Cantor Confusion
in [Math]
|
Prev: Pi berechnen: Ramanujan oder BBP
Next: Group Theory
From: Virgil on 9 Feb 2007 17:32 In article <1171012555.735278.201840(a)j27g2000cwj.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 8 Feb., 13:37, "William Hughes" <wpihug...(a)hotmail.com> wrote: > > On Feb 8, 3:07 am, mueck...(a)rz.fh-augsburg.de wrote: > > > > > > > > > > > > > On 7 Feb., 16:02, "William Hughes" <wpihug...(a)hotmail.com> wrote: > > > > > > On Feb 7, 9:29 am, mueck...(a)rz.fh-augsburg.de wrote: > > > > > > For all for finite natural numbers > > > > > > > > n = |{2,4,6,...,2n}| < 2n > > > > > > > > Now take the limit as n-->oo (the < becomes <= in a limit, > > > > > > > No. n < 2n is always true for natural numbers (which property excludes > > > > > 0). There is no reason to assume that 1/2 becomes 1 in "the limit". > > > > > > > > lim[n-->oo] n = oo, lim[n-->oo] 2n = oo) > > > > > > > > oo <= oo > > > > > > > > There is no contradiction. > > > > > > > Wrong for all finite numbers. > > > > > > And since the limit is not a finite number > > > > > Every number which can appear in this inequality is a finite number. > > > If something can appear in place of n which is not a finite number, > > > then you agree that N contains infinite numbers. > > > > No. The thing that may be able to appear in this inequality > > is the cardinality of E. Either > > > > -The cardinality of E does not exist > > or > > -The cardiality of E is not a finite number. > > > > In neither case does N contain infinite numbers. > > In neither case is the cardinality of E a finite number. > > In neither case is there a contradiction. > > You can refrain from repeating over and over again your dogmas. Why should we be restricted from doing what you do so regularly? isn't that a form of begging the question? > > In potential infinity there is no cardinal number E of an infinite > set, because there are only finite sets. You can refrain from repeating over and over again your dogmas. There are in most set theories non-empty ordered sets which have no maximal members. WE call such sets infinite. What goes on in WM's world is mathematically irrelevant. > > > > The fact that potentially infinite means that the set E > > can become as large as we like is very relevent. > > In any case what we like must be finite because we cannot like wha > cannot exist. The members may need to be finite, but the "number" of them cannot be. > > - the statment "E contains numbers which are larger > > than the cardinal number of E" is false > No. The answer is not so simple. It is everywhere outside of WM's world. > > As has been pointed out before, if you wish to claim > > that only sets with finite cardinality exist, then > > you are free to do so. > > I do not wish it, but the finity of numbers enforces it. Not on anyone who rejects that axiom. > > However, saying that E does not have a cardinal > > does not change the properties of E. E still > > has a sparrow (recall a sparrow is an equivalence > > class under the equivalence relation equitransform > > which generalizes the concept of bijection to include > > potentially infinite sets). This sparrow can be > > compared with other sparrow's including the sparrows > > of finite sets, which are just the finite cardinals. > > There is nothing contradictory about defining > > a sparrow. > > Unless you say that it is a number larger than any natural number. Since he does not claim it to be a number at all, that is not a problem. He say that every set has a sparrow, actually I would prefer something a bit different from sparrows, say wrens, where a wren for any set is merely one of some standard set of wrens which is bijective with the given set. We presume that there is one and only one wren for any set. > > > > Extending the concept of cardinality to include > > potentially infinite sets does not lead to > > a contradiction. > > Unless you say that it is a number larger than any natural number. So if we merely speak of cardinality, or sparrowness or wrenishness and leave "number" out of it entirely there is no problem. That there is a neat bijection between the natural numbers and the sparrows or wrens of finite sets is mere coincidence, and should be ignored.
From: Dik T. Winter on 9 Feb 2007 18:23 In article <ufuos21t7p3slvjsvu3m53af7eavt8c8ek(a)4ax.com> G. Frege <nomail(a)invalid> writes: > On Fri, 9 Feb 2007 13:34:36 GMT, "Dik T. Winter" <Dik.Winter(a)cwi.nl> > wrote: .... > >>> So a statement like "every element is a set in ZFC" is false. .... > It's even possible to "prove" that. ;-) Oh, well, if that is a proof... > >> It is used by most authors of text books on set theory. Talk to them > >> about their errors. I am no interested in what they use to denote the > >> elements (or sets) of their erroneous theory. > >> > > But they give a proper foundation for that statement. You do not. You > > pull that statement out of thin air. > > > Though -I think you will admit that- this doesn't make WM's claim > _false_. Right? No, it does not make his claims false. But it is a common trend in WM's writings to make claims without proper foundations, like definitions. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: G. Frege on 9 Feb 2007 19:07
On Fri, 9 Feb 2007 23:23:15 GMT, "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote: >> >> It's even possible to "prove" that. ;-) >> > Oh, well, if that is a proof... > Sure it is. (I'm serious! :-) >>> >>> [...] they give a proper foundation for that statement. You do not. You >>> pull that statement out of thin air. >>> >> Though - I think you will admit that - this doesn't make WM's claim >> _false_. Right? >> > No, it does not make his claim false. But it is a common trend in WM's > writings to make claims without proper foundations, like definitions. > Yes, I know that. Still, even a blind hen sometimes finds a grain of corn. :-) F. -- E-mail: info<at>simple-line<dot>de |