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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 Ref
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.