Mongre at gmx.de
Wed Aug 15 18:36:56 EDT 2007
What are you thinking of when you think someone might consider
replacement *false*? I offer the following alternatives:
1) It's first-order inconsistent with the other axioms of ZF. That's
an arithmetical claim. I don't think anyone thinks it likely, but
obviously it can't be disproved in a non-question-begging way.
2) Replacement could have been true, but God just didn't bother to
create that many sets. I doubt if this is what you mean.
3) First-order replacement is consistent (with the other axioms of
ZF), but ZF with second-order replacement is unsatisfiable, in the
sense that it *couldn't* have been true. That would be interesting,
and it looks epistemically possible (after all, if there's an
inaccessible, then there are *full* models V_alpha of ZF with
first-order replacement with alpha less than the first inaccessible
which are thus not models of ZF with second-order replacement). But I
think only those of us who are card-carrying second-orderists (think
we) can even understand this possibility.
>I hope the listowner and list memeber will forgive me repeating my
>request, since it has not been answered.
> ``I know there are lots of people who dislike the axiom scheme of
> replacement. They say things like ``it has no consequence for
> ordinary mathematics'' and the like. Unfortunately i have none
> of them handy at the moment, so i have to ask: do any of them
> think that the axiom scheme is actually *false*? Or do they
> merely think that it shouldn't be a core axiom?''
More information about the FOM