FOM: Re: reply to Kanovei

[Kanovei]

> 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 
truth. 

V.Kanovei