits using the same >>>>>>>>>>> method that people do.

Why just imagein? That is fairly easy to make. In some other lanugages
(e.g. Python, Javascript) it is alread in the library or as a built-in
feature.

OK next I want to see the actual Godel numbers and the
arithmetic steps used to derive them.

They can be found in any textbook of logic that discusses undecidability.
If you need to ask about details tell us which book you are using. >>>>>>>>

Every single digit of the entire natural numbers
not any symbolic name for such a number.

Just evaluate the expressions shown in the books.

To me they are all nonsense gibberish.

The books define everything needed in order to understand the encoding >>>> rules.

Encoding nonsense gibberish as a string of digits is trivial.

How one
can convert a proof about arithmetic into a
proof about provability seems to be flatly false.

You needn't. The proof about provability is given in the books so
you needn't any comversion.

So you are saying that the Gödel sentence has nothing
to do with

BEGIN:(Gödel 1931:39-41)
   ...We are therefore confronted with a proposition which
   asserts its own unprovability.
END:(Gödel 1931:39-41)

Nothing is too strong but the connection is not arithmetic.
That "asserts its own unprovability" refers to a non-arithmetic
interpretation of an arithmetic formula.

I want to know 100% concretely exactly what this means,
no hand waving allowed.

It means whatever Gödel wanted it to mean. As the sentence is not
a part of a proof the only clue we have is what Gödel said.

Making arithmetic say anything about provability
seems like making an angel food cake out of lug nuts,
cannot possible be done.

Numbers have features and formulas have features. Therefore it is
possible to compare features of formulas to features of numbers.

This seems to be a type mismatch error. I need to
see every tiny detail of how it is not.

It is possible to compare things of different types. For example,
chairs are not animals but we can compare the numbers of their legs.

--
Mikko

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On 11/3/24 8:11 AM, olcott wrote:

On 11/3/2024 3:10 AM, Mikko wrote:

On 2024-11-02 11:05:20 +0000, olcott said:

On 11/2/2024 3:29 AM, Mikko wrote:

On 2024-11-01 11:53:00 +0000, olcott said:

On 11/1/2024 3:47 AM, Mikko wrote:

On 2024-10-31 14:18:40 +0000, olcott said:

On 10/31/2024 8:58 AM, joes wrote:

Am Thu, 31 Oct 2024 07:19:18 -0500 schrieb olcott:

On 10/31/2024 5:34 AM, Mikko wrote:

On 2024-10-30 12:16:02 +0000, olcott said:

On 10/30/2024 5:02 AM, Mikko wrote:

On 2024-10-27 14:21:25 +0000, olcott said:

On 10/27/2024 3:37 AM, Mikko wrote:

On 2024-10-26 13:17:52 +0000, olcott said:

Just imagine c functions that have enough memory to >>>>>>>>>>>>>>> compute sums
and products of ASCII strings of digits using the same >>>>>>>>>>>>>>> method that
people do.

Why just imagein? That is fairly easy to make. In some other >>>>>>>>>>>>>> lanugages (e.g. Python, Javascript) it is alread in the >>>>>>>>>>>>>> library or
as a built-in feature.

OK next I want to see the actual Godel numbers and the >>>>>>>>>>>>> arithmetic
steps used to derive them.

They can be found in any textbook of logic that discusses >>>>>>>>>>>> undecidability.
If you need to ask about details tell us which book you are >>>>>>>>>>>> using.

Every single digit of the entire natural numbers not any >>>>>>>>>>> symbolic name
for such a number.

Just evaluate the expressions shown in the books.

To me they are all nonsense gibberish. How one can convert a >>>>>>>>> proof about
arithmetic into a proof about provability seems to be flatly >>>>>>>>> false.

The key is selfreference. There is a number that encodes the
sentence
"the sentence with the number [the number that this sentence
encodes to]
is not provable".

Can you please hit return before you reply?
Your reply is always buried too close to what you are replying to. >>>>>>>
We simply reject pathological self-reference lie
ZFC did and the issue ends.

You cannot reject any number from atrithmetic. If you do the
result is
not arithmetic anymore.

I claims that his whole proof is nonsense until you
provide 1200% concrete proof otherwise.

Crackpots claim all all sorts of things. There is no way to change that >>>> so there is no point to try.

All of arithmetic is inherently computable and
any non-arithmetic operation on a number is a type
mismatch error.

There are arithmetic functions and predicates that are not Turing
computable. For example, Busy Beaver.

Not computable because of self-reference is a different class
than not computable for other reasons.

There is no self reference in Busy Beaver. Anyway, not Turing computable
is not Turing computable, whatever the reason.

Computing the square root of a pile of actual mud
is not Truing computable yet does not prove any
actual limit to computation. Likewise for simply
counting to infinity.

Who said it was.

This is just another of your uncounted fallacies that you use to
sidetrack from the fact that you are proving yourself a total idiot.

The Goldbach conjecture seems not computable only because it seems to
require an infinite number of steps.

It seems so but that is not really known.

We do know that an infinite number of steps would
provide the correct answer after an infinite amount
of time. There is no after an infinite amount of time.

Are you sure?

PROOF can't use infinite time, TRUTH can.

Your mind just seems too small by an order of magnatude to understand
that fact.

--- SoupGate-Win32 v1.05
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On 2024-11-03 13:11:09 +0000, olcott said:

On 11/3/2024 3:10 AM, Mikko wrote:

On 2024-11-02 11:05:20 +0000, olcott said:

On 11/2/2024 3:29 AM, Mikko wrote:

On 2024-11-01 11:53:00 +0000, olcott said:

On 11/1/2024 3:47 AM, Mikko wrote:

On 2024-10-31 14:18:40 +0000, olcott said:

On 10/31/2024 8:58 AM, joes wrote:

Am Thu, 31 Oct 2024 07:19:18 -0500 schrieb olcott:

On 10/31/2024 5:34 AM, Mikko wrote:

On 2024-10-30 12:16:02 +0000, olcott said:

On 10/30/2024 5:02 AM, Mikko wrote:

On 2024-10-27 14:21:25 +0000, olcott said:

On 10/27/2024 3:37 AM, Mikko wrote:

On 2024-10-26 13:17:52 +0000, olcott said:

Just imagine c functions that have enough memory to compute sums
and products of ASCII strings of digits using the same method that
people do.

Why just imagein? That is fairly easy to make. In some other >>>>>>>>>>>>>> lanugages (e.g. Python, Javascript) it is alread in the library or
as a built-in feature.

OK next I want to see the actual Godel numbers and the arithmetic >>>>>>>>>>>>> steps used to derive them.

They can be found in any textbook of logic that discusses >>>>>>>>>>>> undecidability.
If you need to ask about details tell us which book you are using. >>>>>>>>>>> Every single digit of the entire natural numbers not any symbolic name

for such a number.

Just evaluate the expressions shown in the books.

To me they are all nonsense gibberish. How one can convert a proof about
arithmetic into a proof about provability seems to be flatly false. >>>>>>>

The key is selfreference. There is a number that encodes the sentence >>>>>>>> "the sentence with the number [the number that this sentence encodes to]
is not provable".

Can you please hit return before you reply?
Your reply is always buried too close to what you are replying to. >>>>>>>
We simply reject pathological self-reference lie
ZFC did and the issue ends.

You cannot reject any number from atrithmetic. If you do the result is >>>>>> not arithmetic anymore.

I claims that his whole proof is nonsense until you
provide 1200% concrete proof otherwise.

Crackpots claim all all sorts of things. There is no way to change that >>>> so there is no point to try.

All of arithmetic is inherently computable and
any non-arithmetic operation on a number is a type
mismatch error.

There are arithmetic functions and predicates that are not Turing
computable. For example, Busy Beaver.

Not computable because of self-reference is a different class
than not computable for other reasons.

There is no self reference in Busy Beaver. Anyway, not Turing computable
is not Turing computable, whatever the reason.

Computing the square root of a pile of actual mud
is not Truing computable

in spite of the lack of any self reference.

--
Mikko

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)

On 2024-11-03 13:15:30 +0000, olcott said:

On 11/3/2024 3:16 AM, Mikko wrote:

On 2024-11-02 11:09:06 +0000, olcott said:

On 11/2/2024 3:37 AM, Mikko wrote:

On 2024-11-01 11:50:24 +0000, olcott said:

On 11/1/2024 3:44 AM, Mikko wrote:

On 2024-10-31 12:19:18 +0000, olcott said:

On 10/31/2024 5:34 AM, Mikko wrote:

On 2024-10-30 12:16:02 +0000, olcott said:

On 10/30/2024 5:02 AM, Mikko wrote:

On 2024-10-27 14:21:25 +0000, olcott said:

On 10/27/2024 3:37 AM, Mikko wrote:

On 2024-10-26 13:17:52 +0000, olcott said:

Just imagine c functions that have enough memory to compute >>>>>>>>>>>>> sums and products of ASCII strings of digits using the same >>>>>>>>>>>>> method that people do.

Why just imagein? That is fairly easy to make. In some other lanugages
(e.g. Python, Javascript) it is alread in the library or as a built-in
feature.

OK next I want to see the actual Godel numbers and the
arithmetic steps used to derive them.

They can be found in any textbook of logic that discusses undecidability.
If you need to ask about details tell us which book you are using. >>>>>>>>>>

Every single digit of the entire natural numbers
not any symbolic name for such a number.

Just evaluate the expressions shown in the books.

To me they are all nonsense gibberish.

The books define everything needed in order to understand the encoding >>>>>> rules.

Encoding nonsense gibberish as a string of digits is trivial.

How one
can convert a proof about arithmetic into a
proof about provability seems to be flatly false.

You needn't. The proof about provability is given in the books so
you needn't any comversion.

So you are saying that the Gödel sentence has nothing
to do with

BEGIN:(Gödel 1931:39-41)
   ...We are therefore confronted with a proposition which
   asserts its own unprovability.
END:(Gödel 1931:39-41)

Nothing is too strong but the connection is not arithmetic.
That "asserts its own unprovability" refers to a non-arithmetic
interpretation of an arithmetic formula.

I want to know 100% concretely exactly what this means,
no hand waving allowed.

It means whatever Gödel wanted it to mean. As the sentence is not
a part of a proof the only clue we have is what Gödel said.

In other words you don't really understand the proof.
You are merely trusting Gödel on faith.

I already told you that you should not use "In other words".
That you do it anyway indicates an attempt of a straw man
deception.

--
Mikko

--- SoupGate-Win32 v1.05
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