Re: Mathematical incompleteness has always been a misconception --- Tarski

On 2/9/2025 4:33 AM, Mikko wrote:

On 2025-02-08 15:32:00 +0000, olcott said:
 

On 2/8/2025 4:45 AM, Mikko wrote:

On 2025-02-07 16:21:01 +0000, olcott said:
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On 2/7/2025 4:34 AM, Mikko wrote:

On 2025-02-06 14:46:55 +0000, olcott said:
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On 2/6/2025 2:02 AM, Mikko wrote:

On 2025-02-05 16:03:21 +0000, olcott said:
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On 2/5/2025 1:44 AM, Mikko wrote:

On 2025-02-04 16:11:08 +0000, olcott said:
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On 2/4/2025 3:22 AM, Mikko wrote:

On 2025-02-03 16:54:08 +0000, olcott said:
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On 2/3/2025 9:07 AM, Mikko wrote:

On 2025-02-03 03:30:46 +0000, olcott said:
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On 2/2/2025 3:27 AM, Mikko wrote:

On 2025-02-01 14:09:54 +0000, olcott said:
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On 2/1/2025 3:19 AM, Mikko wrote:

On 2025-01-31 13:57:02 +0000, olcott said:
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On 1/31/2025 3:24 AM, Mikko wrote:

On 2025-01-30 23:10:18 +0000, olcott said:
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Within the entire body of analytical truth any expression of language that has no sequence of formalized semantic deductive inference steps from the formalized semantic foundational truths of this system are simply untrue in this system. (Isomorphic to provable from axioms).

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If there is a misconception then you have misconceived something. It is well
known that it is possible to construct a formal theory where some formulas
are neither provble nor disprovable.

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This is well known.

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And well undeerstood. The claim on the subject line is false.

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a fact or piece of information that shows that something
exists or is true:
https://dictionary.cambridge.org/us/dictionary/english/ proof

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We require that terms of art are used with their term-of- art meaning and

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The fundamental base meaning of Truth[0] itself remains the same
no matter what idiomatic meanings say.

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Irrelevant as the subject line does not mention truth.
Therefore, no need to revise my initial comment.

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The notion of truth is entailed by the subject line:
misconception means ~True.

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The title line means that something is misunderstood but that something
is not the meaning of "true".

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It is untrue because it is misunderstood.

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Mathematical incompleteness is not a claim so it cannot be untrue.

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That mathematical incompleteness coherently exists <is> claim.

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Yes, but you didn't claim that.
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The closest that it can possibly be interpreted as true would
be that because key elements of proof[0] have been specified
as not existing in proof[math] math is intentionally made less
than complete.

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Math is not intentionally incomplete.

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You paraphrased what I said incorrectly.

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No, I did not paraphrase anything.
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Proof[math] was defined to have less capability than Proof[0].

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That is not a part of the definition but it is a consequence of the
definition. Much of the lost capability is about things that are
outside of the scope of mathemiatics and formal theories.
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When one thinks of math as only pertaining to numbers then math
is inherently very limited.

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That's right. That limited area should be called "number theory",
not "mathematics".
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When one applies something like
Montague Grammar to formalize every detail of natural language
semantics then math takes on much more scope.

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It is not possible to specify every detail of a natural language.
In order to do so one should know every detail of a natural language.
While one is finding out the language changes so that the already
aquired knowledge is invalid.
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When we see this then we see "incompleteness" is a mere artificial
contrivance.

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Hallucinations are possible but only proofs count in mathematics.
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True(x) always means that a connection to a semantic
truthmaker exists. When math does this differently it is simply
breaking the rules.

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Mathematics does not make anything about "True(x)". Some branches care
about semantic connections, some don't. Much of logic is about comparing
semantic connections to syntactic ones.
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Many theories are incomplete,
intertionally or otherwise, but they don't restrict the rest of math.
But there are areas of matheimatics that are not yet studied.
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When-so-ever any expression of formal or natural language X lacks
a connection to its truthmaker X remains untrue.

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An expresion can be true in one interpretation and false in another.

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I am integrating the semantics into the evaluation as its full context.

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Then you cannot have all the advantages of formal logic. In particular,
you need to be able to apply and verify formally invalid inferences.

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All of the rules of correct reasoning (correcting the errors of
formal logic) are merely semantic connections between finite strings:

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There are no semantic connections between uninterpreted strings.
With different interpretations different connections can be found.
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When we do not break the evaluation of an expression of language
into its syntax and semantics such that these are evaluated
separately and use something like Montague Semantics to formalize
the semantics as relations between finite strings then
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it is clear that any expression of language that lacks a connection
through a truthmaker to the semantics that makes it true simply remains
untrue.

 Of course, completness can be achieved if language is sufficiently
restricted so that sufficiently many arithemtic truths become inexpressible.
 

This does not make any sense to me. It is not that truth remains inexpressible. We simply make the system expressible enough that
all of those truths made true through a truthmaker connection to
their formalized semantic meaning can reach this semantic meaning.

It is far from clear that a theory of that kind can express all arithmetic
truths that Peano arithmetic can and avoid its incompletness.
 

LP := ~True(LP) // AKA this sentence is not true
is rejected as a not a truth-bearer.

In the big picture way that truth really works there cannot
possibly be true[0](x) that is not provable[0](x) where x
is made true by finite strings expressing its semantic meanings.
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When one finite string expression of language is known to be true
other expressions are know to be semantically entailed.

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Only if they are connected with (semantic or other) connections that
are known to preserve truth.
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Yes, there must be truth preserving operations.

 More inportant is that there are no other operations.
 

Formal logic fails at this some of the time.
https://en.wikipedia.org/wiki/Principle_of_explosion

 That is not a failure.
 

I can never understand how anyone can be so gullible to
believe that anything besides FALSE logically follows
from a contradiction.
To me this seems the same as"dead rats" being stipulated
as a kind of "live chicken" and all the biologists believe
it because they mindlessly accept whatever they were told.

The only thing that is actually semantically entailed by a
contradiction is FALSE. (A & ~A) ⊨ FALSE

 Yes. And if false is true then everything is true because in ordinary
logic (A ∨ ~A) is a tautology.
 

I fail to understand how anyone could be gullible enough into
being conned into believing that anything besides FALSE is
entailed by a contradiction.

 Indeed. Everybody should understand that a contradiction entails FALSE, too.
 

The received view of POE seems to prove that most people do not understand this.

We simply leave most of formal logic as it is with some changes:
(1) Non-truth preserving operations are eliminated.

 There are none anyway.
 

(A & ~A) ⊨ FALSE thus POE is incorrect.

    A deductive argument is said to be valid if and only if
    it takes a form that makes it impossible for the premises
    to be true and the conclusion nevertheless to be false.
    https://iep.utm.edu/val-snd/

 Yes. That is a feature of any formal logic system when interpreted so that
logical operations satisfy their defining axioms.
 However, that requirement involves semantics so it is not applicable to
a purely formal system.
 

*We correct the above fundamental mistake*
    A deductive argument is said to be valid if and only if
    it takes a form that the conclusion is a necessary
    consequence of its premises.

 Not possible unless you define "necessafy consequence".
 

Modal Logic already defines this.

(2) Semantics is fully integrated into every expression of
language with each unique natural language sense meaning
of a word having its own GUID.

 One kind of semantics. Different interpretations are still possible.
 

The subjective leeway of interpretation utterly ceases to exist
when every GUID semantic meaning is exhaustively defined.
All expressions of lanhguage are comprised entirely of GUIDs.

True[math] can only exist apart from Provable[math] within
the narrow minded, idiomatic use of these terms. This is
NOT the way that True[0] and Provable[0] actually work.

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If you want that to be true you need to define True[math] differently
from the way "truth" is used by mathimaticians.

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We could equally define a "dead cat" to be a kind of {cow}.
Math does not get to change the way that truth really works,
when math tries to do this math is incorrect.

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Math does not care how truth works outside mathematics. But the truth
about mathematics works the way truth usually does.

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Math is not allowed to break these rules without making math incorrect.

 It is. You have no authority to prohibit anything.
 

When math tries to override how truth really works then
math is necessarily incorrect.

My point is much more clear when we see that Tarski attempts
to show that True[0] is undefinable.
https://liarparadox.org/Tarski_247_248.pdf
https://liarparadox.org/Tarski_275_276.pdf

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Tarski did not attempt to show that True[0] is undefinable. He showed
quite successfully that arthmetic truth is undefinable. Whether that
proof applies to your True[0] is not yet determined.

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Tarski is the foremost author of the whole notion of every
kind of truth. "snow is white" because within the actual state
of affairs the color of snow is perceived by humans to be white.

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Many philosophers before and after Tarski have tried to find out what
truth really is and how it works.

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And I am finishing the job.

 Unlikely. Philosohers' job is never finished.
 

Within the limited domain of {expressions of language that
are true on the basis of their meaning} I am finishing the job.

I may have only one month left.
The cancer treatment that I will have next month has a 5% chance
of killing me and a 1% chance of ruining my brain. It also has
about a 70% chance of giving me at least two more years of life.

 Ars longa, vita brevis.
 

--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer