From: vaeth@mathematik.uni-wuerzburg.de (Martin Vaeth) Subject: Re: Axiom of Choice Date: 7 Jul 2000 14:27:15 +0100 Newsgroups: sci.math Summary: [missing] >> It follows from "Shelah's axiom" (which I have mentioned in my >previous post) > >You didn't say what it is, though; would you mind stating it? >And also if you know whether it's consistent with (or perhaps >even follows from) ZF+AD+DC? It is the statement that any subset of a complete separable metric space has the property of Baire (i.e. it differs from an open set only by a meagure set). I used the (not official) name "Shelah's axiom" because Shelah has proved that it is consistent with ZF+DC. I do not know about the connection with AD, although I were surprised if it were not at least consistent. >Embarassingly, I could also use a definition for "singular functional". The "usual" definition is as follows (this is only a sketch; for details see books about Banach lattices, e.g. Luxemburg/Zaanen: Riesz Spaces): Let X be a Banach lattice. Then any functional in the order dual has a unique decomposition into a "regular" part (for l_\infty this corresponds to a functional of l_1) and into a remainder which is the mentioned singular functional. >> So I hope that some of the models in which they use such singular >> functionals to obtain certain properties does not "match" reality. >> However, actually I fear that many physicists then would just throw >> away this model instead of blaming AC for the failure. > >I'm a little confused here -- if there's no such functional, wouldn't >that mean automatically that you have to throw away the model, given >that the model is predicated upon it? Or are you saying that the model >predicts that the behavior of the world is genuinely sensitive to the >existence/nonexistence of such a functional? I mean the latter. Well, either they have to throw away the model or the underlying mathematical logic (say: AC). What I am trying to say is that most physicist would then not have the courage to blame AC. >In the latter case, perhaps on reexamination (and certainly I'm >speculating here) what you'll find is that what it really depends >upon is the existence of such a functional in some definability >class; say, L(R). Given large cardinals, L(R) satisfies AD+DC. > >If such a situation came to pass, one might very reasonably take >the attitude that the "physically real" type-2 objects were the >ones in L(R), but that mathematically AC still held in V (so >as to be able to make the most effective use of the machinery >of set theory). Then when it came time to apply results to >the physical world, you'd just have to remember that only some >type-2 objects, namely the ones in L(R), were directly relevant. Yes, I completely agree with this. However, if it turns out that the only mathematical objects relevant for physics are the constructible ones, it would probably be more *convenient* to just throw AC overboard (actually, this is what I mean when I say that AC does not hold). ============================================================================== From: Mike Oliver Subject: Re: Axiom of Choice Date: Fri, 07 Jul 2000 13:06:57 -0700 Newsgroups: sci.math Martin Vaeth wrote: > [Mike Oliver wrote:] >> You didn't say what ["Shelah's axiom"] is, though; would you mind >> stating it? And also if you know whether it's consistent with (or >> perhaps even follows from) ZF+AD+DC? > It is the statement that any subset of a complete separable metric space has > the property of Baire (i.e. it differs from an open set only by a meagure set). > I used the (not official) name "Shelah's axiom" because Shelah has proved that > it is consistent with ZF+DC. I do not know about the connection with AD, > although I were surprised if it were not at least consistent. This follows from AD; it's just the determinacy of every Banach-Mazur game. To play the Banach-Mazur game for a set A, first I choose a basic open set, then you choose a basic open set inside mine, then I choose one inside yours, and so on. One of us (doesn't matter which) is responsible for making sure the intersection of the sequence is a singleton. Then I win if the point in the intersection is an element of A; otherwise you win. You have a winning strategy if and only if A is meager. I have a winning strategy if and only if A is comeager in some neighborhood. Determinacy means every A has one of these two properties; it's not hard to recover that every set of reals has the p.o.B. >> In the latter case, perhaps on reexamination (and certainly I'm >> speculating here) what you'll find is that what it really depends >> upon is the existence of such a functional in some definability >> class; say, L(R). Given large cardinals, L(R) satisfies AD+DC. >> >> If such a situation came to pass, one might very reasonably take >> the attitude that the "physically real" type-2 objects were the >> ones in L(R), but that mathematically AC still held in V (so >> as to be able to make the most effective use of the machinery >> of set theory). Then when it came time to apply results to >> the physical world, you'd just have to remember that only some >> type-2 objects, namely the ones in L(R), were directly relevant. > Yes, I completely agree with this. However, if it turns out that the only > mathematical objects relevant for physics are the constructible ones, > it would probably be more *convenient* to just throw AC overboard > (actually, this is what I mean when I say that AC does not hold). Not at all, mon vieux -- this is the whole point! Considerations of the wider universe V give us insight into the behavior of L(R) which would be difficult otherwise to come by. For example, the large cardinals in V give us that L(R) satisfies AD. Because L(R) sits inside V in such a nice, stable way (it's a transitive subclass, there's a simple definition with good absoluteness properties, etc), there's very little cost associated with finding out about L(R) by reasoning with V but restricting quantifiers when necessary, rather than refusing ever to consider sets outside L(R).