# undefinability of truth

## Summary

Tarski's Theorem states that no first-order logical theory has a predicate defining the Godel numbers of statements which are true of the natural numbers. In other words, a first-order theory which includes arithmetic can't define its own truth predicate. This is a fundamental limitation on mathematics, because it implies no formal system is powerful enough to define its own semantics.

## Context

This concept has the prerequisites:

- semantics of first-order logic (Tarski's Theorem is about the semantics of first-order logic.)
- Godel numbering (Tarski's Theorem is stated in terms of Godel numbering.)
- representability in arithmetic (The proof of Tarski's Theorem involves representability in arithmetic.)

## Goals

- Know the statement of Tarski's Theorem and why this implies a formal language can't define its own truth predicate

- Prove Tarski's Theorem

## Core resources (read/watch one of the following)

## -Paid-

→ A Mathematical Introduction to Logic

An undergraduate textbook in mathematical logic, with proofs.

Location:
Start of Section 3.5, "Incompleteness and undecidability," pages 234-236

→ A Course in Mathematical Logic

A graduate textbook in mathematical logic.

Location:
Section 7.4, "Tarski's Theorem," pages 327-331

## See also

-No Additional Notes-