An uncountable countable set
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From: Tony Orlow on 19 Aug 2006 12:54 imaginatorium(a)despammed.com wrote: > mueckenh(a)rz.fh-augsburg.de wrote: >> Franziska Neugebauer schrieb: >> >> >>>>>> Therefore there are not infinitely many difference[s] of 1 >>>>>> between natural numbers. >>>>> Your consequent is proven false (see below). Therefore your >>>>> implication is false, too. >>>> You are in error. >>> Where precisely is the error? >> The assertion that infinitely many differences of 1 can be provided by >> finite natural numbers. > > You've been repeating this moronic nonsense for so long it's obviously > hopeless, but just answer me a simple question. In your view of these > "finite natural numbers", I presume 1 is included, and 2, and 3, and so > on. If you were to simply count them, 1, 2, 3, 4, and so on, never > stopping unless you came to an end, do you in fact think you _would_ > reach an end? Any suggestions as to what this end would look like, > given that it would entail a natural number such that adding 1 somehow > failed to happen. > > Brian Chandler > http://imaginatorium.org > Hi Brian, WM, Franziska - By the Dedekind definition you have an infinite set of finite naturals, because there is no discernible end to the finite naturals, and therefore it can be bijected with a proper subset, such as the even naturals. However, WM is only saying the same "moronic nonsense" I've been trying to explain for well over a year. When the set is viewed quantitatively and formulaically, rather than with this rather skewed set-theoretical approach, it's quite obvious that what WM is saying is correct. If the count of elements from 1 up to and including n is always equal to n, which is clearly the case, then aleph_0 is the size of the set of naturals from 1 up to and including aleph_0, whatever aleph_0 is. If your set is the set of all finite naturals, and its size is claimed to be aleph_0, then in essence you are claiming that this is the largest finite natural, despite all your proofs by contradiction that rely on this value being nonexistent. When there are no two elements in the ordered set with infinitely many between them, then the set may be "Dedekind" infinite according to set theory, but in terms of the NUMBER of elements in the set, that quantity is finite. It's time to develop the infinite number systems. Oh, that's right. I already did most of that. WM - you should check out the T-riffics. :) TO
From: Virgil on 19 Aug 2006 12:55 In article <ec7d7r$ath$1(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > Virgil wrote: > > In article <ec5k1i$4gr$2(a)ruby.cit.cornell.edu>, > > Tony Orlow <aeo6(a)cornell.edu> wrote: > > > > > >> By the way, hello! Nice to see you all again, I think. > > > > Does Tony actually think? > > Nice to see you too, Virgil. I don't think I thought not.
From: Tony Orlow on 19 Aug 2006 12:57 mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > >> But I now understand that the series: >> sum{n = 1 .. oo} (-1)^n/n >> does not converge according to your logic. > > For every natural number n we have n/n = 1. But the notation lim n --> > oo is not clear. >> > >> > |{1, 2, 3,..., 2n}| / |{2, 4, 6, ..., 2n}| = 1 + remainings >> > where the remainings are |{n+1,..., 2n}| / |{2, 4, 6, ..., 2n}| = 1 >> >> That makes no sense. As I read it, we have: >> |{1, 2, 3,..., 2n}| / |{2, 4, 6, ..., 2n}| = 1 + remainings = 1 + 1 = 2. > > ? > 1 + remainings = 1 + 1 = 2 > Which error did you see? > >> But I will allow an error here. But I was thinking about: >> |{1, 2, 3, ...}| / |{2, 4, 6, ...}| >> which might be written as: >> lim{n -> oo} |{1, 2, 3, ..., n}|/|{2, 4, 6, ..., 2n}| > > = 1 by mathematics and by set theory. > >> or as: >> lim{n -> oo} |{1, 2, 3, ..., 2n}|/|{2, 4, 6, ..., 2n}| > > = 2 by mathematics but 1 by set theory. > > Regards, WM > =2 by the Inverse Function Rule as well. :) TO
From: Virgil on 19 Aug 2006 13:00 In article <ec7fn3$efv$1(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > However, WM is only saying the same "moronic nonsense" I've > been trying to explain for well over a year. Which may make it a folie a deux, but no less a folie.
From: Tony Orlow on 19 Aug 2006 13:06
mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > >> In article <1155900279.092888.72790(a)m79g2000cwm.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: >> > Dik T. Winter schrieb: >> ... >> > > > > Let's have an easier example. The blocks are 1/2^n high and >> > > > > 1-1/2^n wide. >> > > >> > > > > When you make a stair of them, when you complete, you have a >> > > > > stair with height 1 and width 1. But there is neither a block >> > > > > that is 1 wide, nor a block that is at height 1. On the other >> > > > > hand, for every positive k, there are blocks that are beyond the >> > > > > height of 1-k and beyond the width of 1-k. Both in height and >> > > > > in width the blocks just do not reach the boundary line. And >> > > > > note that in both "infinity is reached". >> > > > >> > > > What you argue is a potential infinity. Infinity, aleph_0, here reduced >> > > > to 1, is present, actually existing, according to Cantor, in width. All >> > > > steps do exist. But, according to him, infinity, here 1, is not >> > > > present in height. >> > > >> > > It is, also according to him. Both in width and in height. >> > >> > Wrong. "Die unendliche Menge der endlichen Zahlen." >> > The set is infinite. The numbers are all finite. >> >> Sorry, I do not understand. There is no block at height 1 and there is >> no block with width 1. The smallest square that contains the complete >> stair has height and width 1, so we can say that the complete stair has >> height and width 1. But the top and right edge are never reached. You >> could say they are "asymptotes". > > Correct. That is potential infinity. That is just what I claim. Width > 1 is *not* reached (at least not without reaching 1 in height). > But according to Cantor width 1 *is* reached while height 1 is not > reached, infinity in number does actually exist infinity in size does > not exist. Yes, this is a contradiction, when viewed quantitatively. It can only be resolved by relegating countable infinity to a finite but unbounded status, or allowing for actual infinite quantities, such as the number of points or reals in the unit interval. I opt for number 2. (Okay Virgil, I set you up. Go ahead an make a "number 2" joke. You deserve it. ;) > You said that a set exists if all its elements do exist. If so, then > all natural numbers do exist and make up an infinite number without one > of them being infinite. Inductively provable: Any set of consecutive naturals starting at 1 always has its largest element equal to size of the set. Is aleph_0 one and not the other? Contradiction. WM's correct. Of course he's still wrong about infinity not existing, but that's okay. He's thinking, and being the cat on the dog pound fence. Meow!! :) > > Regards, WM > :) TO |