An uncountable countable set
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From: Tony Orlow on 1 Oct 2006 16:30 David R Tribble wrote: > stephen wrote: >>> In TO-matics, it is also possible to end up with >>> an empty vase by simply adding balls. According to TO-matics >>> ..1111111111 = 1 + 1 + 1 + 1 + ... >>> and >>> ..1111111111 + 1 = 0 >>> >>> So if you just keep on adding balls one at a time, >>> at some point, the number of balls becomes zero. >>> You have to add just the right number of balls. It is not >>> clear what that number is, but it is clear that it >>> exists in TO-matics. > > Tony Orlow wrote: >> You drew that from my suggestion of the number circle, and that ...11111 >> could be considered equal to -1. Since then, I looked it up. I'm not the >> first to think that. It's one of two perspectives on the number line. >> It's either really straight, or circular with infinite radius, making it >> infinitesimally straight. The latter describes the finite universe, and >> the former, the limit. But, you knew that, and are just trying to have fun. > > You are drawing geometric conclusions that are not warranted. > The Projective Real Line is simply R U {oo}. Adding unsigned oo > to the set allows certain arithmetic operations to be performed > that are undefined in the regular real set. > > But simply adding the limit point oo to the set does not actually make > it a "circle", because oo has no predecessor or successor, and > certain operations like oo+1 and oo+oo are still meaningless within > the set. You do realize that my statements involve a considerable amount of personal reflection, don't you? There is more to the number circle than "proven". In the binary number circle, "100...000" is both positive and negative infinity. > > See: > http://en.wikipedia.org/wiki/Projective_line#Real_projective_line > http://en.wikipedia.org/wiki/Division_by_zero#Real_projective_line > http://en.wikipedia.org/wiki/Extended_real_number_line > > You obviously have something else in mind when you talk about > the "number circle". Perhaps you could actually define it some time? > There are a number of concepts in this area. Get acquainted with those pages, think, and come back and talk.
From: Tony Orlow on 1 Oct 2006 16:32 Han.deBruijn(a)DTO.TUDelft.NL wrote: > stephen(a)nomail.com schreef: > >> I suppose I should clarify this. You can approach the infinite >> using the the limit concept, but you always have to be careful >> when using limits, and you have to be precise about what you >> mean by the limit. > > Okay. But the point is whether there exist infinities that can _not_ > be approached using the limit concept. Obviously they exist, because > how can we approach e.g. the Continuum Hypothesis by employing limts? > > Han de Bruijn > Ummm, we can't. We require IFR. Tony
From: Virgil on 1 Oct 2006 16:43 In article <45201786(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <451df5af(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > >>> In article <451dd5cf(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>> > >>>> Virgil wrote: > >>>>> What Randy is avoiding is TO's fallacious insistence that there must be > >>>>> a last ball removed if one is ever to achieve a state where the balls > >>>>> are all removed. > >>>> When the gedanken specifically states that only one ball is removed at a > >>>> time, what is fallacious about that statement? > >>> > >>> It omits that it is not just any ball which is removed. If one changes > >>> which one is to be removed, one changes the game and the result. > >> Omitting an irrelevant detail does not make a statement fallacious. > > > > Violating one of the rules is not the same as omitting an irrelevant > > detail. And statements which force those violations are fallacious. > > Saying the balls have labels is not a "rule". You are following rules > about the labels which are not stated in the problem itself. I do not know what game TO is playing, but the one presented to me was: Given an empty vase and bijection,f, from {1,2,3,...} to a set of balls. At time 1/2^(n-1) seconds before noon put balls f(10*(n-1)+1) through f(10*(n-1)) into the vase and at time halfway between 1/2^(n-1) seconds before noon and 1/2^(n) seconds before noon remove ball f(n). One was then asked about the contents of the vase at noon. > There were two presented in the original post, by a wonderful > not-even-neophyte with a friend, one of which supposedly emptied, and > the other of which did not, all depending on the labels. What happens if > we don't know what ball we're removing? Can you answer than question? if we don't know we don't know. > No, because much to Han's chagrin, you will not even consider > statistical methods over this set. The original problem did not allow for statistical methods. > > > > >> Only one ball is removed at any > >> time, immediately preceding which 10 balls have been added. You had to > >> have -9 balls in your vase for that to have occurred. > > > > How does 10 - 1 come out -9? > > final removal: x-1=0 > final addition: y+10=x Which in standard arithmetic comes out with 10 - 1 = +9, not 10 - 1 = -9. But TO's arithmetic is funny in lots of ways. > > > > It follows the rules of the gedankenexperiment precisely and exactly. > > Except for order of operations, but what's that matter, right? How have I violated any "order of operations? It is TO who keeps trying to do that.
From: Tony Orlow on 1 Oct 2006 16:54 Han.deBruijn(a)DTO.TUDelft.NL wrote: > Randy Poe wrote: > >> Tony Orlow wrote: >>> Randy Poe wrote: >>>> Tony Orlow wrote: >>>>> Han de Bruijn wrote: >>>>>> Virgil wrote: >>>>>> >>>>>>> In article <d12a9$451b74ad$82a1e228$6053(a)news1.tudelft.nl>, >>>>>>> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: >>>>>>> >>>>>>>> Randy Poe wrote, about the Balls in a Vase problem: >>>>>>>> >>>>>>>>> It definitely empties, since every ball you put in is >>>>>>>>> later taken out. >>>>>>>> And _that_ individual calls himself a physicist? >>>>>>> Does Han claim that there is any ball put in that is not taken out? >>>>>> Nonsense question. Noon doesn't exist in this problem. >>>>>> >>>>> That's the question I am trying to pin down. If noon exists, that's when >>>>> the vase supposedly empties, >>>> Why does the existence of noon imply there is a time >>>> which is the last time before noon? >>>> >>>> It doesn't. >>> I never said it did. When did I say that? >> I was responding to Han, who said that "If noon exists, that's when >> the vase empties". > > HdB never said such a thing. Sorry. At the time, perhaps I cut and pasted, but can't see now from where. However, your retort to Virgil's challenge was that his logic was incorrect because "Noon doesn't exist in this problem." That implies that if noon existed, he might have a point. > >> Noon exists. > > Sure. By dogma. Randy Pope is infallible. See? Your position now is that noon doesn't exist. > >> But in order for the vase to transition from not-empty >> to empty, there would have to be a last non-empty >> moment. That would be the last time before noon. > > Aha. As clear as a klontje. What's that, a type of mud? > >>> I will offer this simple >>> logical argument. If the vase ever became empty, it would be because one >>> ball was removed, >> Hence my continued statement that the vase does not >> "become empty". It is non-empty at certain times and >> empty at others. There is no transitional moment. > > Nature does not jump, Leibnitz said. Leibniz lived in reality. Is that a necessary requirement? > >> Noon is the first moment at which the vase is empty. > >> But noon is not the transitional moment. There's no >> time just before noon where the transition happened. > > Wow ! And _that_ calls himself a physicist ... > > Han de Bruijn > The time was not before that moment, but it was not after it either. Therefore we cannot make any judgments as to when it happened. That's transfinitology for you. Tony
From: Virgil on 1 Oct 2006 17:02
In article <45201ede(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > mueckenh(a)rz.fh-augsburg.de wrote: > > Dik T. Winter schrieb: > >> It is *you* who insists it is a number. In most of my communications > >> with you I put the word number in quotes, because it depends on how you > >> interprete it on whether it is a number or not. > > > > It is me who insists that it is not a representation of a number. > > Well, Wolfgang, that sets us apart, though I agree it's not a "specific" > number. It's still some kind of quantitative expression, even if it's > unbounded. Would you agree that ...333>...111, given a digital number > system where 3>1? Most digital number systems in which 1 and 3 are numbers do not recognize ...111 or ...333 as numbers, so the question is meaningless. > > > > You could come up with that argument for arbitrary numbers, but not for > > unary numbers. what you require is impossible. Either 0.111... is > > larger than any number of the list, then you have to give a position > > which is not covered by a list number or not. > > It's at the "end", you know, the aleph_0th place. :) TO, should write on the board 100 time, the set of naturals does not contain an end (last) natural. > > > > > Take into account that also Cantor's diagonal argument cannot be > > executed in unary representation. The unary representation is capable > > of modelling rational numbers like 0.111 / 0.11111 and even some > > irrational numbers like sqrt(0.11). But it is not capable of modelling > > Cantor's diagonal argument. And it is in clear contradiction with your > > requirement of completely indexing but not covering 0.111... by the > > list numbers. > > That's because Cantor's diagonal argument concerning the reals is not > about the reals, but about symbolic language, as he stated originally. I > was later applied to the reals. Right, for once. > > > > >> > > > And why can't there be more than one number > >> > > > with > >> > > > infinitely many digits? You cannot answer these questions because > >> > > > already one infinite set is a contradiction. > >> > > > >> > > No, I can not answer this question because I have no idea what you > >> > > mean > >> > > with a number with more than omega digits. Consider K = 0.111... . > >> > > What > >> > > is K+1? Can you provide a definition (as I did for K)? > >> > > >> > k + omega is omega. And -k + omega is omega too. There is no well > >> > defined set. > >> > >> In what way is that an answer to my question? Do you understand that > >> 1 + omega = omega != omega + 1? > >> And (as far as I know) -k + omega is not defined for positive k. (With > >> the ordinals addition is defined only between ordinals.) And is not commutative even then. > > > > Of course you can set up a bijection beween the sets > > k + omega = {-k, -k+1, -k+2, ... , 0, 1,2,3,...,} and > > -k + omega = {k+1, k+2, k+3, ...}. > > But that does not mean that both sets have the same number of elements. If one can set up an order isomorphism between two ordinals then they ARE the same ordinal. > > Cardinality is a weak measure of size for infinite sets, the operative > word here being "measure". It is a property of sets, finite or infinite, but until we have a definition of what TO considers a "measure" we have no way of knowing what he is talking about, if anything. Note: there are several different definitions of "measure" in different mathematical contexts, but I suspect that TO's does not conform to any of them. > > Subtract 1 from 0 in binary. In what binary system? There are too many binary systems extant for that to have any universal meaning. |