colonel@gloria.UUCP (Col. G. L. Sicherman) (03/13/85)
> > The same must apply to classical mathematicians. Even though CH is in- > > dependent of the axioms of ZF Set Theory, it is conceivable (although high- > > ly implausible) that someone will some day come up with new methods of > > mathematical reasoning that are *obviously* valid, using which CH can be > > decided. > > Really? I had always thought that there were two "classes" of sets: in > one class, the CH is true, and in the other it is false. That is, just > like there are other geometries in which Euclid's fifth is false, there are > "universes" in which CH is true and other "universes" in which it is false. > If this duality is valid, how can one possibly come up with new methods by > which CH can be decided? Or is this duality valid in the first place? This problem has been thoroughly picked over in the literature. The difficulty is those "obviously" valid axioms. The more abstruse the mathematics becomes, the less meaning the axioms have in the real world. It's been pointed out that even "2+2=4" is false in some natural contexts. For some people the Axiom of Choice is "obviously" true, though our experience of it is limited to finite sets. For some of those people the Axiom of Determinacy is also "obviously" true -- even though it contradicts the Axiom of Choice! The difficulty is that both axioms are natural generalizations of true statements about finite sets. It's safe to say (if it's not, somebody will contradict me!) that ALL of transfinite set theory is moot in the real world, because there are no infinite sets within our experience. -- Col. G. L. Sicherman ...{rocksvax|decvax}!sunybcs!colonel