Cantor Confusion
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From: Virgil on 21 Nov 2006 16:21 In article <45630269.90703(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 11/9/2006 12:17 AM, Virgil wrote: > > > As infinitely many paths pass through each edge no path can "own" more > > than an infinitesimal share of that edge according to that "equal > > sharing" rule. And unless we are in something like Robinson's > > non-standard analysis, less than infinitesimal means zero. > > I wonder. I have to agree with Virgil. Is there something wrong? We will just have to be more careful in future, I suppose.
From: Virgil on 21 Nov 2006 16:24 In article <456304B0.70705(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 11/12/2006 8:28 PM, William Hughes wrote: > > mueckenh(a)rz.fh-augsburg.de wrote: > > > As you stated "Goes on forever" is a property of the natural > > numbers. The set {1,2,3,...} is just the natural numbers. So > > this set must go on forever. > > Being admittedly not very familiar with set theory, I nonetheless wonder > if sets are considered like something going on forever. Sequences do, sets do not. When one represents the members of a set as members of a sequence, as {1,2,3,...} does, that little ambiguity should not really mislead anyone.
From: Virgil on 21 Nov 2006 16:25 In article <4563067B.5050805(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 10/29/2006 10:22 PM, mueckenh(a)rz.fh-augsburg.de wrote: > > > Every set of natural numbers has a superset of natural numbers which is > > finite. Every! > > > I am only aware of the unique natural numbers. If one imagines them > altogether like a set, then this bag may also be the only lonly one. Is that "lonely"?
From: Virgil on 21 Nov 2006 16:30 In article <45630A55.30901(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 11/15/2006 1:49 AM, Virgil wrote: > > mueckenh(a)rz.fh-augsburg.de wrote: > > >> Cantor wrote, for instance to Killing, on April 5, 1895: > >> Was Herr Veronese dar?ber in seiner Schrift giebt, halte ich f?r > >> Phantastereien und was er gegen mich darin vorbringt, ist unbegr?ndet. > >> Ueber seine unendlich gro?en Zahlen sagt er, da? sie auf anderen > >> Hypothesen aufgebaut seien, als die meinigen. Die meinigen beruhen aber > >> auf gar keinen Hypothesen sondern sind unmittelbar aus dem nat?rlichen > >> Mengenbegriff abgezogen; sie sind ebenso nothwendig und frei von > >> Willk?r, wie die endlichen ganzen Zahlen. > >> > >> Briefly: My infinite numbers are founded only on the natural notion of > >> sets. They are as necessary and free of arbitriness as the finite whole > >> numbers. > > > > What Cantor may have said in 1895 need not be binding in 2006. > > May have said? He wrote it. As I do not have access to the original documents, and could not read the German well enough anyway, I cannot be certain of it. > > It was not binding in 1895 either. However, the Utopia by Dedekind and > Cantor was reformed but it never got rid of its basic mistakes. Only in EB's opinion are they mistakes. Others disagree.
From: Randy Poe on 21 Nov 2006 16:30
Lester Zick wrote: > On Mon, 20 Nov 2006 17:40:59 -0500, 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? > > That is an abbreviation not a definition. Definitions are abbreviations. "N" is an abbreviation for "the set of natural numbers". "The set of natural numbers" is an abbreviation for "the minimal set which obeys... [Peano axioms]". The conceptual content is contained in those axioms, which are not definitions and are not abbreviations. - Randy |