Cantor Confusion
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From: mueckenh on 14 Oct 2006 04:36 Virgil schrieb: > In article <1160652854.196211.117740(a)h48g2000cwc.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > William Hughes schrieb: > > > > > > > > > > > > > > > 0.1 > > > > > > > 0.11 > > > > > > > 0.111 > > > > > > > ... > > > > > > > > > > > > > > > > > That is correct. But every element of the natural numbers is finite. > > > > > > Hence every element covers its predecessors. If 0.111... is covered by > > > > > > "the whole list", then it is covered by one element. That, however, is > > > > > > excuded. > > > > > > > > > > > > > > > > Since no one has claimed that '0.111... is covered by "the whole > > > > > list"', I fail > > > > > to see the relevence of a sentence that starts out > > > > > 'If 0.111... is covered by "the whole list"'. > > > > > > > > If every digit position is well defined, then 0.111... is covered "up > > > > to every position" by the list numbers, which are simply the natural > > > > indizes. I claim that covering "up to every" implies covering "every". > > > > > > > Quantifier dyslexia. > > > > Quantifier magic may apply and may be useful at several occasions. But > > to state that in a unary representation of natural numbers the union of > > "up to every" and "every" have different meaning is easily disproved. > > > > > The fact that > > > > > > for every digit position N, there exists a natural number, M, > > > such that M covers 0.111... to position N > > > > > > does not imply > > > > > > there exist a natural number M such that for every digit > > > position N, M covers 0.111... to position N > > > > In case of linear sets we have a third statement which is true without > > doubt: > > > > 3) Every set of unary numbers which covers 0.111... to a finite > > position N can be replaced by a single unary number. > > As stated it is false. You stated so, but your statement is false. Try this: > > 3') Every set of unary numbers which covers 0.111... to a finite > position, n, and no further can be replaced by a single unary number. Why "no further"? There is no reason to insert that. > > > > This holds for every finite position N. If 0.111... has only finite > > positions, then (3) holds for every position. As 0.111... does not > > consist of mre than every position, it holds for the whole number > > 0.111.... > > > False! Of course you will remain at your position. And I will leave you there. I would only be interested to hear an argument, why I should change my position. You don't seem to have an argument. The pure "false" or the silly story of John and his girls is not suitable to convince me. Regards, WM
From: mueckenh on 14 Oct 2006 04:37 Virgil schrieb: > In article <990aa$452e542e$82a1e228$16180(a)news1.tudelft.nl>, > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > > Completed infinity > > does not exist. > > There are more things in heaven and earth, Horatio, > Than are dreamt of in your philosophy. And the set of those things contains also some impossibilities which you are not aware of. Regards, WM
From: mueckenh on 14 Oct 2006 04:38 Virgil schrieb: > In article <1160675140.906009.253460(a)i42g2000cwa.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > this implies > > > > > > for every digit position N, > > > there exists a single unary number, M, > > > such that M covers 0.111... to position N > > > > > > > > > this does not imply > > > > > > there exists a single unary number M such that for every digit > > > position N, M covers 0.111... to position N > > > > Why shouldn't it? > > In general > "for all x there is a y such that f(x,y)" > does not imply > "there is a y such that for all x f(x,y)". > > To establish the latter requires proof over and above the former. I did not state that this be true in general, but it is true in a special case, namely for the covering of linear sets of finite elements. Regards, WM
From: mueckenh on 14 Oct 2006 04:39 David Marcus schrieb: > Han de Bruijn wrote: > > Dik T. Winter wrote: > > > > > In article <1160646886.830639.308620(a)c28g2000cwb.googlegroups.com> > > > mueckenh(a)rz.fh-augsburg.de writes: > > > ... > > > > If every digit position is well defined, then 0.111... is covered "up > > > > to every position" by the list numbers, which are simply the natural > > > > indizes. I claim that covering "up to every" implies covering "every". > > > > > > Yes, you claim. Without proof. You state it is true for each finite > > > sequence, so it is also true for the infinite sequence. That conclusion > > > is simply wrong. > > > > That conclusion is simply right. And yours is wrong. Completed infinity > > does not exist. So _each_ finite sequence "means" the infinite sequence. > > Are you saying that "completed infinity" does not exist in standard > mathematics? If so, please define "completed infinity". If not, then > please define "exist". A good, if no the best source to learn about the different meanings of infinity would be Cantor's collected works. I have only the German version. Therefore I do not post his remarks here. But I recommend you should read it. Regards, WM
From: mueckenh on 14 Oct 2006 04:40
David Marcus schrieb: > Han de Bruijn wrote: > > Dik T. Winter wrote: > > > In article <1160647755.398538.36170(a)b28g2000cwb.googlegroups.com> > > > mueckenh(a)rz.fh-augsburg.de writes: > > > ... > > > > It is not > > > > contradictory to say that in a finite set of numbers there need not be > > > > a largest. > > > > > > It contradicts the definition of "finite set". But I know that you are > > > not interested in definitions. > > > > Set Theory is simply not very useful. The main problem being that finite > > sets in your axiom system are STATIC. They can not grow. Which is quite > > contrary to common sense. (I wouldn't imagine the situation that a table > > in a database would have to be redefined, every time when a new row has > > to be inserted, updated or deleted ...) > > Is your claim only that set theory is not useful or is contrary to > common sense? Or, are you claiming something more, e.g., that set theory > is mathematically inconsistent? It is not useful and contrary to common sense but above all it is mathematically inconsistent. If I remember correctly, you offered to formulate the vase problem in your language. Perhaps you can see the contradiction there. Regards, WM |