On 2024-05-31 05:01:13 +0000, olcott said:
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
That quote does not really say anything as it is only a fragment of
a sentence. However, comparison to the source shows that the only
missing piece is the point at the end of the sentence so the error
is easily corrected.
Note that the mood of that sentence is counterfactual, so nothing
is claimed.
*Formalized as*
x ∉ True if and only if p
That line is Tarski's (2) on page 275, though not as formalization
of anything but as inferred from what was already presented.
where the symbol 'p' represents the whole sentence x https://liarparadox.org/Tarski_275_276.pdf
*adapted to become this*
x ∉ Pr if and only if p // line 1 of the proof
Here is the Tarski Undefinability Theorem proof
(1) x ∉ Provable if and only if p // assumption (see above)
Not just assumption but a consequence of earlier parts of the proof.
Your proof is not the same as Tarski's if you omit those earlier
parts.
(2) x ∈ True if and only if p // Tarski's convention T
Also follows from earlier parts of the proof.
(3) x ∉ Provable if and only if x ∈ True. // (1) and (2) combined
(4) either x ∉ True or x̄ ∉ True; // axiom: ~True(x) ∨ ~True(~x)
(5) if x ∈ Provable, then x ∈ True; // axiom: Provable(x) → True(x)
(6) if x̄ ∈ Provable, then x̄ ∈ True; // axiom: Provable(~x) → True(~x)
(7) x ∈ True
Follows from (3) and (5). Easier to see if you write 5 as
(5') if x ∉ True then x ∉ Provable
if x ∉ True then x ∉ Provable
(8) x ∉ Provable
Also obfious from the reweitten (5).
(9) x̄ ∉ Provable
A x is true, x̄ is false and therefore unporvable.
The expression forming line (1) of the proof is directly derived from
the liar paradox as shown above.
No, it is inferred from earlies parts of the proof.
Therefore, all shown parts of the proof are correct and no
mistake of the Tarski Proof is identified.
--
Mikko
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