An uncountable countable set
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From: Tony Orlow on 21 Aug 2006 22:00 Virgil wrote: > In article <ecb18f$sgk$1(a)ruby.cit.cornell.edu>, > Tony Orlow <aeo6(a)cornell.edu> wrote: > >> Virgil wrote: >>> In article <ec9th3$a4n$2(a)ruby.cit.cornell.edu>, >>> Tony Orlow <aeo6(a)cornell.edu> wrote: >>> >>> >>>> Well, then, since the size of the set in width is always equal to the >>>> max of the set in height, then it would seem that aleph_0 is the LUB on >>>> the set size as well, but that the set never actually achieves this >>>> size. Once it does have some aleph_0th element, that element would have >>>> to be.....aleph_0! But that's not allowed in the set, so how can it be >>>> the size of the set? >>> The same way that the size of the open interval from a to b, with a < b, >>> is of length equal to the difference b-a even though neither a nor b is >>> a member of the set. >> Oh yes! You mean the way sum(n=1->oo: 9*10^-n) has a LUB of 1, but never >> reaches it. Well, good point. Very good. That's true. ;) > > It is only the partial sums, a_k = sum(n=1->k: 9*10^-n), k in N, which > never reach 1. > The limit actually is EXACTLY 1, by reason of how limits are defined. The limit is the least upper bound of the sequence of values, Virgil. The limit of 0.9999... is 1. :)
From: Tony Orlow on 21 Aug 2006 22:15 Virgil wrote: > In article <ecb1o9$h$1(a)ruby.cit.cornell.edu>, > Tony Orlow <aeo6(a)cornell.edu> wrote: > >> Virgil wrote: > >>> Is TO off on his "largest natural" kick again? >>> >> Because it's a set of consecutive naturals starting at 1. > > It doesn't matter where it starts, the issue is whether it ends with a > "largest natural". In standard mathematics it does not. > TO seems to switch postions erratically on the issue. > Let's put it this way. The max and the size of the set are equal. If one exists, the other exists as well, since it's the same. If one does not exist, then neither does the other. I have never said I think there is a largest natural. I have said that some of your assumptions lead to that conclusion. That's not to say I agree with those assumptions. >>>>> >>>>> 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. > > If one can prove that some function is a bijection between some ordinal > and the set to be counted, the counting has been completely achieved by > that proof, at least for those not off in TO's never never land of > TOmania. For those that ascribe to the von Neumann model of the naturals, sure. I don't. >>>> 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. > > > Which is sufficient justification for refusing to allow TO's extension > of induction beyond standard induction. Or, alternatively, sufficient justification for rejecting transfinitology and the notion of a specific boundary between finite and infinite. >>> 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. > > TO's alternative also leads to inconsistencies with the status quo ante, > which is sufficient reason for rejecting it. Huh? You mean, if I reject your axiom system, and concoct another, the fact that yours came first means that contradictions between the two are the fault of mine? Hmmmm..... Nah. If my ideas all fit together and provide at least as robust a system as the status quo, then seniority really doesn't count. >>>>>> 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 > > As TO has shown no talent for recognizing truth, and even seems to have > a talent for avoiding it, his judgements on where mathematical "truth" > lies are, at best, untrustworthy. > Define "truth". > >>>> 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. > > For spiritual needs, one needs something a little less literal minded > than mathematics. What makes you think mathematics is literal? It's certainly not concrete, but very abstract. What the numbers refer to is anyone's guess when they're doing the algebra. That's the beauty. It only becomes literal upon application. > >>> 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. > > The multidimensional point sets of multidimnsonal spaces, such as real > vector spaces, are not ordered and are not orderable in any way > consistent with their geometric properties. Each dimension is ordered, and the dimensions themselves may be ordered, though this does not mean that for every x ad y in the set of n-tuples x>y or x<y or x=y. It's not linear, but it's multi-ordered. > > Nevertheless they are quite important mathematical sets. Without a doubt. > > But TO has to little mathematics to think of them. But that's what IFR and the Finlayson numbers do. (sigh) (woof) TO
From: Virgil on 21 Aug 2006 22:17 In article <ecdofb$i4g$1(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > Virgil wrote: > > It is only the partial sums, a_k = sum(n=1->k: 9*10^-n), k in N, which > > never reach 1. > > The limit actually is EXACTLY 1, by reason of how limits are defined. > > The limit is the least upper bound of the sequence of values, Virgil. > The limit of 0.9999... is 1. :) How does that differ from what I said?
From: Tony Orlow on 21 Aug 2006 22:21 Virgil wrote: > In article <ecb69r$88o$1(a)ruby.cit.cornell.edu>, > Tony Orlow <aeo6(a)cornell.edu> wrote: > >> When you prove this inductively, how much larger is the size of the set >> than its maximum element? Is it always 1 larger than the max element? If >> so, then how does the set size at some point jump to infinity, while all >> elements remain finite? Is there a largest finite, and beyond it a >> point, beyond which everything is infinite? > > This is precisely why TO's extension of induction is not allowed in > standard set theory. What is true for the finite sets does NOT have to > be true for the infinite ones. Well, it does seem that the idea that oo>2 throws a monkey wrench into your works, when it comes to x>2 -> (x^2 > 2*x), that is, if aleph_0>2. Which is false? A) aleph_0>2 B) x>2 -> (x^2>2*x) ;) What is true is ruled by Occam's Razor. Tony
From: Tony Orlow on 21 Aug 2006 22:22
Virgil wrote: > In article <ecb7ae$9qo$1(a)ruby.cit.cornell.edu>, > Tony Orlow <aeo6(a)cornell.edu> wrote: > > >> If, instead of talking about limits, we apply some unit infinity n, then >> we can maintain that 1/n>1/2n, one being half the infinitesimal value of >> the other. We might get in trouble, but I see none ahead, and no one's >> convinced me there's any down this stream. > > No one can convince TO of anything that does not meet the peculiar > criteria of his "intuition", how well proven it may be ever or obvious > to everyone else. his attitude is always " if I don't like it, it isn't > true." > > Such obstinacy would only be justified by a level of genius that TO is > all too obviously lacking. Thank you, Virgil. :) |