Counterintuitions and the well-ordering theorem
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From: Virgil on 17 Nov 2009 03:44 In article <734ce777-8e88-4030-9a79-4fe211ced379(a)w19g2000yqk.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 17 Nov., 05:00, Bill Taylor <w.tay...(a)math.canterbury.ac.nz> wrote: > > > Naturally, to one committed, perhaps unwittingly, to a certain > > particular all-embracing point of view, anything that goes counter > > to it will seem incomprehensible. �I well recall how, in my younger > > days, I was astounded that there could be anything controversial > > about AC - it just seemed so OBVIOUS! � Decades later, I began > > to acquire suspicions, as greater familiarity with math came. > > Similarly, now even more decades later, and still slowly, it is > > finally occurring to me that there are similar suspicions about > > the completion of the entity of (all) the reals. > > It is rather simple. WM is what is rather simple > A "number" that cannot be identified or addressed > so that we can talk about it And which reals are ones which we cannot talk about?
From: Albrecht on 17 Nov 2009 05:43 On 17 Nov., 05:00, Bill Taylor <w.tay...(a)math.canterbury.ac.nz> wrote: > stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > > People skeptical of the power set usuallybalkat the very first > > application that gets you something really new: P(omega), or the > > reals. > > Correct. That (the reals) is the prime source of concern. > (As it has been ever since about 1666 or even before.) > > > So abandoning the power set axiom means, in this case, > > considering the collection of all reals to be a proper class, > > No. This is Cantorian thinking, and fiddling about with > the distinction between set and class doesn't realy change anything. > The problem is that non-Cantorians object to thinking about > them ALL, at the same time, regardless of what formal mechanism > or meta-mathematical slant one wishes to frame it within. > > > But what does that really mean? What does it mean to say that > > the collection of all reals > > They would say there is no such beast. > > > (which exists in the sense that it is a definable class) > > This is already nonsense, in their view. > > Recall that (as I often note) the concept of "definability" is > itself necessarily not definable, or at least not definable in > anything like the same sense, as diagonalization inevitable shows. > It is "extensible", as some philosophical logicians call it. > > > It seems > > to me that we only accept the set/class distinction because > > it is *forced* on us by consistency. > > Certainly, form a Cantorian point of view, it must seem so. > > But Cantorians have allegedly been misled, by the fact that > it is easy an unexceptionable to regard the naturals (et al) > as a completed entity (set, whatever), into thinking that > the same should apply to the reals. The aberration starts with the idea that the naturals form a set. It seems to be clear and today it is a common and well accepted concept. But, in spite of this, this concept is fault. The trouble involves the problem with the ostensible conclusion that "for any x" is the same as "for all x" which in fact doesn't hold in the case of infinite manifolds. The concept of an infinite set involves a logical fault. There is no definite cardinal number of an infinite manifold. Best regards Albrecht > > > I can't understand the motivation for that. > > Naturally, to one committed, perhaps unwittingly, to a certain > particular all-embracing point of view, anything that goes counter > to it will seem incomprehensible. I well recall how, in my younger > days, I was astounded that there could be anything controversial > about AC - it just seemed so OBVIOUS! Decades later, I began > to acquire suspicions, as greater familiarity with math came. > Similarly, now even more decades later, and still slowly, it is > finally occurring to me that there are similar suspicions about > the completion of the entity of (all) the reals. > > If one has NO feeling for the concerns of suspicion involved, > it hardly seems likely that mere debate can produce a change. > > -- Wandering-minded William
From: Herman Jurjus on 17 Nov 2009 05:48 Bill Taylor wrote: > Herman Jurjus <hjm...(a)hetnet.nl> wrote: >>> Quantifying over reals doesn't seem to hold many terrors, >>> (though I could be wrong); but quantifying over sets of them >>> is a whole nother matter. >> Funny; you sound more and more like a (Brouwerian) intuitionist >> the more you talk. > > I understand fully why you would say so. But I renounce > constructivism as a philosophy of math, (while having no qualms > about formal constructiviasm as a way of doing math). Fyi, for me, this is opposite. Constructivism as a(!) philosophy of mathematics is unproblematic to me. But while doing mathematics, I have no problem using LEM; in fact, I prefer mathematics with LEM over mathematics without LEM. > IMHO there is NO trouble at all (philosophically) > with accepting LEM, or with accepting numerical proofs using it, > however existential, (as I've read that Littlewood's Pi/Li does). Let me assure you that rejecting LEM is not what intuitionism is about. >>> It seems that any statement involving sets of reals, must >>> inevitably be interpreted as thinking of *some* set of reals >>> at *some* level. So no, not all reals. >> And quantifying over all reals is -not- similarly problematic, >> in the presence of such 'levels'? > > Is this query merely rhetorical, or do you in fact have > some particular application/example in mind? > If so, I would love to hear of it. Neither. I hoped you would say "yes, it's less problematic", followed by your reason. If the notion 'real' has different levels or different 'versions', I would expect the phrase 'for all reals' to be at least ambiguous. Of course, we could interpret it as 'for all definable reals', but wouldn't that give problems as well, when the predicate 'undefinable' is itself undefinable? -- Cheers, Herman Jurjus
From: Daryl McCullough on 17 Nov 2009 06:53 Bill Taylor says... > >stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: >> So abandoning the power set axiom means, in this case, >> considering the collection of all reals to be a proper class, > >No. This is Cantorian thinking, and fiddling about with >the distinction between set and class doesn't realy change anything. But all it takes to be a class is a criterion for membership. A class is basically a formula with a single free variable. There is no question that the *formula* exists---we can write it down. >> But what does that really mean? What does it mean to say that >> the collection of all reals > >They would say there is no such beast. It exists as a formula, as a criterion for separating reals from non-reals. What else do you expect from a "collection"? I really don't know what you mean by a collection if not that. >Recall that (as I often note) the concept of "definability" is >itself necessarily not definable I don't see how that is relevant. We can certainly talk about definability relative to a language (for example, relative to the language of ZF). That is not problematic. The collection of all reals *is* definable relative to this language. >But Cantorians have allegedly been misled, by the fact that >it is easy an unexceptionable to regard the naturals (et al) >as a completed entity (set, whatever), into thinking that >the same should apply to the reals. I don't think that there is any significant difference between the two cases. What basis could there be for considering the naturals to form a set that wouldn't also apply to the reals? >> I can't understand the motivation for that. > >Naturally, to one committed, perhaps unwittingly, to a certain >particular all-embracing point of view, I'm not *committed* to any particular point of view. I am perfectly willing to play at being a constructivist, or finitist, or definabilist, or whatever. Each point of view can be interesting, in the sense that the self-imposed limitations lead to problems and patterns that don't appear without those limitations. But I don't understand the motive for *believing* in one or the other. I don't really know what it *means* to "not believe that the reals form a set". I know what it means to have rules for set formation that don't allow the power set, but all that means (to me) is that we are restricting our attention to certain kinds of sets. >If one has NO feeling for the concerns of suspicion involved, >it hardly seems likely that mere debate can produce a change. That's a cop-out, it seems to me. -- Daryl McCullough Ithaca, NY
From: Jesse F. Hughes on 17 Nov 2009 07:58
stevendaryl3016(a)yahoo.com (Daryl McCullough) writes: >>If one has NO feeling for the concerns of suspicion involved, >>it hardly seems likely that mere debate can produce a change. > > That's a cop-out, it seems to me. Indeed. At the least, he should make these concerns of suspicion explicit. I can't see that he has presented any such concerns that cast the slightest doubt on the power set axiom. -- Jesse F. Hughes "It is a clear sign that something is very, very, very wrong, as human beings are, well human. Maybe some people think that mathematicians are not, but I disagree. They are human beings." -- James S. Harris |