FOM: Re: reply to Kanovei
kanovei at wminf2.math.uni-wuppertal.de
Wed Mar 3 04:05:40 EST 1999
> Date: Tue, 2 Mar 1999 18:15:58 -0500 (EST)
> X-Sender: cxm7 at pop.cwru.edu
> I believe Con ZFC because I understand
> the axioms (and various extensions of them) fairly well and find them
> persuasive, and people far more expert than I understand them better and
> find them persuasive, and ZFC has worked well under considerable
> investigation over the past 70 years or so.
Some 3 - 5 years ago the following copy of the
credo statement above would be not less convincing:
~~ I believe the great Fermat theorem is false
~~ because I understand arithmetic quite well
~~ and all attempts of people far more expert than
~~ me to prove it over the past 350 or so years failed.
Well, let alone anecdotes.
Yes, an immence number of various applications of ZFC
has not yet produced a contradiction. This makes one to
believe that ZFC will never produce a contradiction.
Note: such a belief, if of a scientist, always contains
a doubt, and in this case, except for trivial doubts,
there is a possibility that ZFC is only feasibly consistent
(that is, the shortest description of a contradiction
would contain near 9^9^9^9^9^9^9 symbols).
But in any case this is your personal matter.
Another case is if you prove a theorem
that some proposition P is true
in such a way that Con ZFC is one of the
assumptions which you take for granted.
If you pretend this to be a mathematical proof
of P, they will not accept this because the
correct form of your result is the following:
"if Con ZFC then P".
> Poincare was quite right. No one actually bases their mathematical
> knowledge on provability in any formal system.
What should this mean ?
The mathematical knowledge is based on provability
in ZFC (or some weaker subsistems like PA) in the
sense that any known mathematical theorem is a
theorem of ZFC. It is different story that many
mathematicians may not know details of ZFC or even
hate ZFC (as perhaps Poincare did). Their success in
mathematics simply shows that ZFC goes beyond human
passions and reflects the hard core of mathematical
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