Full Idea
Second Incompleteness Theorem: If S is a sufficiently powerful formal system, then if S is consistent then S cannot prove its own consistency
Gist of Idea
Second Incompleteness: a decent consistent system can't prove its own consistency
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.165
A Reaction
This seems much less surprising than the First Theorem (though it derives from it). It was always kind of obvious that you couldn't use reason to prove that reason works (see, for example, the Cartesian Circle).