FOM: December 1 - December 22, 1998

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Re: FOM: mathematical certainty



Stephen Cook wrote:
>
> Here is a reply to Andreas Blass's comments on my working definition
> of mathematical certainty as provability in an appropriate formal
> system such as ZFC.
>
...
>
> Of course if ZFC is found to be inconsistent, then this definition
> of mathematical certainty would have to be revised.

I agree with the omitted part (...); cf. the original message of 
Stephen Cook. But I would like to comment on the last two lines.

I think that in the case of inconsistency of ZFC this theory, 
but *not* the above general and sufficiently clear "definition" 
of mathematical certainty will be revised (may be very 
radically!).  Actually, ZFC itself was obtained by such a 
correction of set theory with unrestricted comprehension axiom.
Mathematical certainty (relative to or irrespective to any version 
of set theory) will be again understood in exactly the *same* way, 
as "provability in ANY [VS.] appropriate formal system".  


Vladimir Sazonov



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