[FOM] Replacement

Robert M. Solovay solovay at Math.Berkeley.EDU
Fri Aug 17 06:08:23 EDT 2007

 	Tim Chow seems to think non believers in replacement are as rare 
as hen's teeth. Here are some I've come across.

Vaughn Pratt (his home page is http://shurl.net/565) conjectured in 1992 
that ZFC was inconsistent. Cf. the second posting of http://shurl.net/566.
Here is a quote from that posting:

The proof of mathematics is in the pudding.  As long as people find
existing mathematics useful, and keep discovering new uses for it, no
one's going to complain until someone pins two plane crashes in a row
on the inconsistency of ZFC.  I'm happy to bet ZFC *will* be found
inconsistent by 2012, but only at odds of 100:1.  The experts in that
business (which I am not) for the most part believe ZFC to be

Jack Silver once said to me in conversation that he thought that 3rd order 
arithmetic was inconsistent.

I haven't checked whether or not these are the current views of Silver or 

In Boolos's book of papers "Logic, Logic, and Logic" there is a paper 
"Must we believe in set-theory". As I read it, it casts doubt on full 

Randall Holmes has expressed support for the view that Sigma_2 replacement 
is more reasonable than full replacement. [I don't have a cite for this. 
Perhaps thisview was only expressed in correspondence.] Randall reads this 
list and perhaps can shed more light on his views.

My view on the meaningfulness of arbitrarily complex sentences of the 
language of set theory waxes and wanes. But I have not the slightest doubt 
(a) that Sigma_2 assertions are meaningful and (b) that 
inaccessible cardinals exist. As a Corollary, I believe in the 
consistency of ZFC [and indeed of much stronger systems].

      --Bob Solovay

On Wed, 15 Aug 2007, Timothy Y. Chow wrote:

> Tom Forster wrote:
>> I hope the listowner and list memeber will forgive me repeating my
>> request, since it has not been answered.
> Implicitly, it has probably been answered.  The fact that people ignored
> your actual question probably means that the answer is: No, nobody thinks
> that Replacement is false.
> Unless someone can give an interesting theorem of
> ZFC - Replacement + ~Replacement,
> that is not known to be provable in ZFC - Replacement?
> Tim
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