On 5/7/2024 4:22 PM, Ross Finlayson wrote:

On 05/06/2024 12:36 PM, Ross Finlayson wrote:

On 05/05/2024 03:02 PM, Jim Burns wrote:

I think that your wished.for supplements of
standard.issue quantifiers
can be defined given
standard.issue quantifiers.

For my wish,
I would like everyone to be clear on what
standard.issue quantifiers and variables
mean.

I think that,
way off in that glorious future,
both you and I will be able to be
satisfactorily understood.

And what more could there be
to wish for?

Well, one might aver that extra-ordinary
universal quantifiers are merely syntactic sugar,
yet there's that in the low- and high- orders,
or the first and final, that what they would
reflect of the _effects_ of quantification,
something like

for-any A?
for-each A+
for-every A*
for-all A$

My guess is that 'A' is the ASCIIfication of '∀'
Thus
for-any ∀?
for-each ∀+
for-every ∀*
for-all ∀$

Please use each of ∀? ∀+ ∀* ∀$ in a sentence.


I am familiar with wildcard characters in
their programming.language context. In that context,
they look to me more like variables ranging over
standard.issue.all[1] of a set of characters or
a set of sequences of characters.

Maybe if you used ∀? ∀+ ∀* ∀$
I would see better what you mean.


[1]
| ∀x:B(x) ⇒ B(t)
| ∀x:(B⇒C(x)) ⇒ (B⇒∀x:C(x))
| B(x) ⊢ ∀x:B(x)
| ∃x:B(x) ⇔ ¬∀x:¬B(x)

that it is so that the sputniks or extras
of the quantification in the extra-ordinary,
have that a quantifier disambiguation:
is in the syntax.

Then for the rest of it, like our discussions
on continuous domains and continuous topologies,
i.e. the topology that's initial and final itself,
then these line-reals field-reals signal-reals,
about the integer continuum linear continuum
long-line continuum, ubiquitous ordinals and
extra-ordinary theory, is that these are objects
of the universe of mathematics in the
Hilbert's Infinite Living Museum, of Mathematics.

When considering someone like Paul do Bois-Reymond,
who came up with the diagonal argument and the long-line,
and Mirimanoff, who came up with the axiom of regularity
and also the extra-ordinary, and for example Peano,
with his integers and infinitesimals, then one may well
aver that today's standard is a tenuous sort of course,
that is much more fully enriched by the first sort of
nonstandard function like the Dirac Delta, then into
the greater realm of the superclassical law(s) of large
numbers, and more replete three definitions of
continuous domains, and the Cantor space(s).

That's what I'm talking about.

I think that you are over.estimating
how clear you've been.

I don't see how standard.issue.quantifiers are
enriched by the examples you give.
Standard.issue quantifiers are already rich enough
to describe them, so, huh?


I have had a bit of fun here in sci.math
brewing up non.standard notation for
non.standard quantifiers, among other things.

My non.standard notations are abbreviations for
expressions with perfectly standard quantification.
For example,
the ordinals are well.ordered.
If exists any γ: B(γ)
then, exists first β: B(β)
∃γ:B(γ) ⇒ ∃₁β:B(β)

That's an abbreviation.
∃₁β:B(β) ⇔
∃β:(B(β) ∧ ¬∃α<β:B(α))

I am fond of abbreviating.
But all the interesting stuff is found
as a consequence of the disabbreviated forms.
I once spent an entire course studying
∫ᵟᶿω = ∫ᶿdω

the generalized Stokes theorem https://en.wikipedia.org/wiki/Generalized_Stokes_theorem

It was a very full course.
A lot is crammed into ∫ᵟᶿω = ∫ᶿdω

Are ∀? ∀+ ∀* ∀$ abbreviations of
standard.issue.quantified expressions?

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On 5/8/2024 4:20 PM, Ross Finlayson wrote:

On 05/08/2024 11:40 AM, Jim Burns wrote:

My guess is that 'A' is the ASCIIfication of '∀'
Thus
for-any ∀?
for-each ∀+
for-every ∀*
for-all ∀$

Please use each of ∀? ∀+ ∀* ∀$ in a sentence.

If you don't want to use ∀? ∀+ ∀* ∀$ in a sentence,
that's certainly your choice to make.

It's not a choice which encourages
efforts to understand you.

https://en.wikipedia.org/wiki/Fran%C3%A7ois_Vi%C3%A8te

François Viète, Seigneur de la Bigotière
(Latin: Franciscus Vieta; 1540 – 23 February 1603)

| In his dedication of the Isagoge to
| Catherine de Parthenay, Viète wrote:
|
| "These things which are new are wont in the beginning
| to be set forth rudely and formlessly and must then
| be polished and perfected in succeeding centuries.
| Behold, the art which I present is new,
| but in truth so old,
| so spoiled and defiled by the barbarians,
| that I considered it necessary,
| in order to introduce an entirely new form into it,
| to think out and publish a new vocabulary,
| having gotten rid of all its pseudo-technical terms..."

Consider
| ∀x:B(x) ⇒ B(t)
| ∀x:(B⇒C(x)) ⇒ (B⇒∀x:C(x))
| B(x) ⊢ ∀x:B(x)
| ∃x:B(x) ⇔ ¬∀x:¬B(x)

Is it possible that
several centuries of polishing and perfecting
have given us, in 2024, something which
François Viète had only set out in search of?

I am not a giant.
However, I can stand on giants' shoulders.
Since I can, why shouldn't I?

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On 5/9/2024 3:56 PM, Ross Finlayson wrote:

On 05/08/2024 02:14 PM, Jim Burns wrote:

Consider
| ∀x:B(x) ⇒ B(t)
| ∀x:(B⇒C(x)) ⇒ (B⇒∀x:C(x))
| B(x)  ⊢  ∀x:B(x)
| ∃x:B(x) ⇔ ¬∀x:¬B(x)

Is it possible that
several centuries of polishing and perfecting
have given us, in 2024, something which
François Viète had only set out in search of?

I am not a giant.
However, I can stand on giants' shoulders.
Since I can, why shouldn't I?

Sort of, I suppose.

| I beseech you, in the bowels of Christ,
| think it possible that
| I cannot read your mind.
|
<pseudo.Cromwell>

Like Russell stood on Frege and Peirce,
and von Neumann and Zermelo stood on Mirimanoff,
and Cantor stood on duBois-Reymond, well,
Newton of course is very well-known for
his quote "I stood on people left and right".

| If I have seen further
| it is by standing on ye sholders of Giants.
|
<Newton>

Here it's still "Amicus Plato"

| Amicus Plato — amicus Aristoteles — magis amica veritas
<Newton>
==
| Plato is my friend -- Aristotle is my friend --
| but my best friend is truth.
|
<Newton>

Here it's still "Amicus Plato"
and it's very old-fashioned,
yet every few hundred years at least
it comes back around,
unsurprisingly much the same.

So, ye adherents of Russell's retro-thesis and
semi-Aristotleans of
the "I say" logical positivist variety,
too often thinking that
circa-20'th-century-classical quasi-modal logic
is either classical or full for DeMorgan:
can you get down?

Not.first.false?  Largest.number.ever.

Compare
finite sequences of only not.first.false claims
to
logarithmic slide rules.

When used correctly,
they both give what they're advertised to give.

Doubts that they give that,
to the extent that there are doubts that they give that,
originate from it being less.than.immediately.obvious
that they give what they're advertised to give.

But they do give that,
and it can be shown that they give that,
even if it is challenge and more.than.a.challenge
to _immediately_ show that they give that.

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On 5/9/2024 9:45 PM, Ross Finlayson wrote:

On 05/09/2024 03:55 PM, Jim Burns wrote:

On 5/9/2024 3:56 PM, Ross Finlayson wrote:

On 05/08/2024 02:14 PM, Jim Burns wrote:

Consider
| ∀x:B(x) ⇒ B(t)
| ∀x:(B⇒C(x)) ⇒ (B⇒∀x:C(x))
| B(x)  ⊢  ∀x:B(x)
| ∃x:B(x) ⇔ ¬∀x:¬B(x)

Is it possible that
several centuries of polishing and perfecting
have given us, in 2024, something which
François Viète had only set out in search of?

I am not a giant.
However, I can stand on giants' shoulders.
Since I can, why shouldn't I?

Sort of, I suppose.

| I beseech you, in the bowels of Christ,
| think it possible that
| I cannot read your mind.
|
<pseudo.Cromwell>

Like Russell stood on Frege and Peirce,
and von Neumann and Zermelo stood on Mirimanoff,
and Cantor stood on duBois-Reymond, well,
Newton of course is very well-known for
his quote "I stood on people left and right".

| If I have seen further
| it is by standing on ye sholders of Giants.
|
<Newton>

Here it's still "Amicus Plato"

| Amicus Plato — amicus Aristoteles — magis amica veritas
<Newton>
==
| Plato is my friend -- Aristotle is my friend --
| but my best friend is truth.
|
<Newton>

Here it's still "Amicus Plato"
and it's very old-fashioned,
yet every few hundred years at least
it comes back around,
unsurprisingly much the same.

So, ye adherents of Russell's retro-thesis and
semi-Aristotleans of
the "I say" logical positivist variety,
too often thinking that
circa-20'th-century-classical quasi-modal logic
is either classical or full for DeMorgan:
can you get down?

Not.first.false?  Largest.number.ever.

Compare
finite sequences of only not.first.false claims
to
logarithmic slide rules.

When used correctly,
they both give what they're advertised to give.

Doubts that they give that,
to the extent that there are doubts that they give that,
originate from it being less.than.immediately.obvious
that they give what they're advertised to give.

But they do give that,
and it can be shown that they give that,
even if it is challenge and more.than.a.challenge
to _immediately_ show that they give that.

Ah, good sir, it's certainly to be appreciated rising
to a higher level of rhetoric.

It seems to be a distinctive part of your style
to remove as many clues to what you mean as you can.
Who is appreciating?
Who or what is rising?

Thanks, I've heard that one before. Here it's
also "Amicus Plato, fini".

?== "Dear Plato, finish it"

What would you intend to be conveyed by that, here?

Please don't see my yet not writing terms as for
SX a set and #X an ordinal, while also SX a usual
notation for an ordinal in succession, with regards
to counting, and numbering, where it is so in some
theory that PX, the powerset of X, is, SX.

Are you still talking about quantifiers, here?

As well, please don't see that as a lack of cooperation,
for all the times in all the threads whereas after a
large amount of my proper presentation of correct
reasoning, that you've balked and clammed up,

I doubt that I can haul from my memory
every instance in which I've balked and clammed up,
but I can haul a fair sample.

Broadly speaking,
when I have balked and clammed up,
I have failed at using as an explanation
that which you seem to have intended for me
to use as an explanation.

I have, at times, requested help from you,
such as your using ∀? ∀+ ∀* ∀$ in sentences,
but you seem to be unable to see these requests,
and I have made them less and less often.

I have a few different explanations for
your responses being how they are.
The one you'd dislike least, I think, is that
your intended readers are a group of people
which does not include me, Jim Burns.
The upshot of that and also the other explanations
is that I'll not be getting any farther toward
understanding what you're saying.
Therefore, I balk and clam up.

as what even for a fair-weather formalist and
dreamy intuitionist, must eventually see that
flowing the threads, or arguments as it were,
rhetorically, forensically, here is that I've gotten
around to it.

So, this idea of a proper distinguished syntax for
universal quantifiers as with regards to how they
apply to the various relations, where in a given theory
we may aver that all predicates are relations and as
for vice-versa, that relations are primary, then these
schemes, of quantification, become higher order,
if only an order or so, than the usual syntax where
terms are primary, that it so effects to reflect the
relations as primary, why it is so that these refined
universal quantifiers, are elements of a syntax,
irreducibly.

Formally, ....

Hundreds and hundreds of threads on sci.math and sci.logic,
many last words, ....

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On 5/10/2024 4:12 PM, Ross Finlayson wrote:

On 05/10/2024 03:59 AM, Jim Burns wrote:

[...]

I'd like to suggest a reading from
Dehaene's "The Number Sense",
in Chapter 9 "Of Neurons and Numbers",
in the section
"When Intuition Outruns Axioms".

I found a copy online from
the International Cognition and Culture Institute,
and about page 238.

He explains that there _are_
non-standard models of integers.

Is "When Intuition Outruns Axioms" concerned with
other.than.standard.issue quantifiers?
[1]

If so,
since you are currently holding the talking.stick,
you could use the opportunity to expound on
what Dehaene has to say.

Maybe I should clarify:

I don't say that there aren't non.standard objects.
That's not the same as saying that
there aren't non.standard quantifiers.

I also don't say that there aren't
non.standard quantifiers.
There might not be.
Quantifiers are live near the roots of our logic.
There might be.
Mathematicians are smart.

What I say is
I don't know yet what sort of
non.standard.quantification scheme
you introduced at your "universal quantification"
post.

If you wish I had more to say about your posts
(a big IF, not everyone does)
helping me to understand your posts seems like
an effective strategy for bringing that about.

Just saying.

Again, what correspondence of yours I see,
which is any in response to me, I've replied.

If you have used ∀? ∀+ ∀* ∀$ in sentences,
I have overlooked them.

Surely, it would only be a very small favor to me
to repeat those sentences.
By doing so, you would increase the chances
of me NOT balking and clamming up.

Could you please do so again?

[1]
By other.than.standard.issue quantifiers, I mean
other than those such that:
| ∀x:B(x) ⇒ B(t)
| ∀x:(B⇒C(x)) ⇒ (B⇒∀x:C(x))
| B(x) ⊢ ∀x:B(x)
| ∃x:B(x) ⇔ ¬∀x:¬B(x)

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On 5/10/2024 8:17 PM, Ross Finlayson wrote:

On 05/10/2024 03:26 PM, Jim Burns wrote:

On 5/10/2024 4:12 PM, Ross Finlayson wrote:

On 05/10/2024 03:59 AM, Jim Burns wrote:

[...]

I'd like to suggest a reading from
Dehaene's "The Number Sense",
in Chapter 9 "Of Neurons and Numbers",
in the section
"When Intuition Outruns Axioms".

I found a copy online from
the International Cognition and Culture Institute,
and about page 238.

He explains that there _are_
non-standard models of integers.

Is "When Intuition Outruns Axioms" concerned with
other.than.standard.issue quantifiers?
[1]

If so,
since you are currently holding the talking.stick,
you could use the opportunity to expound on
what Dehaene has to say.

Maybe I should clarify:

I don't say that there aren't non.standard objects.
That's not the same as saying that
there aren't non.standard quantifiers.

I also don't say that there aren't
non.standard quantifiers.
There might not be.
Quantifiers are live near the roots of our logic.
There might be.
Mathematicians are smart.

What I say is
I don't know yet what sort of
non.standard.quantification scheme
you introduced at your "universal quantification"
post.

If you wish I had more to say about your posts
(a big IF, not everyone does)
helping me to understand your posts seems like
an effective strategy for bringing that about.

Just saying.

Again, what correspondence of yours I see,
which is any in response to me, I've replied.

If you have used ∀? ∀+ ∀* ∀$ in sentences,
I have overlooked them.

Surely, it would only be a very small favor to me
to repeat those sentences.
By doing so, you would increase the chances
of me NOT balking and clamming up.

Could you please do so again?

[1]
By other.than.standard.issue quantifiers, I mean
other than those such that:
| ∀x:B(x) ⇒ B(t)
| ∀x:(B⇒C(x)) ⇒ (B⇒∀x:C(x))
| B(x)  ⊢  ∀x:B(x)
| ∃x:B(x) ⇔ ¬∀x:¬B(x)

Well, first of all, it's after pondering that there
is quantifier comprehension artifacts of the extra sort,
as of a set of all sets, order type of ordinals, a universe,
set of sets that don't contain themself, sets that contain
themselves, and so on.

Then, English affords "any, "each, "every, "all".

The -any means for example that "it's always a fragment".
So in this sense the usual universal quantifier is for-each.

Then, for-each, means usual comprehension, as if an enumeration,
or a choice function, each.

Then, for-every, means as a sort of comprehension, where it
so establishes itself again, any differently than -each,
when -each and -every implies both none missing and all gained.

Then, "for-all", sort of is for that what is so "for-each"
and "for-every" is so, "for-all", as for the multitude as
for the individual.

Then, I sort of ran out of words, "any", "each", "every", "all",
then that seems their sort of ordering, about comprehension,
in quantification, in the universals, of each particular.

About sums it up, ...." -- Monday

Date: Tue, 7 May 2024 15:16:27 -0400.

Are there differences in syntax between
'for.any' 'for.each' 'for.every' 'for.all' ?

If your answer, if it ever comes, is "no",
then I do not know what is meant by

quantifier comprehension artifacts of the extra sort

That looks to me like
quantifiers used in several domains, full stop.

If your answer, if it ever comes, is "yes",
then I would like to know different how.
Syntax is pretty intimately entangled with semantics.

If your answer never comes,
why don't I just balk and clam up, because,
without it, I don't have much to say.


I'd like to think of my internet production as
carefully.curated stretches of silence
artfully.punctuated with non.silences,
something in the direction of John Cage's 4'33"

I try to indicate not.having something to say
by not.saying it.

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On 5/11/2024 9:38 AM, Ross Finlayson wrote:

On 05/11/2024 01:18 AM, Jim Burns wrote:

Date: Tue, 7 May 2024 15:16:27 -0400.

Are there differences in syntax between
'for.any' 'for.each' 'for.every' 'for.all' ?

If your answer, if it ever comes, is "no",
then I do not know what is meant by

quantifier comprehension artifacts of the extra sort

That looks to me like
quantifiers used in several domains, full stop.

If your answer, if it ever comes, is "yes",
then I would like to know different how.
Syntax is pretty intimately entangled with semantics.

If your answer never comes,
why don't I just balk and clam up, because,
without it, I don't have much to say.

Now of course such a notion or idea or concept or
pensee or thought didn't just erupt fully-formed,
like Conrad from the tin of corned beef,
that it starts rather more like 'for-any, or, for-all',
about things like "for-any well-founded set, it's a set in
the well-founded universe", then, "for-all well-founded
or non-well-founded sets, they are sets in the set-theoretic
universal set".

Then, it's not necessary to invoke the entire universe of
sets, the entire domain of discourse that is anything that
is a set, though is reasonably brief when in a theory with
only logical sets, logically, sets of sets.

I.e., it applies as closely to "sets of sets", and the n'th order
about quantification, and comprehension.

Let's be clearly understood that I am a formalist,
if though not a nominalist yet a platonist,
because mathematics its truths are discovered
not invented, while our language and terms and
derivations are as yet technique.

So, constructivism is regarded as the rulial in
the standard, and intutionism is that which
revolves in the abductive inference, as what
makes for embracing the fuller dialectic.

Thusly, the "standard" is "our standard",
while what's of interest in the fuller dialectic
is the "extra-ordinary" or "super-standard",
that the "non-standard", must be in these
classes of classes, yet formalist, and rulial
again, in the competing regularities, which
comprise "it", the thing, the universe of the
mathematical and logical objects, a theory,
to which we attain, "A Theory", the theory,
of the things, the theory of every thing.

So, just saying, there's a greater mathematics
than "our standard", with "R, standard", and
modern mathematics as it's usually known,
a paleo-classical post-modern mathematics,
which mathematics owes physics for the
greater context of continuity, convergence,
and the laws of large numbers.

I'm a formalist: and in natural language.

Date: Tue, 7 May 2024 15:16:27 -0400.

Are there differences in syntax between
'for.any' 'for.each' 'for.every' 'for.all' ?

If your answer, if it ever comes, is "no",
then I do not know what is meant by

quantifier comprehension artifacts of the extra sort

That looks to me like
quantifiers used in several domains, full stop.

If your answer, if it ever comes, is "yes",
then I would like to know different how.
Syntax is pretty intimately entangled with semantics.

If your answer never comes,
why don't I just balk and clam up, because,
without it, I don't have much to say.

So, it's door number 3.

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On 5/11/2024 11:47 AM, Ross Finlayson wrote:

On 05/11/2024 07:40 AM, Jim Burns wrote:

[...]

In the logical, the purely logical,
the syntax "is" the semantics.

If what makes logic impure is
to be about something,
then it would make some sense to say that
pure logic has no semantics

...which leads, by default?
to syntax being the missing semantics, I guess?

Sorry, I will not sign your petition.
Syntax and semantics are more different than
cabbages and kings.

It seems to me that
the purest of ultra.pure logic is actually
_about_ claims,
analogous to geometry being _about_ points,
lines, plane.figures, and so on.

It is an unbreakable law that
the sum of the squares of the two shorter sides
of a right.triangle is equal to
the square of the third and longest side.

It is an unbreakable law that
a finite sequence of only not.first.false claims
holds only true claims.

It is an unbreakable law that
Q preceded by P and P⇒Q is not.first.false.

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On 5/11/2024 5:44 PM, Ross Finlayson wrote:

On 05/11/2024 02:05 PM, Ross Finlayson wrote:

On 05/11/2024 12:24 PM, Jim Burns wrote:

[...]

[...]

It's like two inductive analysts were contradicting
each other. One says "base case, subsequent
case, case closed", and the other says "base case,
subsequent case, case not closed".

You just pick one?

Pick the one in a finite sequence of
only not.first.false claims.
It is a true claim.


"Base case, subsequent case" is cisfinite induction.

Today, elsewhere, I gave an argument ==
a finite.sequence of only not.first.false.claims
for cisfinite induction.
Re: because g⤨(g⁻¹(x)) = g(y) [1/2] Re: how
Date: Sat, 11 May 2024 14:49:46 -0400

But there is also transfinite induction, which also
is in a finite.sequence of only not.first.false claims,
claims about ordinals instead of naturals.

The sequence about ordinals is shorter,
and I don't think this is about the sequences themselves,
anyway, but about justification by the existence of
such sequences. I think the shorter will shed more light.


1.
Ordinals are well.ordered.
∃γ: p(γ) ⟹ ∃β: (p(β) ∧ ¬∃α<β:p(α))

2.
Transfinite.induction is true for ordinals
∀β: (̅p(β) ⇐ ∀α<β:̅p(α)) ⟹ ∀γ: ̅p(γ)


(2.) is not.first.false in that sequence
because
(2.) is merely a re.written version of (1.)
¬H ⟹ ¬C for H ⟹ C
∀¬ for ¬∃
C ⇐ H for ¬(¬C ∧ H)
̅p for ¬p


(1.) is not.first.false in that sequence
because
(1.) is not.false (there or anywhere)

If Q is first.false, then Q is false ==
If Q is not.false, then Q is not.first.false.

(1.) "Ordinals are well.ordered" is true
because _that's what "ordinal" means_

(1.) (2.) is a finite sequence of
only not.first.false claims.

Because (1.) (2.) is finite,
if (1.) (2.) held a false claim,
(1.) (2.) would hold a first.false claim.

(1.) (2.) doesn't hold a first.false claim
(1.) (2.) doesn't hold a false claim

(2.) is true.

Transfinite.induction is true for ordinals
∀β: (̅p(β) ⇐ ∀α<β:̅p(α)) ⟹ ∀γ: ̅p(γ)

It's exactly about "not.ultimately.untrue" that
describes how there are "inductive impasses"
that belie their finite inputs.

Please explain "not.ultimately.untrue" and
"inductive impasse". I would welcome examples.

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On 5/11/2024 7:11 PM, Ross Finlayson wrote:

On 05/11/2024 02:44 PM, Ross Finlayson wrote:

On 05/11/2024 02:05 PM, Ross Finlayson wrote:

On 05/11/2024 12:24 PM, Jim Burns wrote:

[...]

[...]

[...]

The case is that induction goes through,
an inviolable law you call it:
does it go all the way through?
Does it complete?

It is complete.
There is no completing.activity,
so I wouldn't say it completes.

Compare to right triangles:
Are all the squares of two shorter sides
summed to the square of the longest side?

That's a tricky question to answer because
there is no summing done.
That relationship between the sides
is simply something true about right triangles.

And it is complete == it is true for each.

We don't typically ask the tricky question
about right triangles.
We ask the tricky question about cisfinite induction
because we imagine it as a process,
which we don't for right triangles.

Cisfinite induction is NOT a process.
Cisfinite induction is an argument,
completely correct or completely incorrect.

See, the contrary inductive analyst just says
"in case you don't have a deductive argument why
something is so, induction is so much shifting-sands
and slippery-slope." He just has "the base case is
you haven't completed induction, and so is the
subsequent case, case closed: case not closed".

When the argument is completed,
induction is completed.

----
There is something completely different
which is also called induction.
The completely.different induction is physics.

Physics.induction is not unbreakable.
Physics.induction isn't cisfinite or transfinite induction.

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On 5/11/2024 9:17 PM, Ross Finlayson wrote:

On 05/11/2024 04:47 PM, Jim Burns wrote:

On 5/11/2024 7:11 PM, Ross Finlayson wrote:

The case is that induction goes through,
an inviolable law you call it:
does it go all the way through?
Does it complete?

It is complete.
There is no completing.activity,
so I wouldn't say it completes.

Compare to right triangles:
Are all the squares of two shorter sides
summed to the square of the longest side?

That's a tricky question to answer because
there is no summing done.
That relationship between the sides
is simply something true about right triangles.

And it is complete == it is true for each.

We don't typically ask the tricky question
about right triangles.
We ask the tricky question about cisfinite induction
because we imagine it as a process,
which we don't for right triangles.

Cisfinite induction is NOT a process.
Cisfinite induction is an argument,
completely correct or completely incorrect.

What I recall of the context of the Pythagorean theorem,

Let's refresh our memories.

ͨₐ🭢🭕🭞🭜🭘ᵇ = ͨₐ🭢🭕ͩ + ͩₐ🭞🭜🭘ᵇ

The right triangle 🞃cab is split into
two right triangles ◥cda ◤adb
by segment a͞d perpendicular to b͞c

🞃cab ◥cda ◤adb are _similar_
[1]
Corresponding sides have equal ratios.

∠acb = ∠dca
∟cab = ∟cda
🞃cab ≚ ◥cda
c͡b/c͡a = c͡a/c͡d

∠cba = ∠abd
∟cab = ∟adb
🞃cab ≚ ◤adb
c͡b/b͡a = b͡a/d͡b

c͡b⋅c͡d = c͡a²
c͡b⋅d͡b = b͡a²
c͡b⋅(c͡d+d͡b) = c͡b² = c͡a² + b͡a²
QED

[1] needs its own proof,
but that can be done, too.

What I recall of the context of the Pythagorean theorem,
was that after algebra already was trigonometry, and
the definitions of the trigonometric functions, for
sine and cosine and tangent, about the opposite and
adjacent and hypotenuse, then as of a right triangle
with its hypotenuse the radius of a unit circle, that
the right angle is as with regards to the abscissa
and ordinates or where the lines drop or slide to
the x or y axis of the usual X-Y coordinate setting
of a circle centered at the origin, it was of the
secondary school's first three years of geometry,
algebra, and trigonometry, or along those lines.

So, we computed a bunch of ready things about
those often with the Pythagorean theorem,
which is as an addition-formula, mostly about
30-60-90 triangles, and, isosceles triangles,
or 45-45-90, then those got used throughout
precalculus and a couple years of calculus
or high school.

I agree that the Pythagorean theorem
gets used in a lot of different ways.

How we know that the Pythagorean theorem
is a fact about each right triangle
has important similarities to
how we know that we cisfinitely.induced claims
are facts about each natural number.

So anyways one time I see a diagram about
Pythagorean triples, those being tuples of
three integers that have a^2 + b^2 = c^2,
and what they'd done was right triangle,
then draw a square as of the square alongside
it, and counting the boxes of the squares of
a b c it's that the boxes of the squares of a
and b equals the boxes of the square of c.

Actually,
that works in the opposite direction.
We know that 3:4:5 is a right triangle
because of the Pythagorean theorem.

If that's not a proof of the Pythagorean theorem
and least it's graphically intuitive for some values,
where of course there are hundreds of known
proofs of the Pythagorean theorem, since the
time of Pythagoras as some even have as from
greater antiquity, then it reminds of things
like Rodriguez formula, Vieta's formulas,
Nicomachus' theorem and formulas,
Pascal triangle and bonomial theorem,
all what are sorts of addition formulas,
like an addition formula of the product
of exponents as the sum of the powers.

So, that Pythagorean triples exist, and it results
that the rightness of a triangle with sides length
the Pythagorean triple can be established without
invoking the Pythagorean theorem, doesn't so
much make it so the other way around, from
induction over Pythagorean triples, without
showing as how all right triangles are somehow
as some congruence to what is some Pythagorean
triple,

We shouldn't want to show that
each right triangle has a Pythagorean triple,
because we know that isn't true.
Famously, Pythagoras executed one of his disciples
for proving that the right isosceles triangle 1:1:√̅2
has no Pythagorean triple.

of the equivalence class of all the triples
and all the congruences to triangles with a
unit length longest side, establishing infinite
expressions, and closures, of completion,
to make a case for the Pythagorean theorem
as via induction from an explication after
the enumeration of Pythagorean triples,
which via inspection have a^2+b^2 = c^2,
as for that it results congruences that
"go to" any given dimensions of a right
triangle.

About the cisfinite and transfinite induction,
and I know it's not the languages fault that
there's the associated psychosexual connotation,
I'm glad you make the point though that
it just is what it is, and, a case for induction
more or less needs some reason its tendency,
to succeed as it were, then that induction
is given its course, then that the course-of--passage,
of what the plain old infinite induction, arrives.

Induction does not arrive,
unless you are talking about arriving at
the end of _expressing the argument_

I.e., it's always "infinite induction", after cause-and-effect,
with that also being induction or a case, mathematical
induction, and there can't be any reasonable counterclaims
or they'd be just as guaranteed as the contradistinct opposite.

So, it makes for a very strong perceived requirement
for deductive reasoning _why_ convergence criteria
exist, besides that "given an infinite expression,
it's an infinite expression".

Here then that's most Zeno's about geometric series,
and then about things like Stirling numbers and of
course the discussions we've been having over the
past few months about the convergence and
the slooowwwly convergent and all this,
the "scaffolding" of the infinite expressions
we've been discussing and at length.

Warm regards

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On 5/12/2024 3:34 PM, Ross Finlayson wrote:

On 05/12/2024 11:21 AM, Ross Finlayson wrote:

On 05/12/2024 10:46 AM, Jim Burns wrote:

[...]

Geometry's "similar" is often related to
"congruency, thoroughly",
where "congruent" means "similar".

You have confused me.
I agree that "similar" is related to "similar"
but I don't see why you tell us this.

ͨₐ🭢🭕🭞🭜🭘ᵇ = ͨₐ🭢🭕ͩ + ͩₐ🭞🭜🭘ᵇ

The right triangle 🞃cab is split into
two right triangles ◥cda ◤adb
by segment a͞d perpendicular to b͞c

🞃cab ◥cda ◤adb are _similar_
[1]
Corresponding sides have equal ratios.

Because triangles, here "similar" means
corresponding sides have equal ratios and
corresponding angles are equal.

You can know that a triangle is a right triangle
if you have the trigonometric functions of its angles,
here as where it doesn't necessarily require
the apparatus of Pythagoren theorem proper,
"its own theory", ....

Pythagoras says
∠cab = 90° ⟹ c͡b² = c͡a² + b͡a²

That means
∠cab = 90° ∨ c͡b² ≠ c͡a² + b͡a²

One side of that disjunction is true
for any triangle.
Thus, we don't need to know it's a right triangle
in order to know Pythagoras is correct.

The most usual tools, of classical constructions:
are: compass and edge.

If 🞃cab CAN be classically constructed
then ∠cab = 90° ⟹ c͡b² = c͡a² + b͡a²

If 🞃cab canNOT be classically constructed
then ∠cab = 90° ⟹ c͡b² = c͡a² + b͡a²

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On 5/14/2024 4:15 PM, Ross Finlayson wrote:

They're not all quite so strong,
the many, many examples
of the balking and clamming,
the actually quite a few very many,
though, these are pretty good.

You don't want to talk about
what I want to talk about.
And there's nothing wrong with that.
Really.

However, it's just as true
in the other direction.

Date: Sat, 11 May 2024 19:47:38 -0400
Message-ID: <a4700775-be6c-46db-ad41-361eee6a3b67@att.net>
<JB<RF>>

The case is that induction goes through,
an inviolable law you call it:
does it go all the way through?
Does it complete?

It is complete.
There is no completing.activity,
so I wouldn't say it completes.

Compare to right triangles:
Are all the squares of two shorter sides
summed to the square of the longest side?

That's a tricky question to answer because
there is no summing done.
That relationship between the sides
is simply something true about right triangles.

And it is complete == it is true for each.

We don't typically ask the tricky question
about right triangles.
We ask the tricky question about cisfinite induction
because we imagine it as a process,
which we don't for right triangles.

Cisfinite induction is NOT a process.
Cisfinite induction is an argument,
completely correct or completely incorrect.

</JB<RF>>

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On 5/15/2024 3:56 PM, Ross Finlayson wrote:

On 05/15/2024 07:10 AM, Jim Burns wrote:

On 5/14/2024 4:15 PM, Ross Finlayson wrote:

They're not all quite so strong,
the many, many examples
of the balking and clamming,
the actually quite a few very many,
though, these are pretty good.

You don't want to talk about
what I want to talk about.
And there's nothing wrong with that.
Really.

However, it's just as true
in the other direction.

Date: Sat, 11 May 2024 19:47:38 -0400
Message-ID: <a4700775-be6c-46db-ad41-361eee6a3b67@att.net>
<JB<RF>>

The case is that induction goes through,
an inviolable law you call it:
does it go all the way through?
Does it complete?

It is complete.
There is no completing.activity,
so I wouldn't say it completes.

Compare to right triangles:
Are all the squares of two shorter sides
summed to the square of the longest side?

That's a tricky question to answer because
there is no summing done.
That relationship between the sides
is simply something true about right triangles.

And it is complete == it is true for each.

We don't typically ask the tricky question
about right triangles.
We ask the tricky question about cisfinite induction
because we imagine it as a process,
which we don't for right triangles.

Cisfinite induction is NOT a process.
Cisfinite induction is an argument,
completely correct or completely incorrect.

</JB<RF>>

You mean "not.ultimately.untrue"?

I don't know.
What does "not.ultimately.untrue" mean?


Is this thing where I ask a question
and you don't answer
an instance of my balking and clamming up?

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On 5/16/2024 5:03 AM, Ross Finlayson wrote:

On 05/15/2024 01:57 PM, Jim Burns wrote:

On 5/15/2024 3:56 PM, Ross Finlayson wrote:

On 05/15/2024 07:10 AM, Jim Burns wrote:

[Cisfinite induction] is complete.
There is no completing.activity,
so I wouldn't say it completes.

Cisfinite induction is NOT a process.
Cisfinite induction is an argument,
completely correct or completely incorrect.

You mean "not.ultimately.untrue"?

I don't know.
What does "not.ultimately.untrue" mean?

It's just an introdunce of introduction,
not contradicted by deduction.

Thank you for answering my question.

'introdunce' looks like a typo or a neologism.
But I get "not contradicted by deduction".

----
_In its correct context_
complete cisfinite induction
is NOT contradicted by deduction.

Outside of its correct context,
we do not assert complete cisfinite induction.
⎛ To be pedantic, some people assert it there.
⎝ The technical term for them is "wrong".


We can say more than that, though.

_In its correct context_
DENYING complete cisfinite induction
IS contradicted by deduction.

Those who deny its completeness there
are, as we say, "wrong".

_In its correct context_
That is the essential qualification.

Context provides _true claims_ about
the objects of cisfinite induction.

If the context is of the correct type,
those true claims can be followed by
only not.first.false claims, bread crumbs
leading us to the statement of induction.

Because of the way in which we arrive at
the statement of induction,
we know it is a true statement.
But the way in which we arrive
must start with the correct context.

Other things, maybe darkᵂᴹ numbers, who knows?
which canNOT be followed by
a finite sequence of
only not.first.false claims
leading us to the statement of induction
are NOT asserted by us to have
complete cisfinite induction available.

You mean "not.ultimately.untrue"?

For complete.cisfinite.induction in correct.context,
I mean "not.first.false", which I take to be
stronger than "not.ultimately.untrue".

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On 5/18/2024 12:09 PM, Ross Finlayson wrote:

On 05/16/2024 09:50 AM, Jim Burns wrote:

[...]

I think that "correct", in context, is the entire
context, which is exactly what deductive inference
contains, explaining when inductive inference either
must complete, or meets its juxtaposition, with
regards to any two forces that balance and align
in symmetry.

So, what you are claiming is that inductive inference
is invincibly ignorant,

I am claiming that inductive inference
is invincibly modest.
Post.inference, we only assert claims about
whatever.it.is we described pre.inference.

Perhaps that doesn't seem modest,
because whatever.it.is is infinitely.many,
but induction holds for infinitely.many cisfinite ordinals
in the same way that geometry holds for infinitely.many
right triangles. Completely.

A given schema for induction has no more correctness,
in its own vacuum, than any other,

Induction on the cisfinite ordinals
⎛ those countable.back.to.0 after only
⎜ those countable.back.to.0
⎝ and also 0
is a theorem.
Theorems are not optional.

and when they're put together and don't
agree, then either they don't, and don't, or
don't, and do.

"Not.ultimately.untrue", ....

Induction on the cisfinite ordinals
is not.first.false in a finite sequence of
only not.first.false claims
which begins "A cisfinite ordinal is ... ".

One can contrive simple inductive arguments
that _nothing_ is so.

An example of such an argument would be clarifying here.

So, I'd say your definition of "correct", isn't,
and is simply a declaration of "relative" and "blind".

No offense meant, of course, it's so that paradoxes
are to be resolved, not obviated.

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On 5/18/2024 4:11 PM, Ross Finlayson wrote:

On 05/18/2024 11:16 AM, Jim Burns wrote:

On 5/18/2024 12:09 PM, Ross Finlayson wrote:

One can contrive simple inductive arguments
that _nothing_ is so.

An example of such an argument
would be clarifying here.

Says nothing,
says nothing,
says nothing,
says nothing,
....

See,
just saying so doesn't make it so,
something that _goes_ has a _place_ to go.

Explain to me how that is an inductive argument.

----
Here is an inductive argument for n⁺¹≠n

BASE CASE.
⎛ Background for finite ordinals
⎝ k⁺¹≠0

for k=0
0⁺¹≠0

STEP CASE.
⎛ Background for finite ordinals
⎝ j≠k ⇒ j⁺¹≠k⁺¹

for j=n⁺¹ k=n
n⁺¹≠n ⇒ n⁺¹⁺¹≠n⁺¹

Therefore,
for each finite ordinal n, n⁺¹≠n
by induction.

----
In the context of finite ordinals,
that is complete, in that
we know by that argument that,
for each finite ordinal n, n⁺¹≠n


There exists a fuller argument which
details _how_ we know that,
the part which we don't often see,
because that part is essentially unchanged
from invocation to invocation of "induction"

Here is the fuller inductive argument for n⁺¹≠n

| Assume a finite.ordinal counter.example nₓ
| nₓ⁺¹=nₓ
|
| ⎛ Background for ordinals
| ⎜ The set of counter.examples
| ⎝ holds a first or is empty.
|
| There is a counter.example nₓ
| There is a first counter.example n₁
| n₁⁺¹=n₁
| k < n₁ ⇒ k⁺¹≠k
|
| ⎛ Background for finite.ordinals
| ⎜ Finite.ordinal n is 0 or
| ⎜ it can be decremented and
| ⎝ each before.ordinal can be decremented or is 0
|
| finite.ordinal nₓ
| n₁ < nₓ
| n₁=0 or n₁ can be decremented.
|
| 1.
| n₁=0
| 0⁺¹=0
| BASE CASE: 0⁺¹≠0
| Contradiction.
|
| 2.
| n₁ can be decremented.
| n₁⁻¹ < n₁
| (n₁⁻¹)⁺¹ ≠ n₁⁻¹
| STEP CASE: n⁺¹≠n ⇒ n⁺¹⁺¹≠n⁺¹
| n₁⁺¹≠n₁
| However,
| n₁⁺¹=n₁
| Contradiction.
|
| Contradiction or contradiction.

Therefore,
a finite.ordinal counter.example nₓ not.exists

For each finite.ordinal n: n⁺¹≠n
Completely.

----
Suppose we want a proof of some property P(n)
for the complete domain of finite.ordinals

We can swap out
proofs of 0⁺¹≠0 and of n⁺¹≠n ⇒ n⁺¹⁺¹≠n⁺¹
and swap in
proofs of P(0) and of P(n) ⇒ P(n⁺¹)
and we will have a correct proof with
a necessarily correct conclusion.

A proof.by.induction is a _general_ form
into which details can be inserted which
make it a correct proof of a _particular_ claim
not unlike a proof.by.contradiction,
in that respect.

What we often _call_ a proof.by.induction is
the details to be swapped into the fuller proof.
It's the _whole_ proof, seen and unseen,
which makes the conclusion invincibly complete.

See,
just saying so doesn't make it so,
something that _goes_ has a _place_ to go.

Yes,
just saying so doesn't _make_ it so.

Even
saying so within
a finite sequence of only not.first.false claims
doesn't _make_ it so.

However,
that allows us to _know_ that it's so.

Proof.by.induction is a telescope, not a ray gun.

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