An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: Dik T. Winter on 24 Jul 2006 19:19 In article <44c4d907$0$17976$892e7fe2(a)authen.yellow.readfreenews.net> Franziska Neugebauer <Franziska-Neugebauer(a)neugeb.dnsalias.net> writes: > mueckenh(a)rz.fh-augsburg.de wrote: > > Dik T. Winter schrieb: > > > >> With the axiom of infinity there are infinitely many finite numbers. > >> And I still do not know the exact difference between actual infinite > >> and potential infinite. But that is (in my opinion) something that > >> is best left to philosphy. > > > > In set theory "infinity" means "actual infinity". Potential infinity > > can neither be counted nor be surpassed. > > _In_ set theory "actual" or "potential" have no meaning. You produce > meaningless verbiage. It is verbiage from philosophy. But Mueckenheim has apparently not read that according to Cantor's own writing, w is not "actual infinity", but "potential infinity". But it is more a philosophical than a mathematical viewpoint. Illuminating is the wikipedia article on it: <http://en.wikipedia.org/wiki/Actual_infinity>. The only thing Mueckenheim is in conflict with is with the axiom of infinity. But he is not able to show inconsistencies within set theory within that axiom because, invariably, in his reasoning the negation of that axiom comes up. So he does not show contradictions in set theory with the axiom of infinity, but in set theory with the axiom of infinity plus the negation of the axiom of infinity. -- 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 R Tribble on 24 Jul 2006 19:47 Mueckenh wrote: > Set theory is theory of the transfinite. Transfinite is actually > beyond the finite. > > Potentially infinite means always finite but without border. Can you show us a set that is potentially infinite?
From: David R Tribble on 24 Jul 2006 19:58 Dik T. Winter wrote: Mueckenh writes: >> But it is obvious for *all* *finite* *unary* number. And only >> finite unary numbers are involved in the second list below: >> >> The *- sum of >> 0.1 >> 0.11 >> 0.111 >> ... >> 0.111... >> >> 0.111... is not a finite unary number. Rather, I would say it is not a >> unary number at all. >> 0.111.. is not a terminating rational number. >> [...] > The list: 0.1 0.11 0.111 ... contains only terminating rational numbers (fractions), all of them of the form: f(n) = sum{i=1 to n} 10^-i, for all n=1,2,3,... This means that every f(i) contains a finite number (n) of 1 digits, and likewise so does f(i+1) that follows f(i). So we conclude that the list is composed entirely of finite terminating fractions. We also conclude, for exactly the same reasons, that the number 0.111111... is not in the list, because it is not a terminating fraction, and because it is not of the form: f(n) = sum{i=1 to n} 10^-i
From: Dik T. Winter on 24 Jul 2006 20:08 In article <1153751354.788807.293880(a)s13g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <1153676088.467784.268720(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > > > In article <1153482158.663876.226910(a)p79g2000cwp.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > > Your hinting at any doubt about the clear meaning of Cantor's > > > > > statement is outrageous. > > > > > > > > Perhaps. In that case his statement was clearly wrong. > > > > > > But nobody seems to have observed that up to now. > > > > Yes, so what? Those papers are even no longer valid in current set theory. > > When developing a theory everybody is liable to take false turns. There is > > no need to note each and every false turn. > > But many have studied Cantor's papers. Couldn't it be that those who > designed modern set theory erred as much as those who studied his > original papers without noting this error? Perhaps the error was noted but no statement about it put in writing? Can you give a comment on Cantor's note that w is not actual infinity but potential infinity? Whatever, the remark was only a side remark in a paper that contained much more interesting things than that. It was neither a theorem, nor was a proof given. > > > > > Wrong. My arguing is directed against the complete existence of > > > > > the set of all naturals, the set of all digits of a real number, > > > > > the actual infinity, the first transfinite number. That all is > > > > > purest Cantor. > > > > > > > > As I said before. You are arguing agains the axiom of infinity. > > > > But your inconsitency proofs show nothing, because in part of your > > > > proofs you always use the negation of that axion. > > > > > > I do not use the negation of the axiom, but accept that every digit of > > > 0.111... can be indexed by a finite natural. Hence every sequence of > > > digits of 0.111... can be covered by a natural. > > > > You omit the "finite" between "every" and "sequence". > > If *every* digit can be indexed, then I need not put "finite" between > "every" and "sequence", because every indexing sequence is finite. A > digit which is not a finite sequence remote from the first digit cannot > be indexed and cannot be covered. Your deleting of "finite" (or "indexing") here is precisely what I said, the negation of the axiom of infinity. Any indexing sequence is finite. Right. But the sequence of *all* digits of 0.111... is neither an indexing sequence, nor a finite sequence. And it is precisely *that* sequence that can not be covered by a finite natural. Remember: there is a set that contains all naturals. There is (clearly) no natural that is the largest in that set. So there is also no last digit in the sequence of digits of 0.111..., where the digits are indexed by naturals. > > > digits of 0.111... can be covered by a natural. (There is no digit by > > > which 0.111... could be distinguished from every natural.) But 0.111... > > > is the union of all of its finite sequences, i.e., of the naturals. > > > > Yes. And so? > > A union of finite sequences is a finite sequence. An arbitrary union of finite sequences is not necessarily a finite sequence. This is similar to: the union of finite sets is a finite set. But this argues that the union of {1}, {1, 2}, {1, 2, 3}, ... is a finite set. Consider the elements of those sets as indices to the elements of the sequences. This is only true when N is finite, but that is the negation of the axiom of infinity. > If you believe that not the whole infinte sequence 0.111... can be > completely covered by a natural number, then please name a 1 in it, > which cannot be covered, be it more at the front edge or more in the > back area. Isn't this is a fair offer? No. Because I never argued such. Suppose I want to cover the n-th digit of 0.111..., I simply take An. That does *not* show that 0.111... can be covered by some An. Name 0.111... K. When K can be covered by any An that would mean: there is an n such that for all p in N, An[p] = K[p] (1) the negation of that is (and that is what I argue): there is no n such that for all p in N, An[p] = K[p] (2) but that does *not* mean: there is a p such that for all n in N, An[p] != K[p] (3) which you seem to imply, by some garbled logic. Pray tell me how you come from (2) to (3). > Don't ask me, by which natural the whole infinite sequence 0.111... > could be covered, because I deny that the whole infinte sequence > 0.111... does actually exist. Yes, you negate the axiom of infinity when trying to find a contradiction. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Franziska Neugebauer on 25 Jul 2006 03:34
Dik T. Winter wrote: > In article <44c4d907$0$17976$892e7fe2(a)authen.yellow.readfreenews.net> > Franziska Neugebauer <Franziska-Neugebauer(a)neugeb.dnsalias.net> > writes: > > mueckenh(a)rz.fh-augsburg.de wrote: > > > Dik T. Winter schrieb: > > > > > >> With the axiom of infinity there are infinitely many finite > > >> numbers. And I still do not know the exact difference between > > >> actual infinite > > >> and potential infinite. But that is (in my opinion) something > > >> that is best left to philosphy. > > > > > > In set theory "infinity" means "actual infinity". Potential > > > infinity can neither be counted nor be surpassed. > > > > _In_ set theory "actual" or "potential" have no meaning. You > > produce meaningless verbiage. > > It is verbiage from philosophy. Nowadays certainly not verbiage from the set theorists. WM cannot successfully "refute" transfinite set theory if he attacks the wrong target. > But Mueckenheim has apparently not read that according to Cantor's own > writing, w is not "actual infinity", but "potential infinity". IMHO wrong target. > But it is more a philosophical than a > mathematical > viewpoint. Illuminating is the wikipedia article on it: > <http://en.wikipedia.org/wiki/Actual_infinity>. > > The only thing Mueckenheim is in conflict with is with the axiom of > infinity. I would like to add - Failure to understand what a definition is - Failure to understand what a proof is - Failure to understand what a contradiction is - Failure to prove claims - Taking Cantor's words for current transfinite set theory > But he is not able to show inconsistencies within set > theory within that axiom because, invariably, in his reasoning the > negation of that axiom comes up. As a "reason of last resort", yes. > So he does not show contradictions in set theory with the axiom of > infinity, but in set theory with the axiom of infinity plus the > negation of the axiom of infinity. I can't recall anything he has /shown/. It merely appears to the reader that WM is arguing as if he presumed such an axiom. F. N. -- xyz |