[FOM] Replacement

[Robert M. Solovay]

Two brief comments.

 	1) It seems to me that beleiving that ZFC is inconsistent is 
stronger than believeing replacement is false. It amounts to holding that 
the idea of replacement [or that of some other axiom of ZFC] is 
incoherent].

 	2) Boolos argues against the existence of a kappa such that kappa 
= aleph_kappa. He waffles a bit and doesn't quite say this is false. But 
he certainly argues that we don't have good grounds to believe in the 
truth of its existence. I haven't reread his paper carefully. But I think 
the instance of replacement he questions would assert that if alpha is an 
ordinal, there is a set consisting of aleph_gamma for those gamma less 
than alpha.

      With the same disclaimer, I don't read him as questioning the notion 
of "set-theoretical truth". He specifically defends number theoretic 
"realism".

 	A personal confession: I find parsing the writings of philosphers 
quite tedious. I hope to abstain from further discussions as to whether 
anyone "believes replacement to be false".

 	--Bob Solovay



On Fri, 17 Aug 2007, Timothy Y. Chow wrote:

> On Fri, 17 Aug 2007, Robert M. Solovay wrote:
>> 	Tim Chow seems to think non believers in replacement are as rare
>> as hen's teeth. Here are some I've come across.
>
> It puzzles me that FOM is having so much difficulty making sense of the
> sentence, "Replacement is false."  Is it really so hard to understand?
> Surely it's equivalent to, "The negation of at least one axiom in the
> schema is true."  And surely this means:
>
> (*) ZFC minus Replacement (or something weaker), plus the negation of
>     of at least one axiom in the Replacement schema, is true.
>
> If X believes that ZFC is inconsistent, this does not suggest to me that X
> believes (*).  If X believes that Replacement is irrelevant to ordinary
> mathematics, this does not suggest to me that X believes (*).  Robert
> Black's suggestion also doesn't sound right to me:
>
>  Replacement could have been true, but God just didn't bother to
>  create that many sets. I doubt if this is what you mean.
>
> Although believers in Replacement might justify it based on some intuition
> about the size of V, there seems to be no reason to think that someone who
> believes that Replacement is false does so because of a size argument.
>
> Finally, the absence of belief in Replacement, or even the active
> withholding of belief in Replacement, is not the same as a belief that
> Replacement is false.
>
> I feel almost silly making all these statements because they are so
> trivial, but the succession of confused posts so far make me feel that
> someone needs to state the obvious.
>
> Unless Thomas Forster really meant to ask if anybody believes that
> Replacement is *inconsistent* with the other axioms of ZFC.  That's an
> entirely different question.
>
> Tim
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