Physics does not allow computers that compute arbitrarily fast, so this is not a good in principle argument. On Aug 24, 2007, at 6:26 PM, Dan Asimov wrote:

Jason asked what[math-fun thinks of constructivist logic.

I'm very "maximal" about what qualifies as set having real existence. Anything that can exist does exist, as far as I'm concerned.

I think asking what can be computed in a finite sense, as the Constructivists do, is a worthy endeavor. But that to me is no reason to ignore all the mathematics that views countable and continuum as sets of unequal size (because of Cantor's diagonal proof, yes).

To try to understand what human beings can and cannot ever know and in what way -- this is a worthy endeavor.

But there must exist some proposition of number theory which can be proved neither true nor false by axiomatic reasoning, despite the fact that checking the proposition over all combinations of integers could, in priciple, be done in a finite amount of time by a computer that could compute arbitrarily fast. And that check would settle the truth of the proposition once and for all.

Do the Constructionsists not mind having to view ostensibly really-true (or really-false, whichever the case may be) propositions as undecidable?

--Dan

_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun