Full Idea
An approximation of Gödel's Theorem imagines a statement 'This system of mathematics can't prove this statement true'. If the system proves the statement, then it can't prove it. If the statement can't prove the statement, clearly it still can't prove it.
Gist of Idea
'This system can't prove this statement' makes it unprovable either way
Source
report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Brian Clegg - Infinity: Quest to Think the Unthinkable Ch.15
Book Reference
Clegg,Brian: 'Infinity' [Robinson 2003], p.202
A Reaction
Gödel's contribution to this simple idea seems to be a demonstration that formal arithmetic is capable of expressing such a statement.