Full Idea
First Incompleteness Theorem: If S is a sufficiently powerful formal system, then if S is consistent then S is syntactically incomplete.
Gist of Idea
First Incompleteness: a decent consistent system is syntactically incomplete
Source
report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.6
Book Reference
George,A/Velleman D.J.: 'Philosophies of Mathematics' [Blackwell 2002], p.161
A Reaction
Gödel found a single sentence, effectively saying 'I am unprovable in S', which is neither provable nor refutable in S.