An uncountable countable set
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From: mueckenh on 25 Jul 2006 08:19 Dik T. Winter schrieb: > > > > > > 0.1 > > > > > > 0.11 > > > > > > 0.111 > > > > > > ... > > > > > > 0.111... > Sorry, you stated "the second list below". As the first list you gave > terminated with "...", I can only conclude that your second "list" > includes 0.111... . That is the extended version, correct. I did not delete it above. > > > 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. > > Indeed, it is not a unary natural number. Perhaps it is the unary > representation of something else, but not of a natural number. As 0.111... according to your assertions can be indexed by the set of naturals and as this set has omega members, 0.111... is the unary representation of omega. Regards, WM
From: mueckenh on 25 Jul 2006 08:22 Dik T. Winter schrieb: > 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". Wrong! Das Zeichen oo, welches ich in Nr. 2 dieses Aufsatzes [hier S. 147] gebraucht habe ersetze ich von nun an durch omegas, weil das Zeichen oo schon vielfach zur Bezeichnung von unbestimmten [d. h. potentiellen] Unendlichkeiten verwandt wird. 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. Wrong. (every digit of n can be indexed) ==> (n is in the list) not (n is in the list) ==> not (every digit of n can be indexed) Regards, WM
From: mueckenh on 25 Jul 2006 08:26 Dik T. Winter schrieb: > 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? Where should he have been written that? > 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. No. It makes use of the fact that every natural is finite. > Any indexing sequence is finite. Right. > But the sequence of *all* digits of 0.111... is neither an indexing sequence, > nor a finite sequence. If it can be indexed, then it must be indexed by finite numbers. >And it is precisely *that* sequence that can not be > covered by a finite natural. Then it cannot be indexed completely. > Remember: there is a set that contains all > naturals. There is (clearly) no natural that is the largest in that set. But everyone is finite. Regards, WM
From: Franziska Neugebauer on 25 Jul 2006 09:52 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > >> mueckenh(a)rz.fh-augsburg.de wrote: >> >> > Franziska Neugebauer schrieb: >> >> > In set theory. It is full of the actual infinite. >> >> >> >> When I ask for "usage of terms" I want you to name contemporary >> >> papers/books on set thery that mention/use these terms. >> > >> > 1976 is too old? >> >> Preface, Wolfgang. You are refering to an elucidation in its preface >> on page 6. How many times does the book mention and use those two >> terms on the remaing pages? > > I am not in possession of the book but have only my notes. Get it and count! > But he frequencay of mentionings may be high though unimporant. How will you know if you don't have it at hand? > The meaning of "infinity" in set theory is clearly the actual one. The other way round. Students first get in touch with plain infinite sets and then read from your posts about the ancient terms. To them the new terms which appear are the ancient ones. Hence from modern set theoretic standpoint one would clearly define: "Actual infite" simply means infinite. > Every expert knows this. Compare Jech's statement. Or compare Hilbert > or Poincar. That is just the clue, and it has not chaned since > Cantor's times. Evidence? > Hilbert (1926, S. 167) ^^^^ > But here is some newer author: Lorenzen: Die endlichen Weltmodelle der > gegenwrtigen Naturwissenschaft zeigen deutlich, wie diese Herrschaft > eines Gedankens einer aktualen Unendlichkeit mit der klassischen > (neuzeitlichen) Physik zu Ende gegangen ist. > Befremdlich wirkt dem gegenber die Einbeziehung des > Aktual-Unendlichen in die Mathematik, die explizit erst gegen Ende des > vorigen Jahrhunderts mit G. Cantor begann. Elucidation not set theory, not logic not maths. F. N. -- xyz
From: Dik T. Winter on 25 Jul 2006 10:45
In article <1153830146.942349.15090(a)s13g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... > > 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. > > Wrong. > > (every digit of n can be indexed) ==> (n is in the list) Prove that statement! You *never* do so. The statement can be roughly translated to (I think): Define: K: for all n in N, K[n] = 1 Ap: for all n <= p in N, Ap[n] = 1 now every digit of K can be indexed, and the list is the list of Ap. Now you state that because of this K must be equal to Ap for some p. But that is clearly false because Ap[p+1] != K[p+1]. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |