An uncountable countable set
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From: Tony Orlow on 20 Aug 2006 11:02 David R Tribble wrote: > Tony Orlow wrote: > David R Tribble wrote: > mueckenh wrote: >>> Learn: Infinitely many stairs require infinite height. But that it not >>> admitted in mathematics. There have to be infinitely many stairs which >>> give an infinitely long staircase ... > > David R Tribble wrote: >>> Yes, infinitely many finite-sized steps produce a staircase of infinite >>> height. That is admitted in mathematics. > > Tony Orlow wrote: >> All too often "admitted" that those infinities are equal. > > Yes, mathematicians accept the proofs that countably infinite sets > have the same cardinality. > > > mueckenh wrote: >>> ... but this staircase is allegedly of finite height. > > David R Tribble wrote: >>> No, the staircase is infinite in height, but each step is finite in >>> height. The distance between any two steps is finite, but the total >>> height of all the steps together (the entire staircase) is infinite. > > Tony Orlow wrote: >> So, the greatest possible value that any natural number can reach, in >> height, is infinite? > > You're almost right. > > The least upper bound of the naturals (and the reals, for that matter) > is infinite (omega). Which is to say that naturals (and reals) can be > as large as you like, but they never actually "reach" their LUB value. > Thus they are all finite. > 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? :) TO
From: Tony Orlow on 20 Aug 2006 11:11 Virgil wrote: > In article <ec9ot6$5a1$1(a)ruby.cit.cornell.edu>, > Tony Orlow <aeo6(a)cornell.edu> wrote: > >> Virgil wrote: >>> In article <ec8hsd$r5t$1(a)ruby.cit.cornell.edu>, >>> Tony Orlow <aeo6(a)cornell.edu> wrote: >>> >>>> Dik T. Winter wrote: >>>>> In article <ec7gek$frg$1(a)ruby.cit.cornell.edu> aeo6(a)cornnell.edu writes: >>>>> > Inductively provable: Any set of consecutive naturals starting at 1 >>>>> > always has its largest element equal to size of the set. >>>>> >>>>> What is the size of the empty set? What is the largest element of the >>>>> empty set? >>>> The empty set is not a set of consecutive naturals starting at 1, >>>> because it does not include 1 as an element, and therefore it does not >>>> stand as a counterexample. However, even in your von Neumann ordinals, >>>> the empty set is 0, and contains zero elements, and could be considered >>>> the 0th in this sequence of sets. You fail to address the inductive >>>> proof that the non-empty sets of this series, of which the naturals is >>>> the limit, always have this property. >>> TO fails to note that induction does not apply. >> Virgil somehow fails to see that it does. >> >>> The form of induction is: >>> Let F(x) be a predicate whose argument, x, is a member of a minimal >>> inductive set S.* >>> If F(x) is true for the first member of S >>> and >>> if whenever F(x) is true then F(successor(x)) is also true. >>> then >>> the principle of induction says that F(x) is true for each member of S. >> Right. Consider S to be the set of all sets of consecutive naturals >> starting at 1. Is N a member of this set? You bet it is. > > Actually, I would bet against N being a member of a subset of N, myself. I'd bet against you. S is the set of all sets of consecutive naturals starting at 1. N is a such a set, and is therefore a member of S. > > N is usually used to represent the set of all naturals. By requiring N > to be a member of S, TO is asserting that there is a set which is a > member of one of its own subsets. No, S is a set of sets, of which N, the set of natural numbers starting at 1, is an element. > > Either that or TO is using N in an unconventional way to mean something > other than the set of all naturals. In which case he should explain what > he means by it. > Oh, Virgil, please try rereading this. S={{1},{1,2},{1,2,3},...}, N={1,2,3,...}. It's a member of S. > > > >> Is this an >> inductive set? > > As described, it may not be acceptable as a set at all. > Sets can include sets as elements. > > >> S is the set of all such sets. N is a member of S. So, it says something >> about N. > > Then TO IS claiming that N is a member of a subset of itself! No. Clear the ghosts out of you head and reread what I wrote. >>> In particular it does not say that F(S) is true. >>> >> No, it says that F(N) is true, since N e S. > > Only if N e S e N e S e N e... S isn't an element of N. What are you talking about? S is a set of sets of naturals. I am not playing ordinal nonsense here. A set of naturals is not the same thing as A natural. > >> 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 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 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. 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. >>> 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.
From: David R Tribble on 20 Aug 2006 11:42 Tony Orlow wrote: >> It would seem to me that any good theory >> of infinite sets could be applied to infinite sets of points, such as >> the reals in (0,1] or those in (0,2], and be able to draw conclusions >> such as that there is twice as many points and twice as much space in >> the second as in the first > Once again, you are confusing 'cardinality' with 'volume' . The first deals with denumeration of sets, while the second deals with geometric distance.
From: Franziska Neugebauer on 20 Aug 2006 11:54 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: [...] >> > Actual infinity does not exist! >> >> I don't discuss Cantorisms. > > Do you discuss whether the cardinal number of omega is a number larger > than any natural number? No. >> > I told you that already several times. >> >> You persistently try to posit Cantorisms. >> >> > But refuse to understand what "to exist" means. >> >> An entity a exists if |= E a. > > It is that easy to establish? Then I hasten to write > |= E agreement between us that transfinite set theory is nonsense. Are pieces of literature nonsense (in general)? F. N. -- xyz
From: David R Tribble on 20 Aug 2006 12:02
Tony Orlwo writes: >> If the formula applies to an infinite number of finites, then does the >> sum of the aleph_0 finite naturals equal (aleph_0^2+aleph_0)/2? > Dik T. Winter wrote: > I think there is a misunderstanding. The formula > sum{i = 1 .. n} i = n * (n + 1) / 2 > holds for every finite number n, so it holds for infinitely many finite > numers n (as there are infinitely many finite numbers n). But we can > not switch to an infinite sum (that is something different). Not without proof that the formula works for infinite n, anyway. But we're still waiting for Tony to provide that proof. Tony Orlwo writes: >> In standard theory, would this not equal aleph_0, and if so, does it make >> sense that sum(n=1->aleph_0: 1) = sum(n=1->aleph_0: n), when n>1 for all >> n>1? The scond would appear to be clearly a larger sum. > Dik T. Winter wrote: > No that does not make sense. sum{n = 1 .. oo} 1 is not defined, the same > holds for sum{n = 1 .. oo} n. If you want to use them you have to > provide a definition for them. But Tony thinks he has provided a definition, based on his "Big'un" number. The problem, of course, is that he simply assumes that arithmetic operations on Big'un work "es expected" without providing any proof of that whatsoever. It's one thing to provide a definition, it's quite another to prove that it is a consistent definition. |