18 Apr
2015
18 Apr
'15
8:14 a.m.
What does "Ex(P(x))! P(x)" mean below? (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. . . .