Cantor Confusion
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From: Lester Zick on 20 Nov 2006 17:11 On 20 Nov 2006 12:00:42 -0800, imaginatorium(a)despammed.com wrote: > >Lester Zick wrote: >> On Mon, 20 Nov 2006 01:39:59 -0500, David Marcus >> <DavidMarcus(a)alumdotmit.edu> wrote: >> >> >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. >> >> I seem to recollect being disingenuous on occasion. >> >> >> 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. >> >> Or people *** ***'* *****. Or people *** ***'* ***** *** ***** **** * >> ****** **** **** ** * *****. Or ****** *** **** ** ********** ** ***** >> *** **** ******* ***** *** ** *********** ******. Or ****** >> *** *** ******* **** demonstrations of truth. Or ****** *** *** >> ********** ** ********** ******* ****** *** **** ** ****** ** ****** >> *** *** ***** ** *** ****** ******** ***** *** *** *** **** ** ****** >> ** **** ***** ** ******. >> >> (Technically *** ******* ** Brian ****** ****** *** **** ** * ******* >> dizinzzzzzzz yyyyyyyyyyyy zzzzz.) > >You spoke, Lester? Very good. Very good indeed. Actually somewhat more >intellectually challenging than your usual level. Very even-handed, >neigh, calm. Except, Brian, you might have used "~v~~" or even "~" instead of "*". Would have indicated a little more ingenuity on your part. You know a little more "model schmodel" kind of originality. >> ~v~~ > >Arf arf! Bark on, Brian. You know, the "Mystery of the Dog who Didn't Bark in the Night"? ~v~~
From: Lester Zick on 20 Nov 2006 17:19 On Mon, 20 Nov 2006 14:35:20 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >Lester Zick wrote: >> On Sun, 19 Nov 2006 17:53:12 -0500, David Marcus >> <DavidMarcus(a)alumdotmit.edu> wrote: >> >Lester Zick wrote: >> >> On Sat, 18 Nov 2006 13:44:58 -0500, David Marcus >> >> <DavidMarcus(a)alumdotmit.edu> wrote: >> >> >Lester Zick wrote: >> > >> >> >> I'm saying that you don't understand what a mathematical definition is >> >> >> but nonetheless want to pretend you do. If a mathematical definition >> >> >> were "just" an abbreviation as you claim you wouldn't have any way to >> >> >> tell one mathematical definition from another. >> >> > >> >> >Why not? Suppose I make the following definitions. >> >> > >> >> > Let N denote the set of natural numbers. >> >> > Let R denote the set of real numbers. >> >> > >> >> >Then I can tell N and R are different because their defintions are >> >> >different. If I write >> >> > >> >> > 0.5 is not in N, >> >> > >> >> >then this means the same as >> >> > >> >> > 0.5 is not in the set of natural numbers. >> >> > >> >> >And, it means something different from >> >> > >> >> > 0.5 is not in R. >> >> >> >> But the problem, sport, is you claim mathematical definitions are >> >> "only abbreviations". Granted I suppose even mathematikers can tell >> >> the difference between N and R in typographical terms. I mean they may >> >> be too lazy and stupid to demonstrate the truth of what they say but >> >> even they can see differences in typography. But in terms of >> >> abbreviations alone we can't really say what the difference is between >> >> N and R because you insist their definitions are "only abbreviations" >> >> and not their conceptual content. >> > >> >You said there was no way to tell two definitions apart. The >> >typographical difference suffices to tell the definitions apart (as you >> >just admitted). >> >> So what exactly is the difference between definitions N and R in >> conceptual terms if definitions are "only abbreviations"? I mean you >> say certain things about definitions which are mutually inconsistent. >> If definitions were "only abbreviations" as you indicate then your >> definitions for N and R would be restricted to those abbreviations N >> and R. Instead you append certain properties to each and pretend that >> they're part of the definitions for N and R which we'll all can see >> are not part of their abbreviations such that your definition for >> definitions is "only abbreviations which are not only abbreviations". >> Obviously this kind of logic extends way beyond your doctoral thesis >> in philosophy but is nonetheless true. > >That's impressive. Either you are trolling or you have completely >misunderstood what people mean when they say "definitions are >abbreviations". Or quite possibly you misunderstand what people mean by "abbreviations". Not quite the same as the sloth and professional turpitude of mathematical definitions you're used to I daresay. N and R are "abbreviations". Whatever you imagine by what you attach to abbreviations are not abbreviations. But please do go on and don't allow me to distract you from your "modern mathematical" definition of abbreviations which I'm quite confindent will afford us many a pleasant evening of muffled laughter. ~v~~
From: David Marcus on 20 Nov 2006 17:21 mueckenh(a)rz.fh-augsburg.de wrote: > Randy Poe schrieb: > > > > I know that at most 10^100 digits of sqrt(2) can be determined, in > > > principle. > > > > In principle, if a is the sqrt(2) to 10^100 digits, then > > 0.5*(a + 2/a) is the sqrt(2) to 2*10^100 digits. > > > > What "in principle" prevents me from calculating 2/a, > > or adding it to 1, or taking 0.5*(a + 2/a)? > > Lack of bits to represent 2/a if a already requires all bits available. > > If you don't believe me, simply try it. By about 330 calculations you > should be able to produce the first 10^100 digits. If you have a slow > computer you will need one second per calculation. So let it run for 5 > minutes and you will know what prevents you from calculating 2/a. You have a funny idea what the words "in principle" mean. Of course, you have a funny idea what almost all words mean. -- David Marcus
From: David Marcus on 20 Nov 2006 17:28 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > > Experience has shown that practically all notions used in contemporary > > > mathematics can be defined, and their mathematical properties derived, > > > in this axiomatic system. In this sense, the axiomatic set theory > > > serves as a satisfactory foundation for the other branches of > > > mathematics. [Karel Hrbacek and Thomas Jech: "Introduction to Set > > > Theory" Marcel Dekker Inc., New York, 1984, 2nd edition, p. 3] > > > > Again: Meet _your_ obligation and define "column[s]". You have > > introduced this notion hence it is up to you to define it. > > No. The notion "column" is well known to anyone having studied the > first semesters math. Further I defined it by means of the EIT as the > union of all initial subsets of N. By "union of all initial subsets of N", do you mean N? > > >> It is curious that you obviously don't like to give precise > > >> definitions of certain notions even when you have explicitly been > > >> asked for. > > > > > > I defined the EIT. A column is a vertical row. > > > > What is a row in the language of (which?) set theory? > > I use the language of mathematics. Was that a joke? Because you are very funny. First, you claim that you are showing how set theory leads to an inconsistency. Then, when asked which version of set theory you are discussing, you switch and say you aren't really talking about set theory. Plus, you play this game of bait and switch repeatedly. Why don't you just admit that you are developing a new type of mathematics because you want to rather than insisting that you are doing it because the existing mathematics is inconsistent? You obviously are not familiar with existing mathematics, since you don't know any of the standard meanings of the many mathematical words you use. -- David Marcus
From: Franziska Neugebauer on 20 Nov 2006 17:28
mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> mueckenh(a)rz.fh-augsburg.de wrote: >> > Franziska Neugebauer schrieb: >> >> > I cannot understand your explanation given there. If you say >> >> > "The cardinality of omega is |omega| not omega", so you must >> >> > have had in mind |omega| =/= omega, [(****)] >> >> >> >> No Way! If you want to misapprehend me do so, but don't confuse >> >> your misapprehensions with theorems of set theory. >> > >> > You wrote: "The cardinality of omega is |omega| not omega." >> >> And what I meant was: "The cardinality of X is |X| not X". I have >> already clarified that my wording was misleading. > > You tried to say "misleading", but your wording was wrong. I will not rectify this once again. > The cardinality of omega is omega as well as |omega|. Under the common definition the cardinality of omega, also written as |omega|, is omega. Anyhow. I still see no support for your pretended interpretation (****), which I did not meant to write and did not write in that form. >> > Shall this sentence of yours express a difference between |omega| >> > and omega or not? >> >> What exactly is so hard to understand? The cardinality of a set X is >> (written) |X| and -- in the case of omega -- equals under the common >> definition to omega. > > Therefore, the sentence "The cardinality of omega is |omega| not > omega" is false? Or is it not false? Perhaps D. Marcus' understanding in <MPG.1fcbb0f0d873ccdd989977(a)news.rcn.com> may help you to cope with my sentence under discussion. >> > (Now I recognize why it is so difficult to convince the proponents >> > sof set theory.) >> >> >> >> > Would it be meaningful to write "Y is X but not X"? >> >> >> >> This is not a "faithful" representation of my sentence you have >> >> objected to because there is no character string which can be >> >> substituted for X to get my original sentence back. >> > >> > If you meant |omega| = omega, then we have X = |omega| = omega. >> >> Let Y = "The cardinality of omega" and >> >> case 1: let X = "|omega|" "Y is X but not X" transforms into >> "The cardinality of omega is |omega| but not |omega|", >> >> case 2: let X = "omega" "Y is X but not X" transforms into >> "The cardinality of omega is omega but not omega". >> >> I did neither write nor mean what case 1 nor case 2 state. > > So you have *not* yet learned that omega = |omega|? My sentence under discussion is _not_ of the pretended _form_ (!) "Y is X but not X" Period. > Because considering this eqation, you effectively wrote "The > cardinality of omega is |omega| but not |omega|". My sentence under discussion was not meant to be interpreted as an equation. F. N. -- xyz |