FOM: a question re. completeness

Stephen G Simpson simpson at
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"
 > proposition. 

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
important notion.

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.

-- Steve

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