Cantor Confusion
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From: mueckenh on 29 Oct 2006 11:26 Dik T. Winter schrieb: > In article <h4n4k25f4lnlsrdfqb6mku1v7eg6gj8v0b(a)4ax.com> Lester Zick <dontbother(a)nowhere.net> writes: > > On Fri, 27 Oct 2006 00:36:24 GMT, "Dik T. Winter" <Dik.Winter(a)cwi.nl> > > wrote: > > >You need transfinity when you want to show that something that holds in > > >the finite case also is valid in the infinite case. Induction will not > > >show that 0.111... is rational, it can only show that all the finite > > >initial parts are rational. And I again note that the notation 0.111... > > >(in the decimals) has only meaning due to the definition of that notation. > > > > However one can certainly show the square root of 2 without > > transfinity through rac construction even though its decimal expansion > > is infinite. > > You need not tell that to me. You should tell that to Wolfgang > Mueckenheim who insists that sqrt(2) does not exist because it is > impossible to know all the decimals in its decimal expansion. It does not exist as a number. It has no b-adic representation. It exists as a geometric entity. It is an idea. Regards, WM
From: mueckenh on 29 Oct 2006 11:31 David Marcus schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > MoeBlee schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > I proved that there are not more real numbers than a countable set has > > > > elements. I proved it in a manner which everybody with moderate > > > > mathematical knowledge can understand. And, what is important, in a > > > > manner completely independent of your special language. > > > > > > Then yours is a proof in an informal context of your own mathematical > > > understanding (and what CLAIM to be a context shared by anyone with > > > moderate mathematical knowledge). And, so, as I've said already about a > > > half a dozen times by now, that is not a proof that set theory is > > > inconsistent. To prove set theory is inconsistent, you must provide > > > prove, IN set theory, a sentence P and a sentence ~P that are sentences > > > in the language of set theory. > > > > > > > It is clear and proven *from outside* that their results > > > > are wrong and, therefore, uninteresting for me. > > > > > > It matters little to me that set theory is uninteresting to you on > > > account of your having convinced yourself that it conflicts with > > > certain ideas of yours that are outside set theory. > > > > > > I view set theory (in any of a number of different formulations) > > > > Therefore I don't see any necessity to use a special language like ZFC > > or NBG, but use mathematics which can be understood by any freshman. > > Since you decline to name any mathematicians who understand what you are > saying and agree with it, please name a freshman who understands it and > agrees with it. After you will have finished your first semester, you will understand the binary tree without external help too, I am sure. Regards, WM
From: mueckenh on 29 Oct 2006 11:37 David Marcus schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > William Hughes schrieb: > > > > > And once again WM deliberately confuses, "all positions are > > > finite", with "there are a finite number of positions". > > > > There is no confusing! Every finite position belongs to a finite > > segment of positions (indexes). If you don't believe that and assert > > the contrary, then try to find a finite position which dos not belong > > to a finite segment (of indexes). > > > > It is simply purest nonsense, to believe that "all positions are > > finite" if "there are an infinite number of positions". > > Before any of us wastes more time on this, please pick one: > > 1. You are making a statement that you say is provable within some > standard mathematical system, e.g., ZFC. > > 2. You are making a statement that is true within your own system. My above statement is true within any system which is free of self contradictions. Regards, WM
From: mueckenh on 29 Oct 2006 11:43 David Marcus schrieb: > Virgil wrote: > > In article <1162067841.239007.74250(a)e64g2000cwd.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > Please name the mathematicians that agree that your argument is correct. > > > > (Han doesn't count, since he says he is a physicist.) > > > > > > Every mathematician whose mind has not yet suffered the drill of set > > > theory you probably were exposed to knows that my argument is correct. > > > > "Mueckenheim" cannot speak for all those people. > > The set of mathematicians who don't know any set theory is probably > pretty small. So, Mueckenheim's statement might be true (vacuously). Most know a bit set theory, but in their study they do not take curses in set theory. In Germany it is not obligatory. I doubt that is different in other countries. 90 % will not be able to tell the ZFC axioms when asked. Most of them will not know that there is no definable well order of the real numbers. Many will not even know what a well order is. They all can be very good mathematicians, and in general they are. Regards, WM
From: Sebastian Holzmann on 29 Oct 2006 12:40
mueckenh(a)rz.fh-augsburg.de <mueckenh(a)rz.fh-augsburg.de> wrote: > > Sebastian Holzmann schrieb: > >> Virgil <virgil(a)comcast.net> wrote: >> > In article <1161883732.413718.244570(a)k70g2000cwa.googlegroups.com>, >> > mueckenh(a)rz.fh-augsburg.de wrote: >> >> What you propose, namely the infinity of ZF without the axiom INF would >> >> not be an advance. But meanwhile you may have recognized that your >> >> assertion (ZF even without INF is not finite) is false. >> > It is, however, quite true that ZF without INF need not be finite. >> It is, more than that, quite true that ZF without INF _is_ infinite > Are you really sure? I am. >> (the axiom schema of separation alone provides infinitely many axioms). > Are you really sure? I am. >> The point is: ZF without INF does not prohibit the existence of infinite >> sets, nor does it force them to exist. > > It prohibits to speak of infinite sets and to recognize such sets. So > one cannot be sure, but you are? Given any model of ZF-INF, one cannot be sure if there are infinite sets. |