Uncountability of the Rationals?
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From: Virgil on 6 Jun 2010 01:56 In article <550ab662-d408-4721-b429-205f17fa5b83(a)k39g2000yqd.googlegroups.com>, Transfer Principle <lwalke3(a)lausd.net> wrote: > IIRC, TO's statement N=S^L means that the number of > strings of length L from a language of size S is > equal to N. Do you mean "the number of strings of length L using an alphabets of S characters is N" ?
From: Transfer Principle on 6 Jun 2010 02:01 On Jun 5, 10:56 pm, Virgil <Vir...(a)home.esc> wrote: > In article > <550ab662-d408-4721-b429-205f17fa5...(a)k39g2000yqd.googlegroups.com>, > Transfer Principle <lwal...(a)lausd.net> wrote: > > IIRC, TO's statement N=S^L means that the number of > > strings of length L from a language of size S is > > equal to N. > Do you mean "the number of strings of length L using an alphabets of S > characters is N" ? Yes.
From: Virgil on 6 Jun 2010 04:32 In article <8141b323-b13b-40a5-9843-314ce48aa6fd(a)j4g2000yqh.googlegroups.com>, Transfer Principle <lwalke3(a)lausd.net> wrote: > On Jun 5, 10:56�pm, Virgil <Vir...(a)home.esc> wrote: > > In article > > <550ab662-d408-4721-b429-205f17fa5...(a)k39g2000yqd.googlegroups.com>, > > �Transfer Principle <lwal...(a)lausd.net> wrote: > > > IIRC, TO's statement N=S^L means that the number of > > > strings of length L from a language of size S is > > > equal to N. > > Do you mean "the number of strings of length L using an alphabets of S > > characters is N" ? > > Yes. IIRC, when L was not finite, TO's interpretation of that equation was considerably non-standard.
From: Brian Chandler on 6 Jun 2010 05:16 Transfer Principle wrote: > On Jun 5, 9:59 am, Virgil <Vir...(a)home.esc> wrote: > > In article > > <db5cbe4b-a8b8-4b6a-ae47-05fa05dd6...(a)i28g2000yqa.googlegroups.com>, > > Tony Orlow <t...(a)lightlink.com> wrote: > > > Yes, that's where I apply N=S^L. > > Which is a wrong now as when first dropped on an unsuspecting world. > > IIRC, TO's statement N=S^L means that the number of > strings of length L from a language of size S is > equal to N. > > I disagree with Virgil that it's "wrong." No, of course the actual meaning of "N=S^L" is correct. The problem is that when Tony says "apply N=S^L" he refers to one of his "proofs" that an infinite set of natural numbers must include at least one number which is itself "infinite" (though he never really defines what this means). I will try to reconstruct the argument, which goes something like: (Tony 'knows' that the set P of all 'natural numbers' must include "infinite naturals", but can't just state it.) So, consider "N=S^L". The number of strings (over alphabet size S) of length L is N. Well, the number of strings (over alphabet size S) of maximum length L is N. (Not exactly true, but close. Failiing to distinguish "finite strings of no fixed maximum length" from "possibly infinite strings" is at the heart of this argumentation technique.) But for there to be an infinite number of strings ("set N = oo"), since S is constant, we have "L=oo". (Confusion and non-sequitur) Therefore a set of strings which is infinite must include some infinitely long strings. Ergo, the set of "natural numbers" includes some "infinite natural numbers". Well, I guess this is a "non-standard theory"... Brian Chandler
From: Rob Johnson on 6 Jun 2010 10:08
In article <Virgil-026A1B.02324906062010(a)bignews.usenetmonster.com>, Virgil <Virgil(a)home.esc> wrote: >In article ><8141b323-b13b-40a5-9843-314ce48aa6fd(a)j4g2000yqh.googlegroups.com>, > Transfer Principle <lwalke3(a)lausd.net> wrote: > >> On Jun 5, 10:56 pm, Virgil <Vir...(a)home.esc> wrote: >> > In article >> > <550ab662-d408-4721-b429-205f17fa5...(a)k39g2000yqd.googlegroups.com>, >> > Transfer Principle <lwal...(a)lausd.net> wrote: >> > > IIRC, TO's statement N=S^L means that the number of >> > > strings of length L from a language of size S is >> > > equal to N. >> > Do you mean "the number of strings of length L using an alphabets of S >> > characters is N" ? >> >> Yes. > >IIRC, when L was not finite, TO's interpretation of that equation was >considerably non-standard. The standard meaning of S^L is the set of functions from L to S. If L and S are finite, TP's definition gives the correct size of S^L. However, strings are sequences of characters, and as such, they are countable. This poses a problem when L is uncountable. Rob Johnson <rob(a)trash.whim.org> take out the trash before replying to view any ASCII art, display article in a monospaced font |