Cantor Confusion
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From: Virgil on 8 Dec 2006 19:13 In article <1165615662.042013.210530(a)l12g2000cwl.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > Again you have provided neither a definition of "number", nor of "grow". > > Are you unable to do so? In common parlance, but that is not mathematics. > > In mathematics functions can grow in relation to their argument, but not > > the entities they denote. > > Functions cannot grow, according to modern mathematics. The expression > "variable" is merely a relict from ancient times when people knew that > the objects of mathematics do not exist in some nirvana but have to be > present in a mind where not everything can be present simultaneously. Then WM excludes the possibility of addition or multiplication functions on merely potentially infinite sets of naturals. So much for WM's arithmetic.
From: Dik T. Winter on 8 Dec 2006 19:15 In article <1165615662.042013.210530(a)l12g2000cwl.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <1165403362.548786.220370(a)16g2000cwy.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > > > Dik T. Winter schrieb: > > > > > > > > Everybody knows what the number of ther EC states is. > > > > > > > > That is *not* what I did ask you. You state that it is simply a matter > > > > of definition how one interprets "to grow" and "number", > > > > > > sure. > > > > > > > and I asked you > > > > to provide definitions. Moreover, the number of the EC states is not > > > > fixed, so you can only state what the number of the EC states is at a > > > > particular point in time. > > > > > > The number of EC states is "the number of EC states". It is simply a > > > notion which can be equal to a natural number. > > > > Again you have provided neither a definition of "number", nor of "grow". > > Are you unable to do so? In common parlance, but that is not mathematics. > > In mathematics functions can grow in relation to their argument, but not > > the entities they denote. And again you do not provide definitions. How many times do I have to ask before you either give definitions or state that you are unable to? > Functions cannot grow, according to modern mathematics. The expression > "variable" is merely a relict from ancient times when people knew that > the objects of mathematics do not exist in some nirvana but have to be > present in a mind where not everything can be present simultaneously. In mathematical parlance "a function grows in relation to its arguments" if the function value grows in relation to its arguments. Most commonly used if domain and range of the function are subsets of the reals. Where "grow" can be defined as: Let a and b be in the domain of f, then for a > b: f(a) >= f(b) and for every b when there is a number > b, there is also a a > b such that f(a) > f(b). (Note that grow is neither monotone increasing, nor monotone non-decreasing in this definition.) -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 8 Dec 2006 19:28 In article <4tu6aeF1580fdU2(a)mid.individual.net> nowhere(a)nowhere.com writes: > Virgil wrote: > > > > Those poor homeless cuts still exist as sets. > > The slickest way of doing Dedickind Cuts is just use the lower half of > the cut. A set of rational numbers s such that s has no greatest element > and such that every element in Rational - s is greater than any element > in s. This is in a way what Baudot dit in his version of the definition of the reals. Take any set of the reals, a majorant is a number larger than each element of the set. Two sets are equivalent if they share all their majorants. The union of a collection of equivalent sets is the lower half of a Dedekind cut. And apart from those two and Cantor (Cauchy sequences), there is also the definition by Weierstrass. That the definitions give isomorph results is fairly easy. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 8 Dec 2006 20:22 In article <45795248.7050801(a)et.uni-magdeburg.de> Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> writes: > On 12/8/2006 12:54 AM, David Marcus wrote: > > Eckard Blumschein wrote: .... > >> According to my reasoning, any really real number is not unique but must > >> rather be void because even the tiniest interval is thought to contain > >> indefinitely not just many rational numbers but indefinitely much of > >> real numbers. > > > > That's gibberish. Trying speaking English. > > In English much refers to uncountables while many belongs to countables. Try speaking English. This is either poetry or nonsense. Take your pick. > >> Therefore, unreachable the very nil on top of the nested > >> intervals has not any significance at all. > > > > More gibberish. Or, maybe poetry. > > No, just 'the' illposed. > I meant: Therefore, the unreachable very zero on top of... As David Marcus wrote. More gibberish. Or, maybe poetry. > >> It cannot even be > >> distinguished from numbers 0- and 0+ left and right from it, > >> respectively, because the diffence is zero. > > > > Are you saying that there are numbers 0- and 0+? Are these rational? > > Real? What are their definitions? > > Limits from the left and from the right. Definitions, please. Why are there 0- and 0+? Why are there not 1- and 1+? > >> So I agree with the > >> Bourbakis perhaps for the first time: 0+ and 0- are indiscriminable in > >> IR. > > > > That's certainly not what Dik said Bourbaki said. > > Should I read Bourbaki? What book? Well, let's see: �l�ments de math�matiques, livre 1, Th�orie des ensembles �l�ments de math�matiques, livre 2, Alg�bre might be a good start. > > What is "nil"? Zero? Are you saying zero is negative? If so, define > > "negative". > > Negative is the opposite of positive. Wrong. You, always talking about trichotomy. Positive is > 0, negative is < 0. Now what is 0 itself? (This is according to non-Bourbaki mathematics.) -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: David Marcus on 9 Dec 2006 01:09
Eckard Blumschein wrote: > On 12/8/2006 12:54 AM, David Marcus wrote: > > Eckard Blumschein wrote: > >> So I agree with the > >> Bourbakis perhaps for the first time: 0+ and 0- are indiscriminable in > >> IR. > > > > That's certainly not what Dik said Bourbaki said. > > Should I read Bourbaki? What book? I doubt it would help. If you really want to learn math (rather than merely posting poetry to sci.math), you should take some math courses. Perhaps the professors can set you on the right track. You don't seem capable of reading a book on your own, yet. -- David Marcus |