Comment by thisiszilff

I don't think PM itself had any contradictions inside of it, rather there are statements composed of the PM langauge that aren't reached from the axioms selected by PM -- that is one of the sentences Godel demonstrates in his proof where he gets one that basically says "this statement has no proof in this system" (so if it does have a proof it is a contradiction, but if it doesn't have a proof then PM is an incomplete system). Someone else in the thread mentioned the continuum hypothesis as something that couldn't be proved from the standard ZFC axioms, so that would be an example of a statement without proof (incompleteness) albeit not in PM. I don't know of any contradictory statements.