This is my layman's understanding of Godel's incompleteness theorems: Any set of axioms powerful enough to decide all its theorems will always have inconsistencies in its theorems (e.g. "if true then false"). To be consistent, the set of axioms must be incomplete (some of its theorems are undecidable).

The following sentences have some of the 'flavor' of Godel's theorems:

"This sentence is false." →if true, then false.

"These are not words." →if true, then irreconcilable with its own truth.