[FOM] paper announcements

[Timothy Y. Chow]

Nik Weaver wrote:
>2.  "Is set theory indispensable?"  A thorough explanation of
>why I feel set theory is not an appropriate foundation for
>mathematics.  Written for a general audience.

An interesting and provocative paper.  A number of the issues raised have 
been debated on FOM before, and I will not revisit them here.  Section 7, 
however, makes a startling claim which I have not seen anyone defend 
before: that ZFC is probably consistent but probably has no models with 
standard integers.  A cogent argument for this claim would be very 
interesting.  Unfortunately, I cannot follow the reasoning.  I quote:

  Now we may believe that ZFC is probably consistent because (1) no 
  inconsistency has been found yet and (2) we have built up some sort of 
  intuition for ZFC which tells us that it is consistent.  I personally 
  find these arguments persuasive but not compelling. They suggest that 
  ZFC probably does have a model.  However, they tell us nothing about 
  whether it has a model with a standard omega.  This seems to me more 
  likely to be false than true.  Given the recursive compexity of ZFC (as 
  measured by its proof-theoretic ordinal, and already suggested by the 
  circularity of the power set of omega mentioned in Section 1) we should 
  not expect that there is such a model absent some special reason to do 
  so.  The presumption should be that ZFC has no such model and hence 
  that there are probably some false statements of first order arithmetic 
  that are provable in ZFC.

  Antiplatonistic belief in the arithmetical validity of ZFC seems to be 
  mainly a matter of faith.  One could argue that the hierarchy of large 
  cardinal axioms exhibits a compelling structure which is evidence for 
  the truth of the arithmetical consequences of these axioms.  Maybe so, 
  but this is at best very indirect evidence and hardly seems very 
  convincing.  At present I think a rational assessment of the evidence 
  would have to conclude that ZFC very likely proves false 
  number-theoretic assertions.

The only argument here that does not reduce to "it's just my personal gut 
feeling" is the one about the recursive complexity of ZFC.  But I can't 
make sense of this argument.  What is it about the proof-theoretic ordinal 
of ZFC that should lead us to expect that ZFC *very likely* proves false 
number-theoretic assertions?

Also, even supposing that there is something about the proof-theoretic 
ordinal of ZFC that should lead us *not to expect that a standard model 
exists*, how do you get from the *absence of expectation* to the 
*expectation of absence*?  That is, why shouldn't we simply be agnostic 
about the existence of a model with standard naturals?  What *positive*
evidence is there that ZFC proves false theorems (as opposed to the mere 
lack of evidence that it doesn't)?

For your argument to have any cogency, I think it would need to be backed 
up by some theorems of the general form, "Let X be a consistent system 
with proof-theoretic ordinal alpha.  If alpha is frabjous, then X proves 
false theorems."  There should also be some theorems that give evidence 
that the proof-theoretic ordinal of ZFC is frabjous.  Do such theorems 
exist?

Tim