[FOM] Replacement

Timothy Y. Chow tchow at alum.mit.edu
Fri Aug 17 13:46:30 EDT 2007

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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