An uncountable countable set
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From: Virgil on 8 Oct 2006 18:23 In article <45296a14(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <452948bf(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > >>> In article <45287184(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>> > > > >>>> Strings with only finite bit positions. > >>> Wrong!!! Strings with only finite bit positions can still have > >>> infinitely many bit positions as there are infinitely many finite > >>> naturals. Finite naturals always have a finite most significant bit > >>> position and only finitely many non-zero digits. > >> Incorrect. If every bit is in finite position, then there is no location > >> in the string where it can be said to have an infinite value. > > > > But there can still be infinitely many bit positions. > > No, that would be a contradiction Not in standard mathematics, but virtually everything in TO-matics is a contradiction. Each n in N is finite but N is not. > since an infinite number of bits > times a positive power of 2 yields an infinite product. Nonsense. Given f:N --> {0,1}, for which n in N is n infinite? > > > > > In order to prohibit infinitely many bit positions one must have a most > > significant bit position as well as a least significant bit postion. > > In the Dedekind sense, yes, That is the only relevant sense in ZFC. > but quantitatively, Garbage! > > > > > A string is only finite if it has two ends, and a digit string without a > > most significant bit position has, at most, one end, and is therefore, > > endless and infinitely long. > > Given quantitative methods, Dedekind's definition of infinite in > insufficient. It may be for TO, but it is admirably sufficient, and equivalent to any other definition of infinite in ZFC. What goes on in TO-matics is irrelevant to mathematics. > > > > To say that each is finite does not require that there be only finitely > > many of them. > > To have a linearly ordered set, with all elements within a finite number > of steps from every other, does. It does not anywhere in mathematics, and whether it does in TO-matics is irrelevant in mathematics. > > > > > That particular conflation of senses has been TO's stumbling block from > > the beginning. he cannot conceive of an endless sequence of naturals > > without having some member itself being endless, despite the many > > contrary examples such as {1/n: n in N} > > I have specifically addressed such cases. You have consistently been a > numbskull in forgetting the fact that each natural is separated from its > closest neighbors by 1 unit. The distance between is irrelevant in mathematics, as long as it is not zero, and what goes on in TO-matics is of no import in mathematics. > > > > Call it a TO-number, if you like, but you do not get to call things > > numbers for others. > > I can declare it as a Peano set. To can, and does, declare all sorts of things, but his declarations only carry weight in TO-matics, never in mathematics. > It's stupid to argue that binary numbers aren't numbers. Do you argue > that ....0000 is anything other than 0? It is whatever To wants it to be, but only in TO-matics, not in mathematics. > > > > Peano has considerably more clout than TO. At least among mathematicians. > > Peano has my respect. If Peano were still alive, I very much doubt that the feeling would be mutual.
From: Virgil on 8 Oct 2006 18:34 In article <45296f69(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> David R Tribble wrote: > >>> Virgil wrote: > >>>>> Except for the first 10 balls, each insertion follow a removal and with > >>>>> no exceptions each removal follows an insertion. > >>> Tony Orlow wrote: > >>>>> Which is why you have to have -9 balls at some point, so you can add 10, > >>>>> remove 1, and have an empty vase. > >>> David R Tribble wrote: > >>>>> "At some point". Is that at the last moment before noon, when the > >>>>> last 10 balls are added to the vase? > >>>>> > >>> Tony Orlow wrote: > >>>> Yes, at the end of the previous iteration. If the vase is to become > >>>> empty, it must be according to the rules of the gedanken. > >>> The rules don't mention a last moment. > >>> > >> The conclusion you come to is that the vase empties. As balls are > >> removed one at a time, that implies there is a last ball removed, does > >> it not? > > > > Please state the problem in English ("vase", "balls", "time", "remove") > > and also state your translation of the problem into Mathematics (sets, > > functions, numbers). > > > > Given an unfillable vase and an infinite set of balls, we are to insert > 10 balls in the vase, remove 1, and repeat indefinitely. In order to > have a definite conclusion to this experiment in infinity, we will > perform the first iteration at a minute before noon, the next at a half > minute before noon, etc, so that iteration n (starting at 0) occurs at > noon-1/2^n) minutes, and the infinite sequence is done at noon. The > question is, what will we find in the vase at noon? With just this > information, it is rather clear that this constitutes an infinite > series, +(10-1)+(10-1)+(10-1)+..., or +9+9+9+..., which clearly diverges > to some infinite number, of balls, that is. Not necessarily. If balls have, like electrons, no individual identity, one can argue as TO does, but if they have individual identities, the result at noon can depend on the order in which the balls are removed. > > However, the gedanken as described contains other information. Each ball > is marked with a unique natural number, so that there is a ball for > every natural. Given this identification scheme, we are given two > possible sequences of inserting 10 and removing 1 which differ only in > the labels on the balls. In both, we insert 1-10 on the first iteration, > 2-20 on the second, etc, or balls 10n+1 through 10n+10 for each > iteration (starting at 0). The difference is in the removal part of the > iteration, where in the first case we remove 1, then 2, then 3, etc, or > ball n+1 in iteration n. In the second case we remove ball 1, then 11, > then 21, etc, or ball 10n+1 in iteration n. Because transfinite set > theory makes no distinction between aleph_0 and aleph_0/10, no > distinction is made between the number of iterations it takes to add all > the balls, and how many it takes to remove them. Erroneously, the > unending sequences of required iterations are considered the same. So, > in the first case, it is reasoned that all balls are inserted before > "noon" and all balls are removed before "noon", and therefore nothing > remains at noon. In the second case, balls 2 through 10 clearly remain, > and all n+2 through n+10, for all n in N, which is an infinite set of > balls remaining. So, apparently, one might surmise that you could > perform experiment 2, have some infinite number of balls in the vase, > and then switch the labels around afterwards and make them all > disappear. It's just silliness. TO's misrepresentation is the "mere silliness" > > If the answer is clear with limited information, then additional > information is irrelevant. But the answer depends on two things, the individuality of the balls, and given that they are individual, the order of removal. If TO assumes that they have no individuality, as he does, then giving them individuality changes the problem. And that different problem may have a different solution. > With them, well, we have a bunch of hocus > pocus, and I can't even really give any mathematical expression to that. I can: Let A_n(t) be equal to 0 at all times, t, when the nth ball is not in the vase, 1 at all times, t, when the nth ball is in the vase, and undefined at all times, t, when the nth ball is in transition. Note that noon is not a time of transition for any ball, though it is a cluster point (limit point) of such times. Let B(t) = Sum_{n in N} A_n(t) represent the number of balls in the vase at any non-transition time t. B(t) is clearly defined and finite at every non-transition point, as being, essentially, a finite sum at every such non-transition point of the finitely many non-zero values. Further, A_n(noon) = 0 for every n, so B(noon) = 0. Similarly when t > noon, every A_n(t) = 0, so B(t) = 0 > It was easy without the labels. "Pay no attention to the man behind the > vase. I am the Great and Powerful Aleph" ;) And it is a different problem without the labels.
From: David Marcus on 8 Oct 2006 19:13 Ross A. Finlayson wrote: > So, ZF has been examined by many and by some found lacking. > > ZF is inconsistent. Cantor's set theory was claimed by him to have a > universe in it, and there's a difference between NBG, and NBG with > classes, there's no universe in ZF, nor ZFC. Please state a specific inconsistency in ZF. -- David Marcus
From: Ross A. Finlayson on 8 Oct 2006 20:31 David
From: Ross A. Finlayson on 8 Oct 2006 20:34
Hi Dave, Have you heard of Burali-Forti, a "paradox"? There's no set of ordinals nor cardinals in ZF. There is a universe, and there's not in ZF. There are smaller sets than the universe that are sufficient for many statements, in terms of quantifying over their elements. There is quite regular use of the word universe in the practice of naive set theory. Basically it's an anti-foundation mindset, and then some. Quantify over sets: the result is not a set. There's nothing else over which to quantify. So, foundation is an exercise in vacuity. In set theory, where basically "pure" set theory means every item is a set, there are only sets, every thing is a set, and there's reason to consider why the transfer principle holds true, that everything is a set. Basically I see a contradiction in the existence of the universal quantifier without a universe. That gets into things along the lines of that infinite sets are irregular, and then some. Ross |