### Single Idea 10128

#### [catalogued under 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic]

Full Idea

The idea behind the proofs of the Incompleteness Theorems is to use the language of Peano Arithmetic to talk about the formal system of Peano Arithmetic itself.

Gist of Idea

The Incompleteness proofs use arithmetic to talk about formal arithmetic

Source

A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.7)

Book Reference

George,A/Velleman D.J.: 'Philosophies of Mathematics' [Blackwell 2002], p.182

A Reaction

The mechanism used is to assign a Gödel Number to every possible formula, so that all reasonings become instances of arithmetic.