Cantor Confusion
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From: Virgil on 20 Nov 2006 02:46 In article <1164004608.458362.311870(a)h54g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > > We cannot construct or define or > > > recognize a well ordering. > > > > We can easily define one: any ordering in which every non-empty subset > > has a first element. > > But you cannot define more than countably many. Definition of "more than countably many" : A set has more than countably many members if there exists an injections of N to it but no surjections from N to it or injections from it to N. > > Please lean what "definable well-order" means technically. We can easily define well-order: any ordering of a set in which every non-empty subset has a first element. > > > > > We have no reason to suppose the we could not recognize one if it were > > presented for inspection. > > We cannot recognize it, because it cannot exist. In ZFC, it must exist. In ZF one cannot prove it impossible. its non-existence in ZF can be no more than an added assumption. > 1) It cannot be defined by a formula. Perhaps. Do you have any proof that it cannot be defined by a formula to present here? > 2) It cannot be listed because of cardinality reasons. I am not quite sure of what you mean by "listed". Would the listing of a countable set of general rules constitute a listing? > 3) It cannot be determined in any other way. (Or do you have an idea?) > Hence, we cannot recognize it. > > > > > It is true that no one has actually constructed any explicit well > > ordering. > > > > > > It is true that no one will ever have constructed any explicit well > ordering. That is a statement of faith only. > > > > > The axiom of infinity does not say anything about ordinal or > > > > cardinal numbers. However, given that the set N exists and > > > > the defnition of ordinal and cardinal numbers, it is easy to > > > > see that if N exists it must have both an ordinal and a cardinal > > > > number. > > > > > > > No. You assume the possibility of a bijection of the set with itself. > > > > The identity function on any set bijects it with itself. And such > > functions are guaranteed, via the axiom of replacement. > > > > > > > That is not proven from the mere existence of the set if we cannot > > > recognize or treat all of its elements. > > > > It is proven in ZF, even without C. > > The recognizability of all elements is not proven. And is not needed if a rule for all cases can be provided, such as x |--> x. > > > > > And even if it had. By means of the equilateral infinite triangle I > > > proved that this cannot be the case. Therefore we have a contradiction. > > > > Since WM assumes properties for his "triangle" which contradict > > themselves, as well as contradicting ZF, ZFC and NBG, among others, he > > is welcome to all contradictions of his own making, but there are no > > other contradictions than those due to his self-contradictory > > assumptions. > > Among other idiocies, WM keeps claiming that in his triangle there is a > > finite line as "long" as the infinite diagonal. > > Either you are a liar or too stupid to understand. I proved that there > is no infinite diagonal. No you did not. At least not in any axiom system yet described. In ZF, at least, any model of such a "triangle" has infinite diagonal. >(The diagonal is a subset of the line ends.) The diagonal contains ALL the infinitely many line ends, so its set of positions a SUPERset of the set of line ends.
From: Virgil on 20 Nov 2006 02:55 In article <1164004869.265383.54640(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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. The existence of a set in such a system as ZF guarantees a bijection of that set with itself, or, equivalently, an identity mapping on itself. Rather like in the category of sets and functions. > > > > > > This _existence_ is "assumed" rightly. The function > > > > B := { <n, n> | n e omega } > > > > is the desired bijection. > > Only if it includes all natural numbers. For any set S, B := { <x,x> | x e S} defines a bijection from S to S. > But there are more than can be > treated (= included in any proof) other than by induction. Not in ZF. > > > That is not proven from the mere existence of the set > > > > It _is_ proven using admissible techniques. > > Who admitted? ZF > It is proven by postulating that it is proven. It is provable from the axioms of ZF. Nothing further is needed.
From: Virgil on 20 Nov 2006 02:59 In article <1164005068.730548.146330(a)j44g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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. While one can define things which turn out not to exist, if one has that something exists, for whatever reason, then it follows that the thing is definable, even when not constructable. > The existence of a set does not guarantee the existence or definability > of a bijection or an identity mapping. It does in ZF > > Piffle. If the natural numbers exist, then the identity map is > > a bijection. > > If it exists, of course. It does in ZF.
From: Virgil on 20 Nov 2006 03:01 In article <1164005396.816569.110500(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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. But royalty all have their fools like WM, and mathematics is the queen of sciences.
From: Franziska Neugebauer on 20 Nov 2006 04:18
mueckenh(a)rz.fh-augsburg.de wrote: > 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. <456068f6$0$97216$892e7fe2(a)authen.yellow.readfreenews.net> F. N. -- xyz |