T.Forster at dpmms.cam.ac.uk
Wed Aug 15 19:02:26 EDT 2007
On Wed, 15 Aug 2007, Robert Black wrote:
> 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.
The question is not `what does Thomas Forster mean?' I don't mean
anything (in the relevant sense!) I'm trying to think myself into the
heads of people who repudiate the axiom scheme of replacement. I wonder
what it is that they believe, what they think its status is.... Maybe
they do believe your (2)....
> 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.
That would - as you say - be an interesting position. Has anyone
ever defended it?
It would be nice to hear from some replacement-deniers, but i suspect we
have none on this list!
v best wishes
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