Robert M. Solovay
solovay at Math.Berkeley.EDU
Fri Aug 17 16:27:29 EDT 2007
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
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
With the same disclaimer, I don't read him as questioning the notion
of "set-theoretical truth". He specifically defends number theoretic
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".
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.
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