[FOM] Replacement

Robert Black 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?''

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