Counterintuitions and the well-ordering theorem
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From: Herman Jurjus on 13 Nov 2009 09:51 Bill Taylor wrote: > Herman Jurjus <hjm...(a)hetnet.nl> wrote: > >> Just in case this wasn't clear before: >> I agree with your assessment of the power set axiom. > > Excellent! We are now TWO voices alone in the wilderness. :) > >> What do you think about the separation axiom schema? >> Should it be restricted to formulas in which all the quantifiers >> are bounded (i.e. 'for all x in V' instead of 'for all x' sec)? > > I'm not fully sure I follow the question, Judging from your answer, you did. > (e.g. what was that "sec" in the last line?) Excuse my French. A translation site suggested that it was used in English, too. Well it is, but only for wine, apparently. It literally means 'dry', and I intended it to mean 'as such' - 'without the addition'. > However, noting that quantifying over sets of naturals > is (encyptically) quantifying over reals, there is still > a considerable difference, seemingly, between quantifying > over reals and quantifying over SETS of reals. > > 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. > It is in this latter that LUB makes > its appearance. Typically, the LUB of a set of reals occurs > at a greater definability level than any of its boundees, > and (presumably) of the set of them. > > In a similar way, Cantor's uncountability theorem has its > content subtly altered, though not if its original statement > is kept in its proper form - that for any list of reals there is > a real not on the list. This now is seen as a rather simple > corollary of the fact that reals occur in distinct levels, > which are indefinitely extensible. > > 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'? > And the consequent > of the statement must allow the possibility of going beyond > that original level. But nevertheless the theory itself > need not, cannot, make any mention of the levels, much as > ordinary ZF never mentions the class V, its intended model. > >> Does that make sense, when throwing out PS? > > I hope it makes *some* sense. Oh, let's not quarrel about that, then. "Hope is the poor man's bread." -- Cheers, Herman Jurjus
From: WM on 13 Nov 2009 10:01 On 12 Nov., 19:23, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Bill Taylor says... > > > > >Herman Jurjus <hjm...(a)hetnet.nl> wrote: > > >> Just in case this wasn't clear before: > >> I agree with your assessment of the power set axiom. > > >Excellent! We are now TWO voices alone in the wilderness. :) > > People skeptical of the power set usually balk at the very first > application that gets you something really new: P(omega), or the > reals. So abandoning the power set axiom means, in this case, > considering the collection of all reals to be a proper class, > rather than a set. As usual you forgot the correct alternative: There is neither a set nor a class of "all reals". There are only some reals (finitely many) which have been defined by individual definitions like pi or e. And there are in total, depending on the point of view, countably many or finitely many reals. Regards, WM
From: Bill Taylor on 16 Nov 2009 23:00 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. > 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: Bill Taylor on 16 Nov 2009 23:18 Herman Jurjus <hjm...(a)hetnet.nl> wrote: > > I'm not fully sure I follow the question, > Judging from your answer, you did. Well, thanks! > > (e.g. what was that "sec" in the last line?) > > Excuse my French. A translation site suggested that it was used in > English, too. Well it is, but only for wine, apparently. > It literally means 'dry', and I intended it to mean 'as such' - > 'without the addition'. Ah. Thanks for the clarification. In English-lanuage books one often sees the appendation "simpliciter" tacked on to such phrases. But this is Latin rather than English, simpliciter, and often seems a bit poncy to me. > > 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). 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). > > 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. > > I hope it makes *some* sense. > > let's not quarrel about that,then. "Hope is the poor man's bread." Again I don't follow, but it sounds positive. I recall (paraphrased) Marx's famous dictum - ** religion is the opium of the underclass .....which has been suggested to be updated to... ** opium is the religion of the underclass .....which is not nearly so positive, alas! -- de-Toxing Taylor *** We must respect the other fellow's religion, but only in the way *** that we assent that his wife is beautiful & his children smart.
From: WM on 17 Nov 2009 01:54
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. A "number" that cannot be identified or addressed so that we can talk about it, meaning the same, does not exist as a part of mathematics. But there are merely countably many means to identify a number. Regards, WM |