Counterintuitions and the well-ordering theorem
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From: Jesse F. Hughes on 3 Nov 2009 08:04 James Burns <burns.87(a)osu.edu> writes: > It might be interesting to point this out to > the next person to prove that the reals are countable -- > that /if/ the reals are countable, then Banach-Tarski > is unavoidable. You really think that such persons are likely to believe that's a consequence? -- Jesse F. Hughes "Well, I guess that's what a teacher from Oklahoma State University considers proper as Ullrich has said it, and he is, in fact, a teacher at Oklahoma State University." -- James S. Harris presents a syllogism
From: Bill Taylor on 3 Nov 2009 23:28 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. > > That isn't true. Real numbers did not have names before Cantor. Oh heavens, of course they did! All the integers and ratiobnals had them, and also all the reals we ever use in applications, like pi, e, sin^(-1)(2/7) and so on. > Points and lines of Euclidean geometry didn't have names. Oh heavens, of course they did! I'll ignore the trivial fact that all Euclid's diagrams had letters Wed Nov 4 17:27:50 NZDT 2009 /users/math/wft13/News> m r 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. > > That isn't true. Real numbers did not have names before Cantor. Oh heavens, of course they did! All the integers and ratiobnals had them, and also all the reals we ever use in applications, like pi, e, sin^(-1)(2/7) and so on. > Points and lines of Euclidean geometry didn't have names. Oh heavens, of course they did! I'll ignore the trivial fact that all Euclid's diagrams had letters on the points and lines, and go straight to Descartes. No-one doubts he was doing Euclidean geometry, and extending it, and all his points were real number pairs, see just above. > To do topology, you start off with "There is a set > with such and such properties", and you work out the consequences. > You don't need to ask: "What is the name of that set?" "What > are the names of its elements?" Oh yes, that's true. Abstract math like topology and group theory aand so on, start off with no names; but as soon as you get to particular examples, they do. > Yes. Pure mathematics should (in my opinion) be a *superset* of > applied mathematics. Whatever our foundation of mathematics should > be, it should at least include the ability to model the sorts of > problems that show up in physics and applied math. OUCH!! Now I recall, that we have had differences of opinion before, on this matter. In effect, it involves which takes "supremacy", physics or math. You claim the former, I claim the latter. > In applied math, > there is no assumption that every object is definable. There is > no reason to. EXACTLY! They simply *are* all definable, so thus, there is no reason to, no reason to make any particular assumption about it! And I extend this view to set theory, sets ought to be definable, so there is no need to, (it may well be counter-productive to) assert this as a formal axiom. It just IS so. We only then need to avoid implicitly (or explicitly, together with claims of reality) other axioms that declare undefinable sets. > >Set theory as applied to physical objects is completely trivial. > No, it's not. Again, the primacy of math over physics. The set theory used in science is utterly trivial compared to that developed in math. > It was *not* at all true, pre-Cantor. Where are you getting that > from? Before Cantor, there was no asumption that all Euclidean > points and lines had names. There was no assumption that every > real number had a finite description. There was no overt assumption, because it was obviosul;y true. See my remarks above about Z, Q, coordinate geometry etc). It was only when peole started to come up withh undefinable objects, (a la Cantor/Zermelo), that others began to notice this profound departure. > At least not since the discovery of transcendental reals. Transcendence has no influence - pi is still perfectly definable. -- Battling Bill
From: Bill Taylor on 4 Nov 2009 00:01 stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > >>> You gave the answer yourself, >>>> in terms of infinte-depth quantifiers > >> But I really don't have an intuition about what infinitely many > >> alternations of quantifiers *mean*, > >Of course you do! > No, I don't. Well, as I say, we are clearly using the above words with differing meanings. > So the whole concept seems insufficiently "tied down". Indeed so. If it had been tied down, we would all be using them now. But such formalization has never been completed. (AFAIK) > Might mean the infinite conjunction of the following statements: > .... > Another meaning is given in terms of strategy functions. > .... > Then in terms of quasistrategies, Your example translations are excellent attempts, but I suspect none of them will capture the precise meaning of what we would want an infinite-depth quantifer to mean, alas. These formalizations are not quite "on", as some previous examples have shown. > My point is that you simultaneously developed 2 different intuitions > about chess (or whatever): (1) Eventually, the game always ends. > (2) There is a winning strategy for one player or the player. > > So I don't see how chess gives you any intuitions about games > that *don't* end. I might well charge you with the same observations about choosing from infinite numbers of sets! > I am curious, though, about the *form* that your intuition > about the existence of a winning strategy took. You really > thought in terms of *functions* from board positions to > your next move? Yes, absolutely. (OC I didn't mentally use the word "function" at that stage of my early high school years,but it would amount to that.) > That's very remarkable to me, if that's > the case. When I played chess, I would pick a certain number > of moves to "think ahead" (and my pitiful brain could never > think more than two or three). On the basis of looking ahead, > I could decide that certain moves are losers. If a win > was sufficiently close at hand, then I could decide that > certain moves were winners. But in the general case, the > best I could say was that certain moves were neither > obviously winners, nor obviously losers. OC this is all very true. But it is observation about the *practical* side of actually playing chess, rather than the theoretical side of what types of positions & moves there might be. > I'm curious as to how the idea that there *must* be a winning > move occurred to you. Presumably, if the win was too many > moves in the future, then you didn't actually *find* this > winning move. Indeed not! Not being a terabyte genius! > So how did you figure that it must exist? Like I say, it just seemed obvious, from considerations of what actually happens in a game; any game. > Well, I would *not* say that my intuition about choice > has anything to do with generalization from finite sets. Well, you obviously believe that, so I doubt anything I could say will budge you. But it is an admission, or claim, which tells its own story. > Now, suppose I have a box that contains a bunch of other > nonempty boxes (it doesn't matter how many). I can imagine > doing the following: for each box, I assign a helper > (that might mean infinitely many helpers, OK OK, I get the picture. It is a mental picture we all had at one time. I daresay I might be able to come up with a picture of "infinitely many game players" moving "infinitely fast" and so on, but what's the point. > There is nothing special about finiteness of the sets > in this intuition. You hope. > >> Using infinite quantifiers, we can't (as far as I know) make > >> the distinction between "Player 1 has no winning strategy" and > >> "Player 2 has a winning strategy". > > >OCN! Because there isn't one,(or so I claim), for tie-free games. > > You don't see that what your saying is circular? Yes, I see your point. But I'm not trying to formally prove anything here, I'm just giving a description of my intuitions, in more than one way. It is VERY like the situation whereby AC was (said to be) justified by reference to "combinatorial" ideas about sets; which (as I noted) turned out to be the same thing said in a different way. Possibly sociologically/didactically useful, but probably no logical help. -- Wittering William
From: Daryl McCullough on 4 Nov 2009 07:10 Bill Taylor says... > >stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > >> >>> You gave the answer yourself, >>>>> in terms of infinte-depth quantifiers > >> >> But I really don't have an intuition about what infinitely many >> >> alternations of quantifiers *mean*, > >> >Of course you do! > >> No, I don't. > >Well, as I say, we are clearly using the above words >with differing meanings. Which words? "infinitely many quantifiers"? >> So the whole concept seems insufficiently "tied down". > >Indeed so. To say that it's not "tied down" to me is synonymous with "we don't know (precisely) what it means". >> Might mean the infinite conjunction of the following statements: >> .... >> Another meaning is given in terms of strategy functions. >> .... >> Then in terms of quasistrategies, > >Your example translations are excellent attempts, but I suspect >none of them will capture the precise meaning of what we >would want an infinite-depth quantifer to mean, alas. Then what *DOES* it mean? That's why I say I don't know what infinitely many quantifiers mean. >> My point is that you simultaneously developed 2 different intuitions >> about chess (or whatever): (1) Eventually, the game always ends. >> (2) There is a winning strategy for one player or the player. >> >> So I don't see how chess gives you any intuitions about games >> that *don't* end. > >I might well charge you with the same observations about >choosing from infinite numbers of sets! I gave you an answer. My intuition about choice has nothing to do with the number of elements in the set. >> Well, I would *not* say that my intuition about choice >> has anything to do with generalization from finite sets. > >Well, you obviously believe that, so I doubt anything >I could say will budge you. I'm talking about *my* intuitions, here. Yes, I doubt that you know more about them than I do. What is special about finite sets is that for such sets a choice function is always *definable*. But definability as a criterion for set existence is *your* thing, not mine. >> Now, suppose I have a box that contains a bunch of other >> nonempty boxes (it doesn't matter how many). I can imagine >> doing the following: for each box, I assign a helper >> (that might mean infinitely many helpers, > >OK OK, I get the picture. It is a mental picture we all >had at one time. I daresay I might be able to come up with >a picture of "infinitely many game players" moving >"infinitely fast" and so on, but what's the point. I certainly am willing to consider such godlike players, and I don't see how the claim that all games are determined follows from it. If your conclusion follows from such considerations, then I would consider that a plausibility argument for AD. As it is, I haven't heard anything that comes close to explaining why you have such an intuition. >> There is nothing special about finiteness of the sets >> in this intuition. > >You hope. I *know*. >> >> Using infinite quantifiers, we can't (as far as I know) make >> >> the distinction between "Player 1 has no winning strategy" and >> >> "Player 2 has a winning strategy". >> >> >OCN! Because there isn't one,(or so I claim), for tie-free games. >> >> You don't see that what your saying is circular? > >Yes, I see your point. But I'm not trying to formally prove >anything here, I'm just giving a description of my intuitions, But you *haven't* given a description of your intuitions. You have simply *stated* that such and such follows from my intuitions, but why it follows, you haven't given any clue. >in more than one way. It is VERY like the situation whereby >AC was (said to be) justified by reference to "combinatorial" >ideas about sets; There is certainly a coherent notion of sets for which AC is true. Are they the real sets, or are they somehow a "nonstandard" notion of sets? I don't know how to answer that. I'm not sure whether the question has a definite meaning. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 4 Nov 2009 07:27
Bill Taylor says... > >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. >All the integers and ratiobnals had them, and also all the reals >we ever use in applications, like pi, e, sin^(-1)(2/7) and so on. > >> Points and lines of Euclidean geometry didn't have names. > >Oh heavens, of course they did! No, they did not! You are completely wrong on this. >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 a line running through A", "A" and "R" are not intended to be unique names. They are labels we are applying *locally*. Tomorrow, I might call a *different* object "A". >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. This is *provable*, Bill. It's a mathematical theorem. Yes, the proof may have not existed before Cantor, but that doesn't mean that it became true when Cantor proved it. It was true beforehand, even if people didn't know it. >All the integers and ratiobnals had them, and also all the reals >we ever use in applications, like pi, e, sin^(-1)(2/7) and so on. That is completely ridiculous. Of course, if a real comes up in an application, then you *give* it a name. That's where "e" came from. That's where "pi" came from. Are you just claiming the tautology that if X is a real, then I am free to name it "Fred", and from then on, I can define it via "that unique real that I have named 'Fred'"? Sure, by that notion, every real can have a name. -- Daryl McCullough Ithaca, NY |