Cantor Confusion
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From: Virgil on 12 Jan 2007 22:34 In article <MPG.2011f913b980b3ab989afa(a)news.rcn.com>, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > Virgil wrote: > > In article > > <11087446.1168647563782.JavaMail.jakarta(a)nitrogen.mathforum.org>, > > Andy Smith <andy(a)phoenixsystems.co.uk> wrote: > > > > > > > > > > Here is a question for you: Suppose you have a > > > > countably infinite > > > > ordered set X. So there is a bijection f:N -> X. Does > > > > it have to be true > > > > that m < n implies that f(m) < f(n)? > > > > > > > At the risk of appearing an idiot, I would say yes. > > > > > > Because X is ordered such that x(n) > x(m) for all n > m. > > > > > > Am I missing something? > > > > Apparently! > > > > Since there is nothing requiring the order of X to be compatible with > > the order of N, and orderings of countable sets exist which are > > incompatible with the natural ordering of N, there is no requirement > > that f be order preserving. > > > > For example, consider X as the set of rationals with the standard > > rational order, for which there exist bijections from N to X which are > > clearly can n to be order preserving > > I think you meant to write "... which clearly cannot be order > preserving". My spell checker sometimes hiccups and turns 'not' into 'n to'.
From: Andy Smith on 13 Jan 2007 00:03 Yes, understood, thank you. I registered that my understanding of the terminology was duff when I saw the answer!
From: David Marcus on 13 Jan 2007 14:26 Andy Smith wrote: > Yes, understood, thank you. You're welcome. > I registered that my understanding of the terminology > was duff when I saw the answer! -- David Marcus
From: cbrown on 13 Jan 2007 15:13 Virgil wrote: > In article <MPG.2011f913b980b3ab989afa(a)news.rcn.com>, > David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > > > For example, consider X as the set of rationals with the standard > > > rational order, for which there exist bijections from N to X which are > > > clearly can n to be order preserving > > > > I think you meant to write "... which clearly cannot be order > > preserving". > > My spell checker sometimes hiccups and turns 'not' into 'n to'. Does that mean you were /really/ talking about the existence of bijections from Not X? Cheers - Chas
From: Andy Smith on 13 Jan 2007 05:42
Chas Brown wrote: > > (Whoops! Actually three: Thanks for taking the time, appreciated, understood. |