Re: Tarski / Gödel and redefining the Foundation of Logic

On 7/25/2024 3:55 AM, Mikko wrote:

On 2024-07-23 14:53:21 +0000, olcott said:
 

On 7/23/2024 3:07 AM, Mikko wrote:

On 2024-07-22 14:40:41 +0000, olcott said:
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On 7/22/2024 3:14 AM, Mikko wrote:

On 2024-07-21 13:20:04 +0000, olcott said:
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On 7/21/2024 4:27 AM, Mikko wrote:

On 2024-07-20 13:22:31 +0000, olcott said:
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On 7/20/2024 3:42 AM, Mikko wrote:

On 2024-07-19 13:48:49 +0000, olcott said:
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Some undecidable expressions are only undecidable because
they are self contradictory. In other words they are undecidable
because there is something wrong with them.

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Being self-contradictory is a semantic property. Being uncdecidable is
independent of any semantics.

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Not it is not. When an expression is neither true nor false
that makes it neither provable nor refutable.

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There is no aithmetic sentence that is neither true or false. If the sentnece
contains both existentia and universal quantifiers it may be hard to find out
whether it is true or false but there is no sentence that is neither.
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 As Richard
Montague so aptly showed Semantics can be specified syntactically.
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An arithmetic sentence is always about
numbers, not about sentences.

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So when Gödel tried to show it could be about provability
he was wrong before he even started?

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Gödel did not try to show that an arithmetic sentence is about provability.
He constructed a sentence about numbers that is either true and provable
or false and unprovable in the theory that is an extension of Peano arithmetics.
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You just directly contradicted yourself.

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I don't, and you cant show any contradiction.
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Gödel's proof had nothing what-so-ever to do with provability
except that he proved that g is unprovable in PA.

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He also proved that its negation is unprovable in PA. He also proved
that every consistent extension of PA has a an sentence (different
from g) such that both it and its negation are unprovable.
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L is the language of a formal mathematical system.
x is an expression of that language.
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When we understand that True(L,x) means that there is a finite
sequence of truth preserving operations in L from the semantic
meaning of x to x in L, then mathematical incompleteness is abolished.

 No, it is not. From the meaning of "formal mathematical system" follows

I am overriding and superseding that. It is ridiculously
stupid to ignore the semantics of a formal expression
when evaluating its truth. Because truth *is* a semantic
property. Haskell Curry shows the correct way to evaluate
the semantic truth of an expression syntactically.
http://www.liarparadox.org/Haskell_Curry_45.pdf
Richard Montague's grammar of natural language semantics
does this same thing.

that whether x is an expression of language L does not depend on semantics
or L is not a language of a formal mathiematical system. In addition,
the system is incomplete if there is a sentence that can be determined
to be true from the meaning of x but cannot be proven in the system.
 

This sentence is not true: "This sentence is not true" is only
true because the indirect reference of the outer sentence escapes
the self-contradiction. Tarski made this same stupid mistake:
Tarski's Liar Paradox from page 248
    It would then be possible to reconstruct the antinomy of the liar
    in the metalanguage, by forming in the language itself a sentence
    x such that the sentence of the metalanguage which is correlated
    with x asserts that x is not a true sentence.
    https://liarparadox.org/Tarski_247_248.pdf
Formalized as:
x ∉ True if and only if p
where the symbol 'p' represents the whole sentence x
https://liarparadox.org/Tarski_275_276.pdf
In my own Minimal Type Theory the self-reference would not
be so clumsy--- x := ~True(x)
--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer