An uncountable countable set
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From: imaginatorium on 4 Nov 2006 06:30 Tony Orlow wrote: > imaginatorium(a)despammed.com wrote: <snip> > > Virgil was muddling everything up, as usual, but the difference between > > a partial and total ordering is basically whether there are pairs of > > elements for which the order is undetermined. The subset relation is a > > very obvious example, where (for example) the set of reals in [0, 1] > > and the set of prime integers cannot be compared, because neither is a > > subset of the other. > Okay, understood. > > "Bigulosity" has never been sufficiently clearly defined to tell, but > > since you get very steamed up about subsets, and since the only known > > coherent claim is that A proper subset of B -> b(A) < b(B), it's > > extremely unlikely Bigulosity could be extended to become a total > > ordering. > > Bigulosity is not based on the subset relation, but on formulaic > mappings and infinite-case induction. Let me just point you to one of your problems here: do you understand the normal set-theoretic definition of a "function" (or mapping)? (Go look it up; I'm not going to type it here) The problem is that at school you learnt about "functions" - first things like x+1, x^2+4x-7, which you later learnt are called polynomials, then things like sin(x), and possibly sinh(x), Bessel functions, and various other exotic varieties. I imagine these all fall into the category of what you call "formulaic". But the general notion of a mapping is unimaginably bigger than these very specific examples, which were chosen (of course) because they are manageable. Any scheme based on "formulaic" anything is not going to apply to almost all cases (in some reasonable sense). A more mundane problem is that in practice your approach to a problem is to intuit the desirable answer, wave your hands, produce a "formula", then start arguing. But anyway... > > Please compare the Bigulosities of the set of polygons with vertices on > > integral x-y coordinates and the set of topologically distinct > > polyhedra. Show your working. (Of course you don't need to come up with > > an "answer" like "ratio of 5 pi^2", but you need to show how such a > > task would be approached. One of the things you still don't seem to > > have realised is that before anything can be "maths" it has to be > > teachable to other people. I don't think anyone but you has the > > faintest idea what Bigulosity is really supposed to be, except in a > > ragbag of specific cases.) > > It consists of a few ideas, and they cover a lot. Could you be slightly more specific: what is "a lot"? I grant you, "formulaic" sequences of integers, and words in the language on n symbols. Anything else? > I don't know where to > start with your topologically distinct polyhedra. Hmm, not polyhedra, I see. > Why don't you give me a rundown of how YOU compare those two sets? OK; how would I think about "counting" the topologically distinct polyhedra? First I'd observe that I could attack from the faces, the edges, or the vertices (F, E, V; and I could remember that there's an obvious vertex-face duality), so I might choose the faces. I'd assemble a list of the first few, out of curiosity (which has to be the driving force here: what are the conditions for a polyhedron to be its own V-F dual, for example?) The minimum number of faces is obviously 4; not less than 3 faces must meet each vertex, and there must be more than one vertex. I make a little list of the number of possibilities: 4: 1 (tetrahedron) 5: 2 (square pyramid, triangular prism) 6: (pentagonal prism, cube, etc. at this point, cheat http://www.research.att.com/~njas/sequences/table?a=944&fmt=4 ) 7: and so on Now I would notice that by "topologically distinct polyhedron" I am referring to a bounded geometrical object; not only bounded, but also discrete, in the sense that if I have one in my hand I know I can count the vertices. A bit of minor handwaving, and I would see that for any number of vertices there will be a limited number of possibilities for arranging that number of vertices into a polyhedron. So I know that I can pick an ordering scheme, and put all of the polyhedra in it. So I can "count" them, in the sense that I know that with my counting scheme there will not be a polyhedron that escapes it. I also notice that this counting sequence will never end, because there is no maximum to the number of vertices. After all, if there were, then given a polyhedron with that number of vertices I could simply pick any face, construct a pyramid on that face, and get a polyhedron with more than the supposed maximum number of vertices, which proves (by contradiction) that my sequence of polyhedra never ends. (This is all incredibly obvious, but I'm afraid I never know which incredibly obvious bits you still haven't grokked.) So with the constraints of the present audience, I would express this by saying that I can see I can "count" the polyhedra, a process which will account for every one of them, given enough time, but I can also see that the process of counting will never end. It's much messier, but I can do a similar thing for the polygons with vertices having integral x-y coordinates. How could I compare the two sets? I don't know - in both cases it's true that I can find a way of counting them, and that the counting will never end. Since it never ends, it's hard to see how I might think that the counting of the polygons was going to be over before the counting of the polyhedra, or vice versa. It's not even possible to say of a process that never ends that after so many counts the processs is "an appreciable way to completion", because there _is_ no completion. So my answer is rather limited: both sets are (ok, dammit, in normal words) countably infinite; there's no way obvious to me that I could regard either as "less" endlessly endless than the other. What's your answer? You appear to be the one claiming to have "numbers" for counting things when the counting never ends: do you have any here? Or are you happy to accept that these two sets, and the set of pofnats can all be put in 1-1 correspondence, or (in normal words again) have the same cardinality? Perhaps this particular fact raises no objections from your intuition module? Notice that of course there are other things one can say about these sets. For example, the number of polyhedrons with n edges [E(n)] is always l
From: Lester Zick on 4 Nov 2006 12:11 On 3 Nov 2006 14:49:55 -0800, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >Lester Zick wrote: >> On 2 Nov 2006 15:26:22 -0800, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >> >> >Lester Zick wrote: >> >> On 2 Nov 2006 11:21:10 -0800, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >> >> >> >> >Lester Zick wrote: >> >> >> I'm perfectly content to stay right here while I refine my definitions >> >> >> and terminology to bring them into better conformance with standard >> >> >> mathematical usage and more appropriate neomathematical forensic >> >> >> modalities. >> >> >> >> Evasion noted. >> > >> >You just quoted yourself, then commented 'Evasion noted'. Nicely done. >> >> Oh well considering the context you dropped somehow I'm not surprized. > >I looked over the context three times. If I excluded anything that has >a material bearing on what I did include and my remark on it, then you >can mention those exclusions specifically. Posts are not discredited by >the mere fact that they don't include all previous quoted matter. >Nobody has an obligation to include all previous quoted matter in every >one of his or her posts. That only makes you a censor not a mathematician. ~v~~
From: Lester Zick on 4 Nov 2006 12:33 On Fri, 3 Nov 2006 17:13:55 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >Mike Kelly wrote: >> Tony Orlow wrote: [. . .] >> What is the set of balls in the vase at noon in the simplified >> experiment? >> What does "noon does not exist" mean? > >Maybe Tony has given up on this discussion. And maybe this discussion just has a truth value of 0. ~v~~
From: David Marcus on 4 Nov 2006 12:35 imaginatorium(a)despammed.com wrote: > Tony Orlow wrote: > > imaginatorium(a)despammed.com wrote: > > > "Bigulosity" has never been sufficiently clearly defined to tell, but > > > since you get very steamed up about subsets, and since the only known > > > coherent claim is that A proper subset of B -> b(A) < b(B), it's > > > extremely unlikely Bigulosity could be extended to become a total > > > ordering. > > > > Bigulosity is not based on the subset relation, but on formulaic > > mappings and infinite-case induction. > > Let me just point you to one of your problems here: do you understand > the normal set-theoretic definition of a "function" (or mapping)? (Go > look it up; I'm not going to type it here) > > The problem is that at school you learnt about "functions" - first > things like x+1, x^2+4x-7, which you later learnt are called > polynomials, then things like sin(x), and possibly sinh(x), Bessel > functions, and various other exotic varieties. I imagine these all fall > into the category of what you call "formulaic". But the general notion > of a mapping is unimaginably bigger than these very specific examples, > which were chosen (of course) because they are manageable. Any scheme > based on "formulaic" anything is not going to apply to almost all cases > (in some reasonable sense). Another aspect of this problem is that over the last couple of centuries, mathematicians realized that the best way to view functions is as mappings, not as formulas. This led to the modern definition of function. And, the fact that the function is sin, not sin(x) (the latter being a number). It seems Tony thinks the old way of viewing formulas as fundamental is better. I suppose it is possible that Tony has discovered something that a couple of centuries of mathematicians have missed, but until I can see a fleshed-out theory, I remain skeptical. > A more mundane problem is that in practice your approach to a problem > is to intuit the desirable answer, wave your hands, produce a > "formula", then start arguing. But anyway... This would seem to imply that it will be a while until we see a fleshed- out theory from Tony. -- David Marcus
From: Tony Orlow on 4 Nov 2006 14:14
David Marcus wrote: > Tony Orlow wrote: >> David Marcus wrote: >>> Tony Orlow wrote: >>>> David Marcus wrote: >>>>> Tony Orlow wrote: >>>>>> David Marcus wrote: >>>>>>> Tony Orlow wrote: >>>>>>>> I am beginning to realize just how much trouble the axiom of >>>>>>>> extensionality is causing here. That is what you're using, here, no? The >>>>>>>> sets are "equal" because they contain the same elements. That gives no >>>>>>>> measure of how the sets compare at any given point in their production. >>>>>>>> Sets as sets are considered static and complete. However, when talking >>>>>>>> about processes of adding and removing elements, the sets are not >>>>>>>> static, but changing with each event. When speaking about what is in the >>>>>>>> set at time t, use a function for that sum on t, assume t is continuous, >>>>>>>> and check the limit as t->0. Then you won't run into silly paradoxes and >>>>>>>> unicorns. >>>>>>> There is a lot of stuff in there. Let's go one step at a time. I believe >>>>>>> that one thing you are saying is this: >>>>>>> >>>>>>> |IN\OUT| = 0, but defining IN and OUT and looking at |IN\OUT| is not the >>>>>>> correct translation of the balls and vase problem into Mathematics. >>>>>>> >>>>>>> Do you agree with this statement? >>>>>> Yes. >>>>> OK. Since you don't like the |IN\OUT| translation, let's see if we can >>>>> take what you wrote, translate it into Mathematics, and get a >>>>> translation that you like. >>>>> >>>>> You say, "When speaking about what is in the set at time t, use a >>>>> function for that sum on t, assume t is continuous, and check the limit >>>>> as t->0." >>>>> >>>>> Taking this one step at a time, first we have "use a function for that >>>>> sum on t". How about we use the function V defined as follows? >>>>> >>>>> For n = 1,2,..., let >>>>> >>>>> A_n = -1/floor((n+9)/10), >>>>> R_n = -1/n. >>>>> >>>>> For n = 1,2,..., define a function B_n by >>>>> >>>>> B_n(t) = 1 if A_n <= t < R_n, >>>>> 0 if t < A_n or t >= R_n. >>>>> >>>>> Let V(t) = sum_n B_n(t). >>>>> >>>>> Next you say, "assume t is continuous". Not sure what you mean. Maybe >>>>> you mean assume the function is continuous? However, it seems that >>>>> either the function we defined (e.g., V) is continuous or it isn't, >>>>> i.e., it should be something we deduce, not assume. Let's skip this for >>>>> now. I don't think we actually need it. >>>>> >>>>> Finally, you write, "check the limit as t->0". I would interpret this as >>>>> saying that we should evaluate the limit of V(t) as t approaches zero >>>>> from the left, i.e., >>>>> >>>>> lim_{t -> 0-} V(t). >>>>> >>>>> Do you agree that you are saying that the number of balls in the vase at >>>>> noon is lim_{t -> 0-} V(t)? >>>>> >>>> Find limits of formulas on numbers, not limits of sets. >>> I have no clue what you mean. There are no "limits of sets" in what I >>> wrote. >>> >>>> Here's what I said to Stephen: >>>> >>>> out(n) is the number of balls removed upon completion of iteration n, >>>> and is equal to n. >>>> >>>> in(n) is the number of balls inserted upon completion of iteration n, >>>> and is equal to 10n. >>>> >>>> contains(n) is the number of balls in the vase upon completion of >>>> iteration n, and is equal to in(n)-out(n)=9n. >>>> >>>> n(t) is the number of iterations completed at time t, equal to floor(-1/t). >>>> >>>> contains(t) is the number of balls in the vase at time t, and is equal >>>> to contains(n(t))=contains(floor(-1/t))=9*floor(-1/t). >>>> >>>> Lim(t->-0: 9*floor(-1/t)))=oo. The sum diverges in the limit. >>> You seem to be agreeing with what I wrote, i.e., that you say that the >>> number of balls in the vase at noon is lim_{t -> 0-} V(t). Care to >>> confirm this? >> No that's a bad formulation. I gave you the correct formulation, which >> states the number of balls in the vase as a function of t. > > Let's try some numbers. > > t = -1, 9*floor(-1/t) = 9, V(t) = 9. > t = -1/2, 9*floor(-1/t) = 18, V(t) = 18. > > Looks to me like V(t) = 9*floor(-1/t) for t < 0. So, > > lim_{t->0-) 9*floor(-1/t) = lim_{t->0-} V(t). > > So, it does seem that what I said you are saying is what you are saying. > Oh. If you express V(t) that way, that looks correct. I thought it was different before. |