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Re: Whitehead & Russell on Reductio A.A. and when Constructivism is
mainstream #200; 2nd ed; Correcting Math
Posted:
Oct 16, 2009 1:04 AM
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On Oct 12, 3:35 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
|Basically is it possible that, e.g. arithmetically, ExPx is
|true without a concrete example? For instance, is it conceivable
|that ~GC is true without the need of citing one single example?
Making this into a precise question might take some
work. Thinking constructively, an arithmetic claim
of the form Ex P(x) just says that there exists an
example, and being an integer it is concrete.
Thinking classically, you could well prove the
existence of an integer satisfying P without being
able to prove that any specific integer satisfies P.
There's a metatheorem ensuring that if you can prove
a statement of the form Ex P(x) where P(x) has all
of its quantifiers bounded (so that whether P(x) is
true is computable) (in a system S taken from some
wide assortment of classical systems) then there is
also a constructive proof (in a corresponding
constructive system S+). Even without a metatheorem
to help, though, one would usually reason (classically)
that the existence of such an x means that we can
find it by checking integers one by one.
If you have a more complicated predicate, there's no
general guarantee. It's conceivable for example that
one could prove (classically) that there are finitely
many Fermat primes without being able to prove
specifically how many there are.
BTW let's please set follow-ups so that sci.physics
doesn't have to put up with this discussion.
Keith Ramsay
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