An uncountable countable set
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From: Dik T. Winter on 20 Aug 2006 08:32 In article <ec8h1q$prm$1(a)ruby.cit.cornell.edu> aeo6(a)cornnell.edu writes: .... > Yes, that's a shame, isn't it? It would seem to me that any good theory > of infinite sets could be applied to infinite sets of points, such as > the reals in (0,1] or those in (0,2], and be able to draw conclusions > such as that there is twice as many points and twice as much space in > the second as in the first, rather than coming to to useless conclusion > that the infinite sets are equal in size. You are looking for measure theory. > Transfinite cardinalities > don't really measure anything, though, do they? They measure something, but not what you want them to measure. > Indeed, they don't form > a field. Tsk tsk. There is not something inherently wrong with a collection of objects not forming a field or a division ring, or whatever. However, if that is the case you must be prepared to find that division is not generally possible. Take for instance the Sedenions, an extension of the Cayley numbers (or Octonians). In the Sedenians general division is not possible because there are zero divisors. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 20 Aug 2006 08:44 In article <ec8hlf$qq8$1(a)ruby.cit.cornell.edu> aeo6(a)cornnell.edu writes: > Dik T. Winter wrote: .... > > Eh? It holds for every finite number but does not hold for an infinite > > number of finite numbers? What do you mean with that? If there are > > infinitely many finite numbers, I would say that as it holds for > > every finite number, it also holds for infinitely many finite numbers. > > 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? 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). > 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. 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. > > With ordering you lose, so not the last. {1, 11, 21, ...} are an example > > of a set that contains 10% of the natural numbers (whatever that may mean). > > I think that what WM is pointing out here is that aleph_0 does not > behave like a normal number when it cannot be divided to determine a > midpoint of the set. This is a problem when viewed quantitatively. He may be trying to point that out, but not in a way that I do understand. Moreover, I have stated over and over again that aleph_0 does not behave like a normal number. It is just people like WM and you that wish that if behaves like a normal number, but it does not do so. I see no problem with that. You can not find a midpoint in the ordered set of natural numbers using a measure derived from the standard measure of the reals. -- 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 08:46 In article <ec8hsd$r5t$1(a)ruby.cit.cornell.edu> aeo6(a)cornnell.edu writes: > Dik T. Winter wrote: > > In article <ec7gek$frg$1(a)ruby.cit.cornell.edu> aeo6(a)cornnell.edu writes: > > > Inductively provable: Any set of consecutive naturals starting at 1 > > > always has its largest element equal to size of the set. > > > > What is the size of the empty set? What is the largest element of the > > empty set? > > The empty set is not a set of consecutive naturals starting at 1, > because it does not include 1 as an element, and therefore it does not > stand as a counterexample. However, even in your von Neumann ordinals, > the empty set is 0, and contains zero elements, and could be considered > the 0th in this sequence of sets. And the next in the sequence is {0} = 1, followed by {0, 1} = 2... -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Tony Orlow on 20 Aug 2006 09:23 Virgil wrote: > In article <ec8gln$oc9$1(a)ruby.cit.cornell.edu>, > Tony Orlow <aeo6(a)cornell.edu> wrote: > >> Dik T. Winter wrote: >>> Let L be the list of all natural numbers >>> Let K be the sequence of digits such that for each p in L, K[p] = 1. >>> What is *wrong* with that definition? >> Hi Dik - >> >> Sorry to interject, but could you tell me how long a sequence K is? If >> it's finite, then you only have a finite list L, and if it's countably >> infinite, then you have an uncountable list of naturals? > > How does having a countable list of naturals require an uncountable list > of naturals? Is K a list of natural numbers? Pay attention. K is the sequence of digits required to represent the naturals in binary. The set of binary strings of length n has size 2^n. If n is aleph_0, then your countably infinite string of digits produces uncountably many possible strings. We've been through this. You cannot cull bits or strings in any way to reduce this uncountably infinite set to produce the countably infinite set of naturals you claim exists. In other words, there is no TYPE of number which could represent the size of K. TO
From: Tony Orlow on 20 Aug 2006 09:25
Virgil wrote: > In article <ec8h1q$prm$1(a)ruby.cit.cornell.edu>, > Tony Orlow <aeo6(a)cornell.edu> 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 > > That would require each point to occupy enough space that only a finite > number of them could fit into any finite interval. WRONG! It requires that some relationship be set up between a particular infinity of points and a finite length. What you suggest would be a finite set of finite elements, not any kind of infinite set. TO |