An uncountable countable set
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From: Tony Orlow on 20 Aug 2006 21:11 Virgil wrote: > In article <ec9th3$a4n$2(a)ruby.cit.cornell.edu>, > Tony Orlow <aeo6(a)cornell.edu> wrote: > > >> Well, then, since the size of the set in width is always equal to the >> max of the set in height, then it would seem that aleph_0 is the LUB on >> the set size as well, but that the set never actually achieves this >> size. Once it does have some aleph_0th element, that element would have >> to be.....aleph_0! But that's not allowed in the set, so how can it be >> the size of the set? > > The same way that the size of the open interval from a to b, with a < b, > is of length equal to the difference b-a even though neither a nor b is > a member of the set. Oh yes! You mean the way sum(n=1->oo: 9*10^-n) has a LUB of 1, but never reaches it. Well, good point. Very good. That's true. ;) Tony
From: Tony Orlow on 20 Aug 2006 21:20 Virgil wrote: > In article <ec9u24$at1$1(a)ruby.cit.cornell.edu>, > Tony Orlow <aeo6(a)cornell.edu> wrote: > >> Virgil wrote: > >>> Actually, I would bet against N being a member of a subset of N, myself. > > >> I'd bet against you. > > Does TO really wish to bet that N IS a member of a subset of N? > > > >> Oh, Virgil, please try rereading this. S={{1},{1,2},{1,2,3},...}, >> N={1,2,3,...}. It's a member of S. > > Since every member of S, at least as represented above, contains a > largest member but TO's N does not contain a largest member, why is > that N a member of that S? > > Is TO off on his "largest natural" kick again? > \ Because it's a set of consecutive naturals starting at 1. > >>> > >>>> Yes, and it's inductively provable likewise that the size of a successor >>>> ordinal is one greater than its max element. That would make aleph_0 1 >>>> greater than the max finite natural. >>> >>> There goes TO's delusion again. What ever makes TO think that a process >>> which cannot not end must end? >>> >> If it doesn't end what makes Virgil think he can tack a count on it? > > I can "tack on " bijections of endless sets by defining those bijections. > And in standard mathematics that is one way of counting. > But since there is no end to counting, there is no particular count. > >> am not saying there IS a max finite ordinal. I am saying that that's >> what the assumption of aleph_0 implies when one is allowed to apply >> inductive proof to the infinite case. > > The only acceptable forms of induction in ZF or NBG are finite > induction and transfinite induction, neither of which has TO shown he > knows how to apply. But I am explicitly suggesting the alternative, that if f(x)>g(x) for all x>y, then it is true for all infinite x. This leads to a clearer set of conclusions. >>>> The only way around this is the >>>> declaration of the limit ordinals, but that's just a philosophical >>>> monkey wrench. >>> it certainly throws a monkey wrench in TO's gearbox, but ZF, ZFC and NBG >>> are unaffected by non-existence of TO's alleged maximum finite natural. >> It creates the problem. > > Not in ZF, ZFC or NBG, and problems occurring in the isolation of > TOmania are irrelevant to mathematics. > Very relevant to mathematical truth, if not the current state of affairs. > >> Nowhere have I ever claimed there IS a max >> finite natural. I am saying that's the implication of your theory, >> unless you throw out induction when n is infinite. > > For those cases we can sometimes invoke transfinite induction, but it > doesn't give any of the results that TO wants. No, transfinitology doesn't satisfy my spiritual needs. >>>>> There is no equivalent model in which the ordinal of each element is >>>>> equal to the element itself as that would require sets to be members of >>>>> themselves, which is impossible for ordinals. > >>>> Don't confuse max with size. Relate them formulaically. >>> Does TO mean that a maximum does not have to be larger than those things >>> of which it is allegedly the maximum? >>> >>> If not, then what does TO mean by "max"? >> Max value in the set vs. size of the set. > > But TO is always going on about how for his version of the naturals they > are the same. When the naturals begin at 1, yes. And otherwise there is an offset involved. And if the inductive set is defined with something other than increment as successor, then there is a different mapping function from the naturals, and this mapping function needs to be inverted and applied over the range under consideration. That's what I'm always going on about. > > What does one do for sets which are not ordered, or ordered sets which > have no max?, both of which are abundant. Examples? Symbolic sets (languages) need not be ordered to be measured, necessarily, and sets which have no distinct range have no distinct size, unless the elements themselves have no relative measure. TO
From: Dik T. Winter on 20 Aug 2006 21:25 In article <ec8if0$rtd$1(a)ruby.cit.cornell.edu> aeo6(a)cornnell.edu writes: > Dik T. Winter wrote: .... > > > Tony Orlow wrote: > > > > A circle is a regular polygon with an infinite number of infinitesimal > > > > sides. The formula for the angle of the vertices of a regular > > > > polygon of n sides is 2*pi*(1/2-1/n). The limit of this formula > > > > as n->oo is pi. > > > > Tony, I see you did agree with Wolfgang Mueckenheim in this thread. But > > I should warn you. According to WM neither pi nor that limit do exist. > > > > (Gives me a deja vu about JSH agreeing with EEE. While the latter proved > > that FLT was false and the former proved that it was right.) > > Oh, how I love being compared to JSH! You misunderstand. What struck me was that again two people in violent disagreement with each other are complimenting each other. > Indeed, I agree with WM's logic concerning the identity relationship > between element count and value in the naturals. He's quite correct in > that regard. Well, you and he are not. The logic is flawed. > I'm a devoted post-Cantorian delver into > infinity. While he takes the argument put forth as proving that the set > of naturals is finite, he does so with the assumption that infinite > naturals cannot exist. And his proof is not a proof. > For my part, I agree that the set of finite > naturals is finite, though unbounded, In that case you are not using standard mathematical terminology. I have no idea what a finite but unbounded set is. > but that there exists an infinite > set of naturals, which includes infinite values. That is alright with me, only, do not call them naturals, because that is extremely confusing. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 20 Aug 2006 21:32 In article <ec9no7$44k$1(a)ruby.cit.cornell.edu> aeo6(a)cornnell.edu writes: > Virgil wrote: > > In article <ec8gln$oc9$1(a)ruby.cit.cornell.edu>, > > Tony Orlow <aeo6(a)cornell.edu> wrote: > > > >> Dik T. Winter wrote: > >>> Let L be the list of all natural numbers > >>> Let K be the sequence of digits such that for each p in L, K[p] = 1. > >>> What is *wrong* with that definition? > >> > >> Sorry to interject, but could you tell me how long a sequence K is? If > >> it's finite, then you only have a finite list L, and if it's countably > >> infinite, then you have an uncountable list of naturals? > > > > How does having a countable list of naturals require an uncountable list > > of naturals? > > Is K a list of natural numbers? Pay attention. K is the sequence of > digits required to represent the naturals in binary. Yes, pay attention. It is no such thing. It is defined as the sequence of digits where each digit position can be indexed by a natural number. There is *no* binary involved, because the only digits used are 1. The whole discussion is about unary representation. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 20 Aug 2006 21:36
In article <ec9pqd$6iu$1(a)ruby.cit.cornell.edu> aeo6(a)cornnell.edu writes: > Dik T. Winter wrote: > > In article <ec8gln$oc9$1(a)ruby.cit.cornell.edu> aeo6(a)cornnell.edu writes: > > > Dik T. Winter wrote: > > ... > > > > What is the wrongness of the definition? Let me refrase: > > > > Let L be the list of all natural numbers > > > > Let K be the sequence of digits such that for each p in L, K[p] = 1. > > > > What is *wrong* with that definition? > > > > > > Sorry to interject, but could you tell me how long a sequence K is? > > > > Can you not determine it? I would think it follows immediately from the > > definition. > > > > > If > > > it's finite, then you only have a finite list L, and if it's countably > > > infinite, then you have an uncountable list of naturals? > > > > What is your problem? From the definition I would say that the size > > of the sequence of digits in K is equal to the size of the sequence > > of natural numbers in L. > > Here is my problem. Given normal combinatorics, there are 2^n unique > bit strings of length n. You have not followed the discussion and so you are telling nonsense. The discussion was about unary representations, so there is exactly one unique strings of length n. Re-read the complete discussion and after that come back again. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |