[FOM] Replacement

Thomas Forster 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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