Cantor Confusion
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From: Dik T. Winter on 11 Oct 2006 21:43 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. -- 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 11 Oct 2006 21:51 In article <452d140b(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> writes: > mueckenh(a)rz.fh-augsburg.de wrote: .... > I don't understand this definition either. You write lim{n=1 .. oo}. Is > that supposed to be a sum over that range, Why should it be a sum? > or do you mean lim(n->oo)? What is the difference? > Does 1 belong there? Why not? > Also, you are looking for a limit of what, a set of > balls from n+1 through 10n? Are you looking for the limit of the *size* > of that set, which would be 10-(n+1)+1, or 9n? Pray re-read, I explicitly state that I am talking about a limit of sets. > If you want to put this > in limit terms, with the corrections I suggest, you have the size of the > set being lim(n->oo: 9n). That's not 0 by any stretch of the imagination. Pray re-read. I am *not* talking about the limit of the size of sets, but about the limit of 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: Dik T. Winter on 11 Oct 2006 22:12 In article <452d14fe$1(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> writes: > Dik T. Winter wrote: > > In article <1160551520.221069.224390(a)m73g2000cwd.googlegroups.com> "Albrecht" <albstorz(a)gmx.de> writes: .... > > > What is the difference to the diagonal argument by Cantor? > > > > That a (to the right after a decimal point) infinite string of decimal > > digits defines a real number, but that a (to the left) infinite string > > of decimal digits does not define a natural number. > > It defines something. What do you call that? If the value up to and > including every digit is finite, how can the string represetn anything > but a finite value? I define it as a string of digits and it does not represent a number. It is only when you give proper definitions of what strings extending infinitely far away to the left represent, that you can talk about what it represents. In common mathematics there is no such definition. That infinite strings to the right define real numbers is entirely due to the *definition* of real numbers. And that infinite strings to the left, within the theory of p-adics, have specified meaning is entirely due to the *definition* of p-adics. (And I may note that in the p-adics there is *no* definition for infinite strings to the right.) In principle, infinite strings are just that. Within some theories you can make consistent definitions for them, but that is all what it means. The only current consistent theories I know (there may be more) is that 0.111... as a decimal representation within the decimal numbers represents 1/9. That is because the sequence 0.1, 0.11, 0.111, ... converges to 1/9 (with precise definitions about what convergence does mean). In a similar way, ...111 represents a number in the n-adics. The reason is that the sequence 1, 11, 111, ... converges. And so that number is 1/(1-n) in the n-adics. Again, with precise definitions about what convergence does mean. -- 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 11 Oct 2006 22:23 In article <MPG.1f973c20b77376049896b6(a)news.rcn.com> David Marcus <DavidMarcus(a)alumdotmit.edu> writes: > Dik T. Winter wrote: .... > > > > Hm. I humbly submit that the probability for a particular rational > > > > number in the range [0,1) the probability to get it when doing a > > > > random choice is 0. > > > > Nevertheless, the sum of all the probabilities is 1. The sum of > > > > countably many 0's is not always 0. > > > > > > I don't follow. Usually, probability measures are countably additive. > > > > I just gave a counterexample. > > Sorry. As I said, I didn't follow your counterexample. If we take > Lebesgue measure on [0,1) as our probability measure P, then P({q}) = 0 > for any rational q, but P({q|q rational and q in [0,1)}) = 0, not 1. I was getting a random selection amongst the rational numbers in [0,1). It is possible that I goofed (I am quite a bit rusty in this field), so perhaps it is not possible to give a normal distribution amongst the rationals in [0,1). -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Poker Joker on 11 Oct 2006 22:40
"Virgil" <virgil(a)comcast.net> wrote in message news:virgil-1F12E7.17222503102006(a)comcast.dca.giganews.com... > In article <TVBUg.27$LU2.7(a)tornado.rdc-kc.rr.com>, > "Poker Joker" <Poker(a)wi.rr.com> wrote: > >> "Virgil" <virgil(a)comcast.net> wrote in message >> news:virgil-C9AC49.20550702102006(a)comcast.dca.giganews.com... >> >> >>I'm sure you still won't understand. >> > >> > The context of "list" in which "any list" occured required such lists >> > to be functions from the naturals to the reals, which The Poker's >> > pseudolists are not. >> > >> > Ergo, the Poker is committing the fallacy of the straw man, which echos >> > the contents of his head quite well. >> >> Virgil obviously can't understand. > > No one need even try to understand what contains as little sense as PJ's > claims. Virgil is so ignorant that she can't write sentences. |