The mistake of the Tarski Proof

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
*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)
(2) x ∈ True if and only if p // Tarski's convention T
(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
(8) x ∉ Provable
(9) x̄ ∉ Provable
The expression forming line (1) of the proof is directly derived from
the liar paradox as shown above.
When Line (2) is combined with line (1) then line (3) derived. Line (3)
partially derived from the Liar Paradox directly contradicts the axiom
on line (5).
Since Line (3) partially derived from the Liar Paradox directly
contradicts the axiom at line (5) this proves the Line(3) is based on a
false assumption. That false assumption was line (1).
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Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer