[FOM] A Defence of Set Theory as Foundations

[Arnon Avron]

On Fri, Oct 14, 2005 at 09:21:55PM -1000, Robert Lindauer wrote:
 
> Personally, I can't imagine the world where someone actually checks the 
> twin prime conjecture by running through all of the integers and 
> checking whether each was divisible, etc.  

Do you have a problem here because you think that
the term "all of the integers" that you have used hereis meaningless, 
or (as I suspect), because (like me) you dont think that it is possible 
to run through  ALL of the integers and check each?
(note that one can classify a certain task as impossible only if 
s/he takes it as  meaningful!). 

  The upshot is that we cannot avoid thinking of *all* the integers. 
Neither do we need to be able to actually run through all the integers in order 
to know that for every integer n we come across it will be the
case that n=n, or that either p(n) or its negation will be true,
or that if we continue to check integers greater than n,  then
either we'll encounter a pair of twin primes or else we shall never 
encounter one. I cant imagine a third possiblity, and if you
do than you have a much better imagination than me.

In short: Classical logic is valid also for potential infinity!

Arnon Avron