Cantor Confusion
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From: Virgil on 20 Nov 2006 18:59 In article <1164058702.803070.89230(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > Again: Meet _your_ obligation and define "column[s]". You have > > introduced this notion hence it is up to you to define it. > > No. Yes! >The notion "column" is well known to anyone having studied the > first semesters math. It is equally well known to those who study Greek architecture. Are yours Doric, Ionic or Corinthian? > Further I defined it by means of the EIT as the > union of all initial subsets of N. What is the EIT? The union of all initial subsets of N is just N itself, as every member of N is a member of at least one such initial subset. > > > > >> 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. WM's version of "the language of mathematics" is somehow never compatible with ZF to ZFC or NBG or NF or any standard foundation for mathematics. He must have learnt it in Babel.
From: David Marcus on 20 Nov 2006 19:09 David R Tribble wrote: > david petry schrieb: > >> Perhaps we should replace "absolute truth" with "culturally neutral > >> truth", or in other words, truth without any cultural, religious, or > >> philosophical bias. [...] Thinking about this question > >> leads most of us to believe that there is a core of mathematics which > >> every such civilization will accept. > > mueckenh wrote: > >> without axioms, yes. For instance: I + I = II (after translating "+" > >> and "="). Therefore I call this an absolute truth. > > David R Tribble schrieb: > >> Which axioms are you using to describe the "+" and "=" operators? > > mueckenh wrote: > > Axioms? For which purpose? > > For the purpose of making sense of the phrase "I + I = II". > > > Do you think the symbols constituting the > > words constituting the axioms are easier or clearer to understand than > > the symbols "+" and "="? > > Without words that define those symbols, the symbols are > meaningless. I can say that "x =& y", but without any definition > of "x", "=&", and "y" it is just a meaningless string of symbols. > > > Take an apple and then another apple. Show > > the apples first apart and then together. Repeat with oranges or > > fingers or mixed objects, possibly. That defines all that is needed. > > So then we can take yours words there as axiomatic definitions of > "+" and "="? Or are you still claiming that "I + I = II" means > something without axiomatic definitions? Since WM has his own meaning for the word "definition", he may have his own meaning for the word "axiom" (or "axiomatic"). -- David Marcus
From: David Marcus on 20 Nov 2006 19:15 Virgil wrote: > In article <1164058702.803070.89230(a)m7g2000cwm.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > Franziska Neugebauer schrieb: > > > > Again: Meet _your_ obligation and define "column[s]". You have > > > introduced this notion hence it is up to you to define it. > > > > No. > > Yes! > > > The notion "column" is well known to anyone having studied the > > first semesters math. > > It is equally well known to those who study Greek architecture. > Are yours Doric, Ionic or Corinthian? > > > Further I defined it by means of the EIT as the > > union of all initial subsets of N. > > What is the EIT? That's WM's Equilateral Infinite Triangle, i.e., the list with line n having length n. Haven't you been paying attention? :) > The union of all initial subsets of N is just N itself, as every member > of N is a member of at least one such initial subset. > > > > >> 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. > > WM's version of "the language of mathematics" is somehow never > compatible with ZF to ZFC or NBG or NF or any standard foundation for > mathematics. He must have learnt it in Babel. Nor is it compatible with that of any historical mathematician, despite his frequent quotes from Cantor's papers. -- David Marcus
From: Virgil on 20 Nov 2006 19:16 In article <1164059499.319149.184780(a)k70g2000cwa.googlegroups.com>, 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. The > cardinality of omega is omega as well as |omega|. Technically speaking, omega is an ordinal and |omega| = aleph_0 is a cardinal. They are distinct in the sense that here is nothing requiring a cardinal to to have any internal ordering in general, though by some definitions cardinals may be ordered sets. Any set of cardinals will be ordered, but nothing in the general properties of cardinality ( that two sets have the same cardinal if and only if there is a bijection between them) requires a single cardinal to be an ordered set. It is only when chooses something like the particular definition that a cardinal of a given set is the ordinally smallest ordinal that bijects with the given set, that any individual cardinal needs to be ordered at all, much less well-ordered. > > > > > 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? On the sense that the cardinality of omega does not need to be a well ordered set, it is true. > > So you have *not* yet learned that omega = |omega|? What is your definition of the cardinality of a set? the truth of "omega = |omega|" depends on that definition.
From: Virgil on 20 Nov 2006 19:24
In article <MPG.1fcbf909c89f84c3989993(a)news.rcn.com>, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > Lester Zick wrote: > > 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. > > And, we can let "U.S." stand for the United States. Is this > abbreviation/definition/whatever true or false, in your opinion? There is no point in trying to communicate with Zick, as he is not the least bit interested in communicating. Your best response to his blather is kill filing him. |