[FOM] Re: Alleged proof of inconsistency of ZF
Bryan Ford
baford at mit.edu
Thu Apr 29 11:02:29 EDT 2004
Tim,
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,
Bryan
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