In article <email@example.com>, "Gene Ward Smith" <firstname.lastname@example.org> wrote:
> Gerry Myerson wrote: > > > I think Norm would say, 1) you can't form the set (and Norm's reasons > > are given in the article), and 2) you don't need to - there's no good > > mathematics you can do with the completed infinite set that you can't > > do without it. > > Norm is also opposed to axioms. Without axioms, how do we know when we > are "forming" an infinte set? If I state Euclid's theorem on the > infinitude of primes, am I "forming" a set? Am I forming a set just by > referencing the integers at all? If Norm won't give a set of axioms he > finds acceptable, we can't very well say that measureable cardinals > contradict his foundations for mathematics, because he hasn't really > given a foundation. He has, in fact, claimed that infinite sets are > metaphysics; but if they are metaphysics, he's not talking mathematics > at all, but metaphysics. In which case, so what? What do his > metaphysical beliefs have to do with mathematics?
All these questions are better directed to Norm than to me, but I'll make believe I know what he is on about, and answer thus:
If I remember right, Euclid never said "there's an infinitude of primes." He just said, "given any prime, there's a bigger one." You and I are accustomed to interpreting the second as meaning the same thing as the first, but I think that until quite recent times, mathematicians didn't. They didn't accept "the completed infinity," and they were still able to develop the theory of numbers, prove the quadratic reciprocity theorem, the four squares theorem, etc. If you state the theorem the way Euclid did, you are not forming an infinite set, and you can get on perfectly well that way.
As for measureable cardinals, I'm guessing the question of their existence doesn't interest Norm one way or the other, on the grounds that the stability or otherwise of the Sydney Harbor Bridge is unlikely to depend on the outcome. The real world questions mathematics sprang from do not depend on these abstractions, nor on the axiom systems in which they are debated.
> > More ad hominem. Norm accepts the axioms of group theory as the > > definition of what a group is, and has no problem with them because > > he can construct (finite) models of them. > > In which case, he can hardly say he is rejecting axioms, and ought to > step forward and say what his proposed axioms are. Would ditching the > axiom of infinity do it? If not, what would?
I don't know, and if you really want to know, you could try asking him. But perhaps he is only saying, "the axioms of group theory define an interesting set of structures, which I can construct, and which help me answer questions about physics, while the axioms of set theory only help me study set theory, which is not where the real value of mathematics is."
-- Gerry Myerson (email@example.com) (i -> u for email)