Generalizing Tarski’s 1933 Formal Correctness formula to every formal system:

∀X True(X) ↔ φ(X)

becomes

∀L∀X True(L,X) ↔ φ(L,X)

Material Adequacy

This means that the objects satisfying φ should be exactly the objects that we would intuitively count as being true sentences of L, and that this fact should be provable from the axioms of the metalanguage.

Completing the RHS of this formula such that Material Adequacy is also satisfied:

φ(L,X)

becomes

∃Γ ⊆ Axioms(L) ∃Ψ ⊆ WFF(L) (Sequence(Γ,Ψ) ⊢ X)

Sequence(Γ,Ψ) indicates a sequence of Axioms followed by a sequence of WFF

**[TRUE]** ∀L∀X True(L, X) ↔ ∃Γ ⊆ Axioms(L) ∃Ψ ⊆ WFF(L) (Sequence(Γ,Ψ) ⊢ X)

**[FALSE]** ∀L∀X False(L, X) ↔ ∃Γ ⊆ Axioms(L) ∃Ψ ⊆ WFF(L) (Sequence(Γ,Ψ) ⊢ ~X)

**[~TRUE]** ∀L∀X ~True(L, X) ↔ ~∃Γ ⊆ Axioms(L) ∃Ψ ⊆ WFF(L) (Sequence(Γ,Ψ) ⊢ X)

The {assign alias} operator: “≡” is added to predicate logic to allow an expression to refer directly to itself.

“This sentence is not True.”

LP ≡ ∀L ∈ Formal_Systems ~True(L, LP) // ~True is shown above

Expanded by the definition of ~True(L, X) becomes this:

LP ≡ ∀L ∈ Formal_Systems ~∃Γ ⊆ Axioms(L) ∃Ψ ⊆ WFF(L) ( Sequence(Γ,Ψ) ⊢ LP)

For all element of set L of Formal_Systems there does not exist a sequence of Axioms Γ of language L at the beginning of a sequence of WFF Ψ of language L that proves this sentence.

Since neither the above expression of language nor is negation is satisfied in any formal system the above expression is semantically incorrect.

According to the author of the 1931 Incompleteness Theorem the above conclusion of semantic error applies equally to his proof:

ON FORMALLY UNDECIDABLE PROPOSITIONS OF PRINCIPIA MATHEMATICA AND RELATED SYSTEMS I by Kurt Gödel Vienna

The analogy between this result and Richard’s antinomy leaps to the eye; there is also a close relationship with the “liar” antinomy,14

14 Every epistemological antinomy can likewise be used for a similar undecidability proof.

Copyright 2016, 2017, 2018 by Pete Olcott