Counterintuitions and the well-ordering theorem
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From: Daryl McCullough on 6 Nov 2009 06:54 Bill Taylor says... > >stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > >> >stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > >> >> That isn't true. Real numbers did not have names before Cantor. > >> >Oh heavens, of course they did! > >> No, they did not. *Some* reals had names. > >Oh. It would have been clearer if you'd prefaced the above with >"some" or "all" . No matter. Go ahead with your evidence that >(pre-Cantor) there were any reals that did not have names. >By "name", OC, I mean a definition. Cantor proved this fact. But it didn't *become* true when Cantor proved it. So you are the one who is making a claim that is provably false. Your argument is there were no reals that people had reason to talk about before Cantor that had no definition. That's certainly true, and almost tautological---if you have reason to talk about a specific real, then you are motivated to give it a name. Prior to Cantor there was no naming *scheme* that assigned a name to each real. Your counter-argument is that you can name all reals by a hierarchy of schemes (perhaps indexed by computable ordinals). Well there was no such infinite hierarchies of schemes prior to Cantor, either. So your claims make no sense at all. Unless you are just saying the trivial fact that prior to Cantor, people never studied any *particular* real unless that real had a finite description. Of course, that's tautologically true. How could it be otherwise? If I tried to get people to be interested in studying the properties of the real 2.126745..., it would take an infinite length of time to tell people which real I was talking about, and then there would be no time to study it. The fact that humans communicate using finite noun phrases is certainly true. Cantor didn't change that. -- Daryl McCullough Ithaca, NY
From: Peter Webb on 7 Nov 2009 12:08 "Bill Taylor" <w.taylor(a)math.canterbury.ac.nz> wrote in message news:accf8622-d756-425f-9ccf-b34ea8bef1c6(a)u36g2000prn.googlegroups.com... > stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > >> >stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > >> >> That isn't true. Real numbers did not have names before Cantor. > >> >Oh heavens, of course they did! > >> No, they did not. *Some* reals had names. > > Oh. It would have been clearer if you'd prefaced the above with > "some" or "all" . No matter. Go ahead with your evidence that > (pre-Cantor) there were any reals that did not have names. > By "name", OC, I mean a definition. > >> > I'll ignore the trivial fact that all Euclid's diagrams had letters > >> You don't understand the difference between a *variable* and a >> unique name? When someone says "Let A be a point, and let R be > > OK OK, it was just a flippancy. I won't make any more! > >> >No-one doubts he [Descartes] was doing Euclidean geometry, and >> >extending it, and all his points were real number pairs, see just above. >> >> Not every real has a name, then not every pair of reals has a name. > > So, we agree as regard geometry - that it comes down to naming reals. Point of order. All Reals constructible with straightedge and compass are name-able; they are named by their construction. Indeed they are a tiny subset of the algebraic numbers.
From: Daryl McCullough on 7 Nov 2009 12:36 Peter Webb says... >Point of order. All Reals constructible with straightedge and compass are >name-able; they are named by their construction. Indeed they are a tiny >subset of the algebraic numbers. Yes, that's right. But a lot of Euclidean constructed started with an *arbitrary* initial state: Given a line and a point not on that line, it's possible to draw a second line through that point parallel to the first line (or whatever). Was there an assumption that such starting points were constructible? Was there an assumption that for an arbitrary pair of line segments, the ratio between their lengths was a constructible real? Maybe the ancient Greek geometers did believe that at some point. -- Daryl McCullough Ithaca, NY
From: Bill Taylor on 9 Nov 2009 22:04 Isn't it remarkable how threads can wander!? This thread started out with a (perhaps disingenuous) enquiry as to the alleged non-obviousness of well-ordering. After a brief detour through the anecdote about AC WOT and Zorn, (walking into a bar?), it then transferred its attention to AD, the axiom of determinacy. After lingering there for some time, it has now devolved into a debate about Powerset, let's call it PS, and what it could possibly mean for it to be false. Now I am in a quandary. I might defend its possibility of being false, but this would necessitate going OUTSIDE Cantorian set theory. And as almost everyone else here feels inescapably INSIDE Cantorian set theory, pre-Z, it might be called, I cannot defend it! All the objections and incredulity expressed about this are from confirmed Cantorians, which is most mathies. Including myself (!), at least for operational purposes. What *is* pre-Z ? It is informal set theory, Cantor style, which means, effectively, the basic set theory of Boole and perhaps earlier, and perhaps up to Peano; and Cantor's own unique original ideas. And what were these? NOT (not chiefly) AC or well-ordering, as much as he was determined that universal well-orderability should turn out to be true; not that, IMHO. The essential Cantorian departure was POWER SET. PS. (IMHO). Most of the slick uniformity of C20 math is derived from this. And a little more is derived from AC. It was Zermelo who turned pre-Z into Z in 1908(?), or rather into ZC, as Zermelo regarded AC as "obvious", and sufficient to prove wellorder, as it obviously is. Truly, as one book title has it, we ought to speak of "Zermelo's Axiom of Choice". Z is formalized Z, that is, axiomatized pre-Z. ZC is Zermelo's set theory, not Cantor's. Cantor's set theory is basic (Boolean) set theory plus PS. ZFC later introduced F, a further mild addition, in 1920 or so. But it is intriguing that mathematicians and mathematical logicians have always slightly reversed history and regarded ZF (which never arose historically!) as an important stepping-stone on the way to ZFC. And such was the utter slick and encompassing nature of ZF(C) that mathematicians from 1920 onwards almost uniformly adopted the new ideas wholesale. Occasionally with doubts expressed about AC, but NEVER with any doubts about PS. PS was "obviously" true, and needed to ensure lots of other stuff like the reals, R, general Cartesian products etc, and students were introduced to it so smoothly (usually without even being aware of its axiomatic nature), that no-one ever noticed what a huge dead fish they were swallowing! Only in recent decades, has it become more suspicious again. And even then only to math loggies, rather than mathies proper. As Halmos(?) said, "The Axiom of Choice is unique in its ability to trouble the conscience of the working mathematician. This conscience-troubling, as I noted, is only apparently due to AC, but really, the blame lies with PS, and such is the intimacy with which mathies enjoy with Z(F(C)) that they never even notice. And indeed, react with utter incredulity to the suggestion that it be in some way false, or at least dangerous and dark. Mathematics, now, runs on ZFC the way Freecell runs on Windows. So what would math be with a non-Cantorian set theory as its platform or operating system? Most people might say, hugely different, but I think they would be wrong. Obviously some stuff that is intimately entwined with set theory would have to change, but not greatly. We could still have Cartesian Products, ordered multiplets, sequences, continuous functions, and the whole panoply of C19 math - differential equations, optimization, analysis, tensors, etc etc virtually unchanged, without PS. No-one has ever tried this (outside constructivism), mostly because, "Why bother?" - a comment often levelled at constructivists as well, intriguingly. But it could be done, without too much trouble, as well. And finally, what about the reals, R ? They would certainly look a lot different, or rather, the way we standardly handle them would be somewhat different. Essentially, we would have to abandon our habits of theft over honest toil, a la Russell, and take a lot more constructive, or rather *definitional* approach to them. They would have to be built up slowly from basic unexceptionable ones, through increasingly "artificial" (but necessary for the handling of l.u.b etc) levels of posterior definability, exactly mirroring the building up of recursive ordinals to any level below but NOT including w_1^CK (which need not, does not, exist outside Cantorian Z). Such a study has not yet been done, though small steps along the way have been made here and there. If this system were ever worked out in full - WHEN this system is worked out in full - it will be seen as a much more ontologically reliable system than Z, though admittedly far more cumbersome, and thus always likely to be ignored by working mathematicans. And it is on this (as yet incomplete) basis, that I say with arrogant confidence, that the more egregious absurdities following from AC are simply *false*, though this would take a lot of proving, or even explicating. And even more radically, I declare that Powersets of infinite sets, at least in their Cantorian conception, simply *will not exist* - and will not be needed. Like everyone else, I adore ZF because of its uniformizing slickness, but must reluctantly concede that it sometimes produces falsities/absurdities; but these are almost entirely due to the tacking on of AC - that uniquely conscience-troubling axiom! I have endeavoured to indicate why it might be so unwittingly troublesome. -- Basic Bill
From: Aatu Koskensilta on 10 Nov 2009 04:42
Bill Taylor <w.taylor(a)math.canterbury.ac.nz> writes: > This thread started out with a (perhaps disingenuous) enquiry as to > the alleged non-obviousness of well-ordering. Bill, I will get back to you on your mumblings about the powerset axiom, definability, what not, later. Here I'd just like to note this thread started out with a (perfectly ingenuous though possibly not very serious-minded) enquiry as to the alleged /counter-intuitiveness/ of the well-ordering theorem, this enquiry prompted by my (possibly erroneous and arbitrary) notion that people's statements about counter-intuitiveness, evidence, etc. often are almost totally arbitrary -- that is, it is often impossible to get any explanation whatever of e.g. what intuitions are contradicted by this or that.(I don't recall if it's been mentioned already, but Sol Feferman's paper _Mathematical Intuition vs. Mathematical Monsters_ is an enjoyable and relevant read in this context.) -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |