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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 selecte

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.

tildedave 6y ago
The continuum hypothesis is more like Euclid's parallel postulate than a Gödel sentence - assuming ZFC consistent there are models with CH true and CH false (the cardinality of the continuum doesn't have to be the first uncountable cardinal). Everything gets qualified with "assuming ZFC consistent" or "assuming Peano consistent" because any inconsistent theory proves any statement. More of a proof technicality than anything too profound. There is a construction of a model of Peano arithmetic, so it is consistent, as long as you accept the system used in the proof: https://en.wikipedia.org/wiki/Gentzen%27s_consistency_proof https://en.wikipedia.org/wiki/Gentzen%27s_consistency_proof Not sure if this sheds light on the parent commentator's question ... the terminology can be quite tricky.