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

On 2/8/25 9:45 PM, olcott wrote:

On 2/8/2025 4:28 PM, Richard Damon wrote:

On 2/8/25 10:32 AM, olcott wrote:

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.

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But no one has been claiming that, so you are just fighting strawmen.
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The problem is these links can be infinite, and proofs must be finite.
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Math is only incomplete when it is intentionally defined
in such a way to make it incomplete.
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*See if you can understand this*
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On 2/8/2025 9:51 AM, Ross Finlayson wrote:
 > Instead of just usual model theory and axiomatics
 > and "what's true in the logical theory", "what's
 > not falsified in the scientific theory", you can
 > have a theory where the quantity is truth, and
 > then there's a Comenius language of it that only
 > truisms are well-formed formulas, then the Liar
 > "paradox" is only a prototype of a fallacy...
>

 In other words, you admit that you don't under how the logic you are trying to talk about works, so you just lie and make up stuff that you think sounds good.