Counterintuitions and the well-ordering theorem
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From: Daryl McCullough on 31 Oct 2009 09:55 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. It's a limit of finite quantifiers, but in what sense of "limit"? I can guess at several *different* things that the infinite quantifier expression *might* mean, but the possibilities don't seem equivalent. So the whole concept seems insufficiently "tied down". For example, to say Ex_1 Ax_2 Ex_3 ... Phi([x_1, x_2, ...]) Might mean the infinite conjunction of the following statements: 1. Ex_1 [x_1} can be extended to an infinite sequence s such that Phi(s) 2. Ex_1 Ax_2 [x_1,x_2] can be extended to a sequence s such that Phi(s) etc. In terms of games, this would be formalized by "there is a strategy on the part of the first player such that following that strategy, he never loses at any finite stage. Following that strategy, he could still lose, but you can only know that by looking at the whole infinitely many moves played by both players. Another meaning is given in terms of strategy functions. If we define play(f,g) to be the infinite sequence resulting from the first player following strategy f, and the second player following strategy g, then the infinite quantifier expression Ex_1 Ax_2, ... Phi([x_1,x_2,...]) might mean Ef Ag Phi(play(f,g)) It could *also* mean the quantifier-switched version: Af Eg Phi(play(f,g)) We can also formulate "strategy" in terms of sets, instead of functions. Those are called "quasistrategies". When a player follows a quasistrategy, the quasistrategy doesn't give a unique move at each stage, but gives a set of possible moves, and the player just picks one. Then in terms of quasistrategies, Ex_1, A_x_2, ... Phi([x_1, x_2, ...]) could be formalizes as: Exists quasistrategy Q1, Forall quasistrategies Q2, forall infinite sequences s: s is consistent with the first player following quasistrategy Q1, and is consistent with the second player following quasistrategy Q2, and Phi(s). It could also be formalized as the quantifier-switched version of the above: Forall Q2, Exists Q1, blah, blah (same as above). >(Else I have a quite different understanding of the words you use.) >The mere fact that you *came up* with the idea, independently of me, >shows you have a fair idea of what it means. The germ of the idea is: generalize quantifiers to the case of infinitely many alternations of quantifiers. That doesn't uniquely pin down what it is we are talking about. >> >> It's not *obvious* that chess or checkers has a >> >> winning strategy; >> >>>It IS.(That is, a winning strategy or a drawing strategy for both) >> >> >It was obvious to me even before I started high school. >> >> I'm sure you never played infinite games before high school. > >I's talking about CHESS, you goose, when I mentioned high school! :) My point is that you simultaneously developed two 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 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? 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. 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. So how did you figure that it must exist? >> I don't agree. Once you introduce infinite games, the intuition >> disappears completely for me. > >Well, I could say exactly the same thing about choice, >which you seem to find no problems with. >How are we to resolve this quagmire!? Well, I would *not* say that my intuition about choice has anything to do with generalization from finite sets. The size of the set has nothing to do with it, as far as I can see. My intuition about sets is that they are like boxes, which can contain things (possibly other boxes). If a box is nonempty, then I can reach in and pull something out of the box. It doesn't matter how many things are in the box, and it doesn't matter whether those things have unique names, or whether they can be well-ordered or whatever. 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, but if I'm going to imagine infinitely many boxes, I can imagine infinitely many helpers). On my signal, each of my helpers reaches into his corresponding box and pulls something out. I put this collection together into one box. In terms of sets, I conclude: If there is a set of nonempty sets, then it is possible to get a new set that contains one element from each set. There is nothing special about finiteness of the sets in this intuition. >> 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? There are two different claims being made: (1) The infinite quantifier notation is adequate to capture all distinctions that are relevant. (2) There is no distinction between "Player 1 has no winning strategy" and "Player 2 has a winning strategy". I agree that *if* you accept (1), then (2) follows, and that *if* you accept (2), then (1) follows. But you can't use (1) as an argument for the truth of (2), and then turn around and use (2) as an argument for the truth of (1). I would think that you need an *independent* argument for (2) (one that doesn't involve infinite quantifiers) in order to justify the use of infinite quantifiers in the first place. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 31 Oct 2009 10:14 Bill Taylor says... > >On Oct 29, 12:02 pm, Herman Jurjus <hjm...(a)hetnet.nl> wrote: > >> > I don't see why you would believe it. >> >> I can't say it better, at this stage, than i already did. Sorry. >> >> (Can you /explain/ /why/ you think the Jordan curve >> should definitely be true? I can't.) > >Now THAT is a wonderful question, and I would dearly love >to hear a response from some of the people it is directed toward! Here's my way of thinking: We have a rubber band (infinitely stretchable and bendable) that is initially a circle. We place the rubber band on the floor, and place a tiny ant inside the circle made by the rubber band. Is there any way for the ant to get out of the region enclosed by the rubber band without breaking the rubber band? (We want this problem to be two-dimensional, so we ignore the possibility of it climbing over or under the rubber band) Clearly not. Now, imagine the ant pushing or pulling spots on the rubber, distorting it away from a circle. Does that help? No. Imagine painting the inner surface of the rubber band red and the outer surface green. No matter how the ant pushes or pulls the rubber band, no matter how he distorts its shape away from a circle, nothing he can do can ever change the fact that he only has access to the red surface, and to be out, he must be able to touch the green surface. Now, to mathematize this picture, we let the rubber shrink to zero thickness and height, so that it can be represented by a closed curve, and let the ant be a mathematical point, and the ant's trajectory in trying to get becomes another curve. For any closed curve that is obtainable by stretching and bending from a circle, there is no nonintersecting curve connecting the inside to the outside. -- Daryl McCullough Ithaca, NY
From: Nam Nguyen on 31 Oct 2009 11:27 Frederick Williams wrote: > Nam Nguyen wrote: >> I'm a bit off topic here but since it's still about >> (counter)intuition so I'm going to ask. >> >> Let 2ZFC be a formal system written in a language >> L with 2 epsilon relations: L = L(e,e'). > > Assuming that e and e' have intended interpretations, what are they? They each would have the same usual set-membership interpretation: but same interpretation of 2 different kinds of mereological concepts. Together they'd make 2ZFC a 2-dimensional set theory. (So potentially we could have n-ZFC). An "rudimentary" example would be in the area of mind-vs-body. A physical body part could be said to be a member of another body part, thus one epsilon symbol. But a thought could be said to be part of (or reflected by) another thought, thus another epsilon symbol. But that's just one intuition. There might be more from others. > >> The axioms >> of 2ZFC are the following: >> >> - A set of axioms that has formulas using only e and >> all together would be a ZFC axioms (for e). >> >> - A set of axioms that has formulas using only e' and >> all together would be a ZFC axioms (for e'). >> >> - An additional axiom: >> >> Axy[(Au[~(uex)] /\ Av[~(ve'y)]) -> (x=y)] >> >> (This says the 2 empty sets be identical.) >> >> How would intuition perceive such a multiple epsilon relation >> sets? How would AC's and AD's connotations be impacted by 2ZFC? > >
From: Nam Nguyen on 31 Oct 2009 11:37 Nam Nguyen wrote: > Frederick Williams wrote: >> Nam Nguyen wrote: >>> I'm a bit off topic here but since it's still about >>> (counter)intuition so I'm going to ask. >>> >>> Let 2ZFC be a formal system written in a language >>> L with 2 epsilon relations: L = L(e,e'). >> >> Assuming that e and e' have intended interpretations, what are they? > > They each would have the same usual set-membership interpretation: but same > interpretation of 2 different kinds of mereological concepts. Together > they'd make 2ZFC a 2-dimensional set theory. (So potentially we could have > n-ZFC). > > An "rudimentary" example would be in the area of mind-vs-body. A physical > body part could be said to be a member of another body part, thus one > epsilon symbol. But a thought could be said to be part of (or reflected by) > another thought, thus another epsilon symbol. > > But that's just one intuition. There might be more from others. Another example would be SR but I'd think in this case instead of having just n epsilon symbols, we might need countably infinite of them. (But haven't given this "example" too much thought though). > >> >>> The axioms >>> of 2ZFC are the following: >>> >>> - A set of axioms that has formulas using only e and >>> all together would be a ZFC axioms (for e). >>> >>> - A set of axioms that has formulas using only e' and >>> all together would be a ZFC axioms (for e'). >>> >>> - An additional axiom: >>> >>> Axy[(Au[~(uex)] /\ Av[~(ve'y)]) -> (x=y)] >>> >>> (This says the 2 empty sets be identical.) >>> >>> How would intuition perceive such a multiple epsilon relation >>> sets? How would AC's and AD's connotations be impacted by 2ZFC? >> >>
From: Bill Taylor on 1 Nov 2009 03:24
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. Now "explicit" and "naming" are very vague terms, so we must determine our philosophy of sets, to try and illuminate them. And mine is - the whole IDEA of a set, is that one can say WHAT THE MEMBERS ARE. 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. So, to be a set, IMHO, means there is (in principle, implicitly, or whatever other caveat one cares about) a determining property for what constitutes an element or not. Now, we all know, there are many ways of doing this, not all of them very explicit at all. But at least, in "normal" math, they exist in the background. But for the sets declared/created by AC, this informal criterion is cast to the winds. There is no longer the slightest hint, pretense or intuition of what is a member and what isn't. Sure, there are plenty of intuitions etc as to what PROPERTIES the set itself may have - but that isn't enough for existence, IMHO, because a set with those properties may or may not actually exist. But until one has a handle on the members, the set does not EXIST. No set-hood without element-hood. Interestingly, this idea, that elements must be (in some sense) logically prior to the set, is at the heart of almost all the methods of addressing "the paradoxes", which are easily seen to be caused by attempts to put a set on the same footing as its members. Sadly, the math world drops this principle like a hot potato once it has done its minimal work. I feel this is bad. Please note:- I'm not against set theorists using whatever bunch of (consistent) axioms and principles and ideas they want, for abstract set theory, just please stop pretending it has any bearing on the existence or otherwise of basic concerns about N, R and their immediate neighborhood. Rant over. But see my sig... ====== W.Taylor ====== ** No set-hood without elementhood! ** No set-hood without elementhood! ** No set-hood without elementhood! |