Re: The key undecidable instance that I know about

On 3/9/25 4:08 PM, olcott wrote:

On 3/9/2025 2:24 PM, Richard Damon wrote:

On 3/9/25 1:15 PM, olcott wrote:

Is the Liar Paradox True or False?
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LP := ~True(LP)
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?- LP = not(true(LP)).
LP = not(true(LP)).
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?- unify_with_occurs_check(LP, not(true(LP))).
false.
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Its infinitely recursive structure makes it neither true nor false.
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>
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The liar's paradox isn't an "undecidable" instance, as "undecidable" is about a problem that has a true or false answer that can not be computed for every case.
>

 Tarski thought that is was undecidable and anchored his
whole proof in it.
 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

Note, he says to construct the antinomy of the liar in the METALANGUAGE representing the statement x in the LANGUAGE. Thus "x" is *NOT* the liar, but something that with the additional information of the metalanguage can be reduced to it.
You are just showing you don't understand how formal logic works, and how meta-systems/langauges relate to the base system/language.

 Formalized as:
x ∉ True if and only if p
where the symbol 'p' represents the whole sentence x

So? x is some sentence in the language, like say Godel's G that says there does not exist a number g such that g satisfies a particular primative recursive relationship.
Clearly such a statement isn't what you just tried to specify it as.

 He never shows the above intermediate step before
he arbitrarily swaps True for Provable on line (1)
of his proof in the first paragraph of this proof.

Because he refers to the work he did earlier that shows it to be a valid transformation in the system.

 (1) x ∉ Provable and only if p
    In accordance with the first part of Th. I
   we can obtain the negation of one of the
   sentences in condition (ex) of convention
   T of § 3 as a consequence of the definition of
   the symbol 'Pr' (provided we replace 'Tr' in
   this convention by 'Pr').
   https://liarparadox.org/Tarski_275_276.pdf
 

Right, so look at Theorem I
The problem is you don't understand the logic it uses, in part because you just don't understand how Formal Logic works, which is why you think your fraud of changing the definitions was ok to do.