Wait. Not(provable(consistency)) is provable in PA? Then run that through the above.
Not(provable(consistency)) is provable in PA?
I'm not sure that this is true. I can't find anything that says either way, but there's a section on Godel's second incompleteness theorem in the book "Set theory and the continuum hypothesis" by Paul Cohen that implies that the theorem is not provable in the theory that it applies to.
This thread is for asking any questions that might seem obvious, tangential, silly or what-have-you. Don't be shy, everyone has holes in their knowledge, though the fewer and the smaller we can make them, the better.
Please be respectful of other people's admitting ignorance and don't mock them for it, as they're doing a noble thing.
To any future monthly posters of SQ threads, please remember to add the "stupid_questions" tag.