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From: David R Tribble on 20 Oct 2005 19:39 David Kastrup said: >> Wrong. N is a different set from N\{1}, but that does not mean that >> the sets have different cardinality. To illustrate this for the >> really dense people with a simpler example: 2 is in {2} and not in >> {1}, and that means that the set {2} is different from the set {1}, >> even though both sets have the same cardinality, namely 1. > Tony Orlow wrote: > What about {1}/{1,2,3}? I think that has a negative number of elements, > don't you David? It has an unidentifiable number of elements.
From: Dave Rusin on 20 Oct 2005 19:34 In article <dj934r$a5e$1(a)news.msu.edu>, <stephen(a)nomail.com> wrote: >There are no strings. There is no largest element. Is there, at least, a spoon?
From: Virgil on 20 Oct 2005 22:22 In article <MPG.1dc1c0b757ad81cd98a518(a)newsstand.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > Virgil said: > > For what set is TO assuming that each member of its power set can be > > represented by a single bit in an infinite sequence of bits? > I never said that. If each subset of *N is to be represented by an infinite binary sequence of digits with 1 in some position representing the presence of a member *N and 0 representing its absence, then one element sets must be represented by strings with one 1 in them. So that, while he may not have been aware of it, TO was saying precisely that. That TO is often unaware of the menning of what he is saying has long been apparent. > > If that set is the set of all Dedekind infinite binary strings, which is > > uncountably infinite, or any set bijectable with it, then TO's > > assumption is false. Unless we are operating in TOmatics where an > > "infinite string of bits" can contain an uncountable number of bits. > What is the limit on the number of bits? One bit for every real number in the > entire real interval. How's that? To do that in a string, TO must not only well-order the reals, but order them isomorphically to the set of finite naturals. > > > > > > > > > I got a little confused, > > > but you still haven't proven anything like > > > the impossibility of a bijection with the power set. > > > > We have to those who are not so permanently confused. > > > > > > > In other cases > > > bijections are performed without regard to such discrepancies. > > > > The 'discrepancy' is that when mapping a set to its power set, there > > must always be a member of the codomain which is not the image of > > anything in the domain. > > > > When that happens, whatever one does have, it is not a bijection. > So, which element of the power set does not have a natural mapped to it? {x in S:x not in f(x)}
From: Virgil on 20 Oct 2005 22:34 In article <MPG.1dc1c10dcf2f969e98a519(a)newsstand.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > Virgil said: > > > > Since 'more' in the context above means that there cannot be any > > bijection, TO is now claiming that where there *are* reasons why no > > bijections can exist, there also are no reasons why bijections > > cannot exist. > I have said repeatedly and consistently that bijection alone does not > indicate equality of size for infiite sets. You know that, > dorkenburger. TO is, as usual, missing the point! The issue is whether where bijection is impossible, equality of size can still occur. Does TO claim that? If so, TO should give us an example of it occuring.
From: Virgil on 20 Oct 2005 22:40
In article <MPG.1dc1c16b9636802398a51a(a)newsstand.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > > > > Since the argument is not about sizes but about lack of any > > surjection/bijection from any set X to its power set P(X), size in TO's > > sense, is not an issue, but bjiection is. > > > > > It's clearly not. I just see a bijection between them. > > > > TO accepts the proof of no bijection then immediately turns around and > > claims to see a bijection. > No, I accept the result that the power set is larger, but not that a > bijection > is impossible. You need to focus. Try ginseng. As usual TO is off point. No one has claimed that the sets are of equal size, unless it be TO, so that equality of size has never been the issue. But there has been stated and proved that there is never a surjection, much less an injection, from any set to its power set, and that IS the issue. So far, it is an issue that TO has carefully avoided. |