[FOM] Re: Disaster?

[Neil Tennant]

> Let's suppose again, though, that the proof [of 0=1 in PA] is
> sufficiently small.
> 
> In that case, I strongly suspect that the contradiction would *not* be
> shrugged off by most mathematicians.  There would be only finitely many
> instances of the first-order induction axiom used, and the arguments
> would use standard principles of classical logic, leading to any desired
> (first-order number-theoretic) conclusion whatsoever.  What would people
> do?
> 
> There would presumably be some effort to push the idea behind the
> contradiction as far as possible, to see just how few axioms and
> logical principles were needed to get a contradiction.  Then, something
> would have to be rejected.  Some might reject classical logic, but only
> if some replacement logic were available that would avoid the problem.

What possible "replacement logic" might there be?

Intuitionistic logic won't do, since if classical arithmetic is
inconsistent, then so is intuitionistic arithmetic. (Any
philosophical or foundational argument for intuitionistic rather than
classical arithmetic would have to appeal to considerations other than the
fear that classical arithmetic might be inconsistent.)

Relevant logic arguably won't do either, since it ought to be a condition
of adequacy on the choice of any relevant logic that any proof of
inconsistency should be relevantly provable.

Neil Tennant