Counterintuitions and the well-ordering theorem
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From: Daryl McCullough on 1 Nov 2009 08:36 >AC produces sets, by fiat of existence, which simply DON'T EXIST! >Non-measurable sets in R,free ultrafilters on N, partitions of R^3 >into non-parallel lines - all these things simply don't exist, >in that no explicit set can be named, with those properties. The reason I have little sympathy for this point of view is that I don't see that traditional mathematics ever makes the assumption that everything has an explicit name or description. Yes, some things do---namely naturals, but points in a topological space? The *names* of those points are completely irrelevant. It's an amazing accomplishment of ZF and the concept of the cumulative hierarchy that we can come so close to an explicit taxonomy of all the objects that will ever come into play in any mathematical argument, but I don't see that the existence of such a standard model has any relevance to what mathematicians *do* with sets. Instead, they start with certain basic sets that are *not* defined, except implicitly by axioms governing them, and then they apply set-theoretic operations on *those* sets to get new sets. To me, the astounding claim about set theory is that, whatever your basic objects are --- paths of particles, wavefunctions, games, strategies, probabilities, trees, computations --- we can describe and reason about them using set theory. Having explicit names for the basic objects is just *not* necessary in order to use the techniques of set theory. >Now, I do not mean by this, that one >must be able to prove whether or not some particular >putative element is in it or out of it. But at least >*a condition* for membership should be available. i.e. > >{ x | phi(x) } > >should be taken as a model or paradigm for what a set actually IS. I sort of agree with this, but I don't feel that it is appropriate to assume that the only possible conditions phi are ones that are definable using pure set theory. For example, if we are trying to formulate Newtonian physics using set theory, then we can talk about such sets as "the set of all past locations of the particle". That is a set of triples of real numbers, and it has an associated condition, but the condition is not given by *set* theory, it is given by the premise of the problem being analyzed using set theory. If we are talking about sets of naturals, then "the set of phone numbers of mathematicians" is a perfectly good set of naturals, even though the condition defining that set isn't a set-theoretic condition. Now, in the case of finite sets of naturals, we can always *find* a set-theoretic condition that is extensionally equivalent. We can cook up a formula Phi(x) expressed in the language of arithmetic, such that Phi(x) holds only of those x-s such that x is a phone number of a mathematician. But such an exercise would be pointless---the existence of such a Phi would be completely irrelevant to anything we want to do with such a set. In a sense, the insistence that all mathematical objects have an explicit description is more radical than constructivism. Constructivists take the principle that everything should be constructable as an obligation on the part of someone trying to *prove* something. It only counts as a proof if everything can be made explicit and computable. But constructive mathematics does *not* assume that nothing *exists* unless it is constructable. Some constructive mathematicians may believe that, but such an assumption is not itself something that can be used in a constructive proof. The claim: "every real number is computable" is not constructively provable, so it can't be assumed in a constructive proof. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 1 Nov 2009 11:02 Bill Taylor says... > >Herman Jurjus <hjm...(a)hetnet.nl> wrote: > >> What are your /reasons/ for rejecting AC, > >Well! That would be a whole thread in itself. >And I don't wish to bore the old hands YET AGAIN. >So I'll try to be brief. > >AC produces sets, by fiat of existence, which simply DON'T EXIST! > >Non-measurable sets in R,free ultrafilters on N, partitions of R^3 >into non-parallel lines - all these things simply don't exist, >in that no explicit set can be named, with those properties. There is something a little strange about this philosophical position: If you assume that all sets have an explicit "membership criterion" Phi(x), then choice is *automatically* true. You can well-order sets by their defining formulas. The only way to get a nonempty set of reals *without* a choice function is if contains some undefinable reals. -- Daryl McCullough Ithaca, NY
From: Bill Taylor on 2 Nov 2009 01:29 stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > The reason I have little sympathy for this point of view is > that I don't see that traditional mathematics ever makes the > assumption that everything has an explicit name or description. OC it doesn't. It never needed to. Before Cantor everything DID. > It's an amazing accomplishment of ZF and the concept of the > cumulative hierarchy that we can come so close to an explicit > taxonomy of all the objects that will ever come into play in > any mathematical argument, but I don't see that the existence > of such a standard model has any relevance to what mathematicians > *do* with sets. I don't follow this sentence at all, sorry. > Instead, they start with certain basic sets > that are *not* defined, In pure set theory they all are. They are the empty set. > To me, the astounding claim about set theory is that, whatever > your basic objects are --- paths of particles, wavefunctions, That is all applied mat, applied set theory. We are (I thought) speaking of pure set theory, as applied to (at most) other pure math objects like N & R. > >{ x | phi(x) } > > >should be taken as a model or paradigm for what a set actually IS. > > I sort of agree with this, Well! There's something. > but I don't feel that it is > appropriate to assume that the only possible conditions > phi are ones that are definable using pure set theory. Set theory as applied to physical objects is completely trivial. I don't think it has any place in this debate. > If we are talking about sets of naturals, then "the set > of phone numbers of mathematicians" is a perfectly good No, it's perfectly bad! > In a sense, the insistence that all mathematical objects > have an explicit description is more radical than constructivism. It's NOT an "insistence", as you call it, but merely an observation of standard (pre-Cantor) math. And I feel that set theory should be conducted along the lines of those traditions. At least, if it is to make traditional sense. Anyway, I was asked for my view, and I gave it. -- Bill of basics
From: Bill Taylor on 2 Nov 2009 02:02 stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > >AC produces sets, by fiat of existence, ... > > ... - all these things simply don't exist, > >in that no explicit set can be named, with those properties. > > There is something a little strange about this philosophical > position: If you assume that all sets have an explicit > "membership criterion" Phi(x), then choice is *automatically* > true. You can well-order sets by their defining formulas. AHA! A very astute observation. This is the kind of thing that had me bothered for a long time. And it is all bound up with the UNDEFINABILITY of "DEFINABILITY", as I have noted before. All sets will be definable, but this notion cannot be delimited in advance. So e.g. once you have a bounded set of reals, it must have a least upper bound, BUT this lub will not be definable at the same level - if the reals in the set are definable at level "a" and below, then the lub will be most likely be at level a+1. There is no end to the levels, they can be notated exactly as any initial segment of the recursive ordinals. But not of the whole set itself - that would be declaring existences BY FIAT again. omega_1^CK itself is purely a creature of ZF, with its all-encompassing power set operation, (without which it cannot be defined.) So the levels of definibility proceed endlessly onward, without (sans Cantor) a definite or upper bound. This sort of property is called "extensible" in philosophical logic circles, and is the topic of the paper I alluded to earlier. -- Well-ordered William ** Definibility is itself undefinable!
From: Butch Malahide on 2 Nov 2009 03:01
On Nov 2, 12:29 am, Bill Taylor <w.tay...(a)math.canterbury.ac.nz> wrote: > stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > > The reason I have little sympathy for this point of view is > > that I don't see that traditional mathematics ever makes the > > assumption that everything has an explicit name or description. > > OC it doesn't. It never needed to. Before Cantor everything DID. What objects of Euclid's geometry have explicit names or descriptions? |