FOM: a question re. completeness
Stephen G Simpson
simpson at math.psu.edu
Wed Feb 4 08:24:20 EST 1998
Michael Detlefsen writes:
> This is a remark on Steve's suggestion for how to understand my
> question. I think your question IS interesting Steve. I don't think
> it's my question,
Fair enough, my question is a different question. My question makes a
lot of sense to me, although it's not completely precise. I'm not
really sure I understand the thrust of your original question.
> I'm also a little befuddled at someone's finding my definition of
> consistency-completeness unclear (because of its use of a
> counterfactual conditional) but not finding one unclear that
> appeals to the notion of a "mathematically natural"
You are absolutely right: "mathematically natural" is not a
mathematically precise concept. That's why I used scare quotes. But
the informal notion "mathematically natural" is clearly important in
evaluating f.o.m. research, as for instance when we say that the
Paris-Harrington independent statement is more mathematically natural
than G or Con(PA). My question leverages off of this informal but
A correction: In formulating property C, I need to stipulate that S is
a mathematically natural proposition that is independent of T.
> One last question: Do you find it unclear to say 'G's being
> provable in PA would make PA inconsistent'?
Yes, because in fact G is not provable in PA (assuming PA is
consistent), so G's being provable in PA is a counterfactual, or
perhaps a "hypothetical", as US politicians would say.
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