**How the “x” variable referenced in Tarski’s proof is defined: **

Tarski Undefinability Proof (pages 247-248)

**Tarski’s Undefinability Theorem proof: **

Tarski Undefinability Proof (pages 275-276)

Direct quotes from page 275-276 **// indicates paraphrase**

the symbol ‘Pr’ which denotes the class of all provable sentences of the theory under consideration

…

We can construct a sentence x … which satisfies the following condition:

It is not true that x ∈ Pr if and only if p

…

(1) x ∉ Pr ↔ p **// p ↔ ~Provable(x)**

Where the symbol ‘p’ represents the whole sentence x.

…

We shall now show that the sentence x is actually undecidable and at the same time true.

…

If we denote the class of all true sentences by the symbol ‘Tr’ then – in accordance with convention T – the sentence x which we have constructed will satisfy the following condition:

(2) x ∈ Tr ↔ p **// p ↔ True(x)**

From (1) and (2) we obtain immediately:

(3) x ∉ Pr ↔ x ∈ Tr **// ~Provable(x) ↔ True(x)**

We can derive the following theorems from the definition of truth

(4) either x ∉ Tr or ~x ∉ Tr **// ~True(x) ∨ ~True(~x)**

(5) if x ∈ Pr, then x ∈ Tr **// Provable(x) → True(x)**

(6) if ~x ∈ Pr, then ~x ∈ Tr **// Provable(~x) → True(~x)**

From (3) and (5) we infer without difficulty that

(7) x ∈ Tr **// True(x)**

and that

(8) x ∉ Pr **// ~Provable(x)**

In view of (4) and (7) we have

(8a) ~x ∉ Tr, **// ~True(~x)**

which together with (6) gives us the formula:

(9) ~x ∉ Pr **// ~Provable(~x)**

The formulas (8) and (9) together express the fact that x is an undecidable sentence; moreover from (7) it follows that x is a true sentence.

Copyright 1936 Alfred Tarski

Paraphrase Simplifications Copyright 2018 Pete Olcott

The Concept of Truth in Formalized Languages, Alfred Tarski 1936