Re: Tarski / Gödel and redefining the Foundation of Logic

On 7/22/2024 3:14 AM, Mikko wrote:

On 2024-07-21 13:20:04 +0000, olcott said:
 

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.

 I don't, and you cant show any contradiction.
 

Gödel's proof had nothing what-so-ever to do with provability
except that he proved that g is unprovable in PA.

A proof is about sentences, not about
numbers.
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The Liar Paradox: "This sentence is not true"

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cannot be said in the language of Peano arithmetic.

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Since Tarski anchored his whole undefinability theorem in a self-contradictory sentence he only really showed that sentences that
are neither true nor false cannot be proven true.

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By Gödel's completeness theorem every consistent incomplete first order
theory has a model where at least one unprovable sentence is true.
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https://liarparadox.org/Tarski_247_248.pdf // Tarski Liar Paradox basis
https://liarparadox.org/Tarski_275_276.pdf // Tarski proof

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It is very simple to redefine the foundation of logic to eliminate
incompleteness.

 Yes, as long as you don't care whether the resulting system is useful.
Classical logic has passed practical tests for thousands of years, so
it is hard to find a sysem with better empirical support.
 

When we show how incompleteness is eliminated then this also shows
how undefinability is eliminated and this would have resulted in a
chatbot that eviscerated Fascist lies about election fraud long
before they could have taken hold in the minds of 45% of the electorate.
Because people have been arguing against my correct system of reasoning
we will probably see the rise of the fourth Reich.

Any expression x of language L that cannot be shown
to be true by some (possibly infinite) sequence of truth preserving operations in L is simply untrue in L: True(L, x).

 That does not help much if you cannot determine whether a particular
string can be shown to be true.
 

Every element of the set of human knowledge can be proven true
by a finite sequence of truth preserving operations. Also every
line can be proved to be false by this same basis.
The Heritage Foundation is the author of Project
2025 and a staunch Trump ally could only find 1546
cases of voter fraud in the last ten years.
#ElectionFraudLies
Even the Heritage Foundation agrees
Never any evidence of election fraud
that could possibly change the results:
Only 1,546 total cases of voter fraud
https://www.heritage.org/voterfraud
Trump is just copying Hitler's "big lie"

Tarski showed that True(Tarski_Theory, Liar_Paradox) cannot be defined
never understanding that Liar_Paradox is not a truth bearer.

 However, every arithmetic sentence is either true or false.
 

The same diagonalization proof that Gödel used works on
the arithmetization of the Tarski proof. Diagonalization
never shows why g is unprovable in PA, it only shows that
g is unprovable in PA.
The Tarski proof shows why x is unprovable in the Tarski Theory
(because x is self-contradictory in the Tarski Theory)
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://www.researchgate.net/publication/315367846_Minimal_Type_Theory_MTT
In my own Minimal Type Theory the self contradiction
is much easier to see: LP := ~True(LP)
--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer