[FOM] Re: Alleged proof of inconsistency of ZF

Bryan Ford baford at mit.edu
Thu Apr 29 11:02:29 EDT 2004


I appreciate your looking at the paper and giving it the limited endorsement 
that the author has "at least a basic grasp of the issues involved". :)  I 
wasn't planning to mention the draft in a wider form like this until more of 
my colleagues here at MIT had had a chance to look at it (a few have so far, 
but not thoroughly) - but I also understand that uploading a paper to the 
arXiv inherently constitutes an invitation to public scrutiny, so I'm not 
complaining.  Now that the cat is out of the bag, though, I would like to 
give a little context and necessary disclaimers. :)

In this draft I'm trying to explore some of the interesting things you can 
construct through the use of impredicative reasoning.  The line of argument, 
as it currently goes in the paper, does lead to the conclusion that ZF is 
inconsistent (because it proves itself consistent) - but what I'm more 
interested in is the essential characteristics of this particular 
impredicative construction itself.  It is entirely possible (most would 
probably say certain :)) that there is an essential flaw in the argument such 
that the final conclusion doesn't follow.  But it seems to me that there 
could still be something interesting or revealing about the impredicative 
construction developed in the argument, or whatever fragment of it remains 
standing after scrutiny.  I hope at any rate that interested readers will 
find the argument to be diligent and rigorous enough so that finding that 
essential flaw at least provides a little real mathematical sport. :)

As a heads-up, I should mention that Prof. Solovay from Berkely was already 
kind enough to read the paper and found a number of fixable errors, which I 
will correct in an updated version that will hopefully go to the arXiv today.  
All but one are inessential to the argument and have to do with my 
formalization of logic; the remaining one is that Theorem 4.12 relies on a 
restriction on the class of "universes" that I recently eliminated from the 
most recent drafts, mistakenly thinking it was no longer necessary.  
(Specifically, in the latest draft I say that _any_ set can be a "universe"; 
for 4.12 to go through it turns out necessary that universes be sets with 
transitive membership: if x is in U, then x is a subset of U.  But the whole 
proof was designed around this assumption in the first place, so adding that 
back in shouldn't affect the remainder of the argument, knock on wood.)  
Interested readers may wish to wait until tomorrow's arXiv update before 
reading the paper, or else keep the above in mind.

Thank you very much for your time and consideration,

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