Cantor Confusion
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From: David Marcus on 20 Nov 2006 01:39 imaginatorium(a)despammed.com wrote: > Lester Zick wrote: > > On Sat, 18 Nov 2006 18:31:05 +0000 (UTC), stephen(a)nomail.com wrote: > > >David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > > >> Lester Zick wrote: > > >>> On Thu, 16 Nov 2006 02:02:49 -0500, David Marcus > > >>> <DavidMarcus(a)alumdotmit.edu> wrote: > > >>> >Please give a specific example of something that you think is absurd or > > >>> >a contradiction. I don't know what you mean by "containment of sets and > > >>> >subsets". > > >>> > > >>> Well as I recollect Stephen seems to think infinite sets are proper > > >>> subsets of themselves. > > > > > >> Are you sure that is what Stephen thinks? > > > > > >I see Lester has resorted to lying. > > That was Stephen... I must say I've always been puzzled by accusations > of "lying" in this group. In particular, how can a claim to "seem to > recollect.." be lying? Depends on whether the person's memory is really hazy or they are being disingenuous. > Of course, one may be annoyed by people who > jumble up what one says, but... Or, people who don't listen. Or, people who don't learn. -- David Marcus
From: mueckenh on 20 Nov 2006 01:41 Franziska Neugebauer schrieb: > > We cannot construct or define or recognize a well ordering. > > Irrelevant to your claim. In set theory there are no "equal rights" for > things which are different. The existence of a well-order does not guarantee the constructibility or definability of a well-order . The existence of a set does not guarantee the existence or definability of a bijection or an identity mapping. > > This _existence_ is "assumed" rightly. The function > > B := { <n, n> | n e omega } > > is the desired bijection. Only if it includes all natural numbers. But there are more than can be treated (= included in any proof) other than by induction. > > > That is not proven from the mere existence of the set > > It _is_ proven using admissible techniques. Who admitted? It is proven by postulating that it is proven. > Regards, WM
From: mueckenh on 20 Nov 2006 01:44 William Hughes schrieb: > > > The axiom of infinity says that the set N exists. Your iia says > > > > > > "we cannot recognize or treat all of its elements" > > > > This is no contradiction to the axiom. Compare the proof that the real > > numbers can be well-ordered. We cannot construct or define or > > recognize a well ordering. > > The fact that we cannot construct a real ordering does not > stop us from proving such an ordering exists LOL. I know. > and using > the existence of such an ordering. The existence of a well-order does not guarantee the constructibility or definability of a real ordering. The existence of a set does not guarantee the existence or definability of a bijection or an identity mapping. > > > >And even if we could, the > > axiom of infinity does not prove that infinite ordinal and cardinal > > numbers are numbers, i.e., that they stand in trichotomy with each > > other. > > Check the definitions of ordinal and cardinal below. Look for > the term trichotomy. Fail to find the term trichotomy. Draw > the obvious conclusion. My conclusion is: You don't know this term. Nevertheless, you will agree if you learn what trichotomy means: a < b or a = b or a > b for any two numbers a and b. > > There are even models without a bijection to the natural numbers. > > Piffle. If the natural numbers exist, then the identity map is > a bijection. If it exists, of course. Regards, WM
From: mueckenh on 20 Nov 2006 01:49 Franziska Neugebauer schrieb: > > The reaction of William confirms that my ideas are understandable. So > > your reaction does not concern my writings but rather your means of > > reception. > > You may even cite D. Marcus as your witness: There is a proverb in Germany: Sage mir, mit wem du umgehst, und ich sage dir, wer du bist. Therefore, I wouldn't like to become too familiar with fools. Regards, WM
From: mueckenh on 20 Nov 2006 01:52
Franziska Neugebauer schrieb: > >> >> The cardinality of omega is |omega| not omega. [1] > >> > > >> > Kunen's "Set Theory" defines |A| to be the least ordinal that can > >> > be bijected with A. So, with this definition, |omega| = omega. > >> > >> You are absolutely right. > > > > Well learned (after all)! > > > > Now try to understand the next step: > > > > If omega exists, then |omega| =/= omega & |omega| = omega. > > Sorry how did you arrive at > > |omega| =/= omega (*) Look a the first line [1], written by yourself. Regards, WM |