[FOM] A Defence of Set Theory as Foundations
aa at tau.ac.il
Wed Oct 19 07:07:03 EDT 2005
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!
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