Cantor Confusion
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From: Lester Zick on 20 Nov 2006 17:30 On Mon, 20 Nov 2006 19:54:32 +0000 (UTC), stephen(a)nomail.com wrote: >David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >> Lester Zick wrote: ><snip> > >> That's impressive. Either you are trolling or you have completely >> misunderstood what people mean when they say "definitions are >> abbreviations". > >Lester misunderstands pretty much anything anyone says. Lester certainly misunderstands pretty much everything which is not demonstrably true. Occupational hazard Lester expects when one deals with truth. Which Lester also expects doesn't include pretty much everything Stephen says. > I suppose >that is what happens when someone perversely uses their own personal >vocabulary for too long. Probably the very difficulty David is experiencing with his private definitions for "truth" and "abbreviations". ~v~~
From: David Marcus on 20 Nov 2006 17:33 Virgil wrote: > In article <1164037587.537732.324270(a)j44g2000cwa.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Here are some explanations: > > A list is a (injective) sequence. > > A list is a function from N to some set in which the listed objects are > to be found. While one often wishes to have such functions injective, > there is no inherent reason why an object may not be listed more than > once is a list, and there are lists allowing this. Perhaps we should be grateful that WM actually gave a definition of something (and one we can understand) rather than take him to task for using an unusual meaning for a word. It only took us a few months to convince him that not everyone knows that he means injective when he says "list". -- David Marcus
From: David Marcus on 20 Nov 2006 17:40 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? -- David Marcus
From: Franziska Neugebauer on 20 Nov 2006 17:42 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. Why not stating it if it is so obvious? > Further I defined it by means of the EIT as the > union of all initial subsets of N. Does anybody keep track of your definitions? Where can I look them up? Do you mean "initial subsets" or "initial segments" according to the definition in http://mathworld.wolfram.com/InitialSegment.html ? >> >> 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. What is a possible formal definition of the set which represents a row? F. N. -- xyz
From: David R Tribble on 20 Nov 2006 18:36
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? |