An uncountable countable set
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From: mueckenh on 26 Jul 2006 16:41 Franziska Neugebauer schrieb: > > T. Jech is one of the leading set theorist. > > He says up to the advent of set theory there was no use for actual > > infinity. > > 1. This utterance is not set theroretic use. It explains and distinguishes what infinity means in set theory in comparison with analysis. Aren't you even able to understand such simple texts? It has *nothing* to do with his wife. > 2. There is also no use of "actual" vs. "potential" infinity after > the advent of set theory either. You continuously contradict yourself. Compare Hilbert's statement, which he uttered *after* the advent of set theory. You had the chance to read it most recently and to learn something about that topic. Hilbert (1926, S. 167). "Will man in Kürze die neue Auffassung des Unendlichen, der Cantor Eingang verschafft hat, charakterisieren, so könnte man wohl sagen: in der Analysis haben wir es nur mit dem Unendlichkleinen und dem Unendlichgroßen als Limesbegriff, als etwas Werdendem, Entstehendem, Erzeugtem, d.h., wie man sagt, mit dem potentiell Unendlichen zu tun. Aber das eigentlich Unendliche selbst ist dies nicht. Dieses haben wir z.B., wenn wir die Gesamtheit der Zahlen 1,2,3,4, ... selbst als eine fertige Einheit betrachten oder die Punkte einer Strecke als eine Gesamtheit von Dingen ansehen, die fertig vorliegt. Diese Art des Unendlichen wird als aktual unendlich bezeichnet." > If you can't take criticism you'd better not post. Critizism is welcome, senseless gossip of ignorants is not. Regards, WM
From: mueckenh on 26 Jul 2006 16:52 Dik T. Winter schrieb: > > But every natural is finite. Every digit which can be indexed by > > naturals is *by definition* in the true list (not the extended list) > > and every digit which cannot be indexed is not in the true list. So > > your attempt to explain that a number which is not in the true list > > nevertheless could be indexed is completely unfounded. > > Let's state that more proper. The index of every digit that can be > indexed is in the true list (of natural numbers). Because this is the list of all indices. > But later you use > the word "index" to mean something completely different. You are > stating there that if every digit of a number can be indexed the number > *itself* is in the list. But that does not follow. You need a proof > for that. It does not *follow* but it is he definition of the true list. This list contains all naturals, hence all indices. It is the complete set N. Therefore there is no proof required (and possible). > And there is no such proof. In order to save set theory with its "infinite set of finite numbers" you are forced to assert that the list of all natural numbers does not contain the list of all natural numbers. > As I stated earlier, every > *terminating* number of which all digits can be indexed through numbers > in the list is itself in the list. That is trivial. The last digit > index position is the number itself. But that is *not* true for a > non-terminating list, as it does not have a last index position. But every index number has a last position. Regards, WM > > K is nothing else than every digit. > > Right. > > > Hence your claim > > unavoidably implies that K can be covered. > > Wrong. Ridiculous. > > > (every digit of n can be indexed) ==> (n is in the list) > > Where is the *proof* of this? It is the definition of the list. Every n is finite. N is infinite. My list contains only finite numbers. My list is infinite. My list is N. More indices are not available. > > The left hand states: > for all n in N, K[p] = 1 > your conclusion reads: > there is an n such that An = K > As I said before, this is true *only* if N is finite. But N is not finite > by the axiom of infinity. My list is *not* finite. But every n in it is finite by its definition. You intermingle size and Anzahl. > > > not (n is in the list) ==> not (every digit of n can be indexed) > > This is indeed a correct conclusion from the first. But as the first is > unproven (and wrong) the conclusion is also unproven (and wrong). It is the *definition* of N that all n e N are in N. The list is N. Every unary representation of n e N is finite!!!!! Regards, WM
From: mueckenh on 26 Jul 2006 16:56 Virgil schrieb: > > > > But every natural is finite. Every digit which can be indexed by > > naturals is *by definition* in the true list (not the extended list) > > The "true list" of all indices is the one represented by the unending > "0.111...", since every ending list is too short. My true list is unending. Every n is finite. N is infinite. My list contains only finite numbers. But my list is infinite. My list *is* N. More natural numbers are not available. More indices are not available. > > and every digit which cannot be indexed is not in the true list. So > > your attempt to explain that a number which is not in the true list > > nevertheless could be indexed is completely unfounded. > > Thus "muecken" acknowledges the infiniteness of the set of indices( > naturals) and simultaneously declares it finite. You intermingle magnitude and number of numbers. Regards, WM
From: mueckenh on 26 Jul 2006 17:09 Dik T. Winter schrieb: > In article <1153829288.635491.96140(a)m73g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > ... > > I need not count to w. If a column contains one 1 then the *+sum is 1. > > This is a definition like that of Cantor's list. > > Depends. I have no idea (and it is very dissimilar), whatever way I > look at it, I find: > (*+){k = 1 .. n} Ak = An > and I can not substitute oo for n (so as to include all Ak), because there > is no Aoo. Therefore you cannot need and cannot use it. It is simply not available. > > > > > But each one will be finite. > > You mean the index of the digit and the An. Yes. > > > (every digit of n can be indexed) ==> (n is in the list) > > Where is the proof? It is even an equivalence by *definition*. The list contains all natural numbers. All indexes *are* natural numbers and vice versa. > There is an easy proof of: > (n is in the list) ==> (every digit of n can be indexed) > but not of the converse. You cannot index other digits than those which are indexes. All indexes are all naturals. All naturals are by definition in he true list. > > > > > The last digit of 0.111... has no ordinal number. > > > > > > Indeed, but the reason is that there is no last digit. w is still a > > > potential infinity, you can not count to it (for one thing, it has > > > no predecessor). > > > > Why does the left column of my extended list differ from the right one? > > There are the same numbers given, only their representation is > > different. > > You do not show here what you are referring to, but I think I understand > what you mean. Look at the list. The ordinality of the number 1 2 3 .... n is the same as that of the unary number in the corresponding line 0.111...1 (with indices 1,2,3,...,n) as long as n is a natural number. But the ordinality of 1 2 3 .... w differs from the ordinality of the unary of the corresponding line 0.111... This proves that there is no possibility to interpret w as the number which counts the naturals. > > The least ordinal of a class is also its cardinal number. If you say N > > has the cardinal number omega you are not wrong today. Even Cantor did > > so at his later times. > > Perhaps. I do not know whether that is really true for all sets. But what > you stated is still wrong. Formally aleph-0 is defined as the equivalence > class of sets that biject with the set of naturals. w is defined as the > order type of N (and because N is well-ordered), it also becomes the > ordinal number of N. You need proof to show that all ordinal numbers of > a class (what is a class?) A class of numbers is a set of ordinals of sets with same cardinality. Their cardinality in newer set theory (and already by the late Cantor) is expressed by the least ordinal. The cardinality of sets w , 2w, w^2, .... is expressed by aleph_0 or by omega. >are for sets that are in the same equivalence > class with respect to bijection. But perhaps I am wrong, but I think that > today the distinction between w and aleph-0 is made more clear than in > Cantor's later times. On the contrary! Cantor distinguished very clearly in all his written papers. But as G. KOWALEWSKI: Bestand und Wandel, Oldenbourg, München (1950) 202 reports, he used the modern notation at his later times. Regards, WM
From: mueckenh on 26 Jul 2006 17:11
Virgil schrieb: > In article <1153863159.167151.152210(a)h48g2000cwc.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Michael Stemper schrieb: > > > > > >Is July 11, 2002 contemporary enough? Thomas Jech: Set Theory > > > >Stanford.htm, Stanford Encyclopedia of Philosophy: Until then, no one > > > >envisioned the possibility that infinities come in different sizes, and > > > >moreover, mathematicians had no use for "actual infinity." > > > > > > Okay, so your own reference shows that philosophers don't think that > > > mathematicians use the term "actual infinity". That sounds correct. > > > > T. Jech is one of the leading set theorist. > > He says up to the advent of set theory there was no use for actual > > infinity. > > Does he say that there is no use for actual infinity within current set > theory? I doubt it. He says that there is only use for actual infinity in (current and uncurrent) set theory Regards, WM |