The key aspect of my proof is that I provide axiom of Truth (3) that correctly decides that some expressions of language such as the formalized Liar Paradox are either ill-formed or false. We evaluate these as not true.

Truth Predicate Axioms

(1) ∀F ∈ Formal_Systems ∀x ∈ WFF(F) (True(F, x) ↔ (F ⊢ x))

(2) ∀F ∈ Formal_Systems ∀x ∈ WFF(F) (False(F, x) ↔ (F ⊢ ~x))

(3) ∀F ∈ Formal_Systems ∀x ∈ WFF(F) (~True(F, x) ↔ ~(F ⊢ x))

Formalizing the Liar Paradox in this way:

True(F, G) ↔ ~(F ⊢ G)

it becomes equivalent to Tarski’s third equation:

3) x ∉ Pr ↔ x ∈ Tr

By Truth axiom (3) we substitute ~True(F, G) for ~(F ⊢ G)

deriving True(F, G) ↔ ~True(F, G) ∴ the Liar_Paradox is false in F.

This causes the Tarski Proof to fail at his third equation.

**The above can only be properly understood within the context**

of the following four pages of the Tarski Paper:

of the following four pages of the Tarski Paper:

http://liarparadox.org/247_248.pdf

http://liarparadox.org/Tarski_Proof_275_276.pdf

**Tarski Undefinability Theorem Succinctly Refuted**

https://www.researchgate.net/publicatio ... ly_Refuted