Re: what follows from denying an axiom

george says...

In what sense COULD something "be false" IF it is
TRUE IN ALL MODELS of ZFC?!?

Okay, let me back up. If I say that a sentence of set theory
is *true* without saying what model I'm talking about, then
I mean that the statement is justified by the closest thing
we have to an explicit standard "interpretation" of set theory,
which is the iterative hierarchy. All the axioms of ZFC are
true, according to this interpretation, so it's very hard for
me to imagine what it would mean for the completeness theorem
to be false.

But suppose that somebody introduced some other concept of
the universe of sets, maybe inspired by Quine's New Foundations,
or some other alternative set theory. Then if this new conception
of sets became standard, then I could certainly make sense of the
claim that "The completeness theorem is false, but con(ZFC) is true".
That implication may in fact be a theorem of this alternative set
theory.

--
Daryl McCullough
Ithaca, NY

.

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