On Sat, Apr 18, 2015 at 10:14 AM, Dan Asimov <asimov@msri.org> wrote:
What does "Ex(P(x))! P(x)" mean below?
Sorry, a line break would have made this clearer. There exists an x such that P(x). Exclamation point added for emphasis, not for factorial. Then the next sentence starts "P(x) is true for one of those 'infinite integers'... Andy
(Sure, Ex(P(x)) just means there exists x such that P(x), but what about the "! P(x)" part?)
Thanks,
Dan
On Apr 17, 2015, at 7:14 PM, Andy Latto <andy.latto@pobox.com> wrote:
. . .
Another way to look at why I'm "biased against" PH + ~PA is that while it's consistent, it isn't "omega-consistent". That is, there are predicates P such that all of
~P(0), ~P(S(0)), ~P(S(S(0))), ~P(S(S(S(S(0)))))...
are theorems, but so is Ex(P(x))! P(x) is true for one of those "infinite integers" that exist in the weird non-standard models of the integers, but not in the model Peano was thinking about when he invented the axioms. . . .
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