An uncountable countable set
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From: mueckenh on 24 Jul 2006 10:14 Dik T. Winter schrieb: > In article <1153674914.649461.71180(a)m73g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > In article <1153478871.303029.5360(a)i3g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de 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. > > Eh, can we please keep to the subject. You state you have a list that > only finite numbers are involved in the "list" above. I state that 0.111... > is not a finite unary number. Pray show that 0.111... is a finite unary > number. (I neither have stated that it is a terminating rational number, > because it is not.) The pure list stretches only from top to the three ... The last element does not beong to it. My reply concerned your statement that 0.111... was not a unary number at all and the difference between decimal and unary interpretation you could perhaps try to derive from your observation. Regards, WM
From: Franziska Neugebauer on 24 Jul 2006 10:28 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. F. N. -- xyz
From: mueckenh on 24 Jul 2006 10:29 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? > > > > > 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 > > 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. > > > 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. 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? 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. But if you insist on it, then you should support your belief. Regards, WM
From: mueckenh on 24 Jul 2006 11:09 Franziska Neugebauer schrieb: > 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. Obviously you have not the foggiest idea about set theory. Here are few quots from many: Fraenkel, Abraham A., Levy, Azriel: "Abstract Set Theory" (1976) p. 6 the set of all integers is infinite (infinitely comprehensive) in a sense which is "actual" (proper) and not "potential". One may doubt whether this example really illustrates the abyss between finiteness and actual infinity. p. 240 Thus the conquest of actual infinity may be considered an expansion of our scientific horizon no less revolutionary than the Copernican system or than the theory of relativity, or even of quantum and nuclear physics. Potential infinity existed already long before set theory. Regards, WM
From: Franziska Neugebauer on 24 Jul 2006 12:11
Dear Wolfgang! mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > >> 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. > > Obviously you have not the foggiest idea about set theory. I'm not keen on ancient set theory, that's true. > Here are few quots from many: Fraenkel, Abraham A., Levy, Azriel: > "Abstract Set Theory" (1976) > p. 6 the set of all integers is infinite (infinitely comprehensive) in > a sense which is "actual" (proper) and not "potential". 1. Page 6 (six)! What's the name of the Chapter? "Preface"? 2. "in a sence". You know what that means? An eidetic elucidation. 3. What this sentence states is: omega exists. 4. Where do Fraenkel/Levy use (!) the aforementioned terms _mathematically_? > One may doubt whether this example really illustrates the abyss > between finiteness and actual infinity. One may also doubt that you understood my objection. _In_ (not in the elucidation of) set theory there is no meaning of "actual" or "potential". > p. 240 Thus the conquest of actual infinity may be considered an > expansion of our scientific horizon no less revolutionary than the > Copernican system or than the theory of relativity, or even of quantum > and nuclear physics. > > Potential infinity existed already long before set theory. Where is the mathematical use of the aforementioned terms? F. N. -- xyz |