Cantor Confusion
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From: Dik T. Winter on 12 Oct 2006 10:31 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. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 12 Oct 2006 10:36 In article <1160650371.242557.284430(a)h48g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <1160578706.221013.145300(a)c28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > > > So the definition I gave for a limit of a sequence of sets you agree > > > > with? Or not? I am seriously confused. With the definition I gave, > > > > lim{n = 1 .. oo} {n + 1, ..., 10n} = {}. > > > > > > Sorry, I don't understand your definition. > > > > What part of the definition do you not understand? I will repeat it here: > > > What *might* be a sensible definition of a limit for a sequence of sets of > > > naturals is, that (given each A_n is a set of naturals), the limit > > > lim{n = 1 ... oo} A_n = A > > > exists if and only if for every p in N, there is an n0, such that either > > > (1) p in A_n for n > n0 > > > or > > > (2) p !in A_n for n > n0. > > > In the first case p is in A, in the second case p !in A. > > Pray, read the complete definition before you give comments. > > I do not believe that definition (2) is of any relevance. It is. > Cantor uses Lim{n} n = omega witout much ado. That is not a limit of "sets". > In his first paper he uses even Wallis' symbol oo. What should there > require a definition, if all natural did exist? > This is what I use and write in modern form: Lim {n-->oo} {1,2,3,..,n} > = N. Yes, and that fits my definition. On the other hand, how would you define lim{n --> oo} {n, n+1, ...}? > Not this expression is meaningless but the assumption behind it, the > complete set N. In your not so humble opinion. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 12 Oct 2006 10:34 In article <1160649760.068206.172400(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... > > > a) How many different natural numbers can you store using a maximum of > > > 100 bits? > > > b) What is the largest natural number you can store with a maximum of > > > 100 bits? > > > > What is the relevance? > > To inform the set theorist about the possible existence of sets with > finite cardinality but without a largest number. Interesting but in contradiction with the definition of the concept of "finite set". So you are talking about something else than "finite sets". -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Han de Bruijn on 12 Oct 2006 10:41 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. Han de Bruijn
From: Han de Bruijn on 12 Oct 2006 10:49
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 ...) Han de Bruijn |