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Feb 7, 2015 at 17:15 comment added Pace Nielsen This is an excellent answer, with a lot of questions that I hope some other people here might be able to address.
Feb 7, 2015 at 12:46 comment added Joel David Hamkins Yes, except that since it has complexity $\Pi^1_1$, the natural context for independence results might be second-order number theory. The methods of reverse mathematics could conceivably have something to say about it.
Feb 7, 2015 at 9:57 comment added arsmath The Koethe conjecture is a conjecture in ordinary mathematics, so the relevant set of axioms is ZFC.
Feb 7, 2015 at 4:40 history answered Joel David Hamkins CC BY-SA 3.0