FOM: Some thoughts on "Realism"
shipman at savera.com
Mon Jun 19 10:59:27 EDT 2000
Peter Schuster wrote:
> >To maintain that the twin prime conjecture is indeterminate, you don't
> >have to deny the determinacy of the property "primeness", nor do you
> >have to deny ontological status to any integers. You just need to deny
> >that the SET of integers exists as a completed whole.
> Doesn't it suffice to reject the idea that all properties/subsets
> of integers are decidable/detachable?
> I cannot see why one had to deny the integers "as a completed whole".
Rejecting the idea that all properties/subsets are decidable/detatchable is
not enough, because the particular property being discussed here, primeness,
is a very concrete, computationally straightforward property (strictly
speaking, we are using the property of being a prime which is the successor
of the successor of another prime, but that's just as concrete).
Tennant suggested that rather than rejecting the integers as a completed
whole, one could just deny that the universal quantifier guarantees
determinacy of truth-value. In other words, we could grant, for each n, the
determinacy of the statement "there is a pair of twin primes above n", but
deny the determinacy of the statement "for all n, there is a pair of twin
primes above n". I have trouble with this notion--it is hard to see how
this is not simply denying that the set of integers exists as a completed
whole, because "the set of integers exists as a completed whole" means, to
me, that the set of integers is something you can quantify over with no loss
-- Joe Shipman
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