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Just to be clear. This implies that it's possible to write some specific statement (using PM axioms) that contradicts itself? Is there a readable example of thi

by ko27 6y ago

Just to be clear. This implies that it's possible to write some specific statement (using PM axioms) that contradicts itself? Is there a readable example of this statement without using Gödel numbers but instead just with the axiom statements?

thisiszilff 6y ago
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.
ProfHewitt 6y ago
Yes, PM would be made inconsistent by including the [Gödel 1931] proposition I'mUnprovable for reasons explained elsewhere in this discussion. Fortunately, the rules on orders of propositions make it impossible to construct proposition I'mUnprovable in PM.

by ko27 6y ago

thisiszilff 6y ago

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.

ProfHewitt 6y ago

Yes, PM would be made inconsistent by including the [Gödel 1931] proposition I'mUnprovable for reasons explained elsewhere in this discussion. Fortunately, the rules on orders of propositions make it impossible to construct proposition I'mUnprovable in PM.