Uncountability of the Rationals?
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From: Virgil on 16 Jun 2010 15:11 In article <1ab17b0f-0144-453e-8bad-31aa2852ae0b(a)b35g2000yqi.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > If you prefer, for set H: > > 0eH > xeH -> -(2^x)eH ^ (2^x)eH Or, better, 0eH xeH -> -(2^x)eH /\ (2^x)eH where "^" exponentiates and "/\" conjoins.
From: Virgil on 16 Jun 2010 15:13 In article <87ocfbyq04.fsf(a)phiwumbda.org>, "Jesse F. Hughes" <jesse(a)phiwumbda.org> wrote: > Tony Orlow <tony(a)lightlink.com> writes: > > > On Jun 16, 2:15�am, Transfer Principle <lwal...(a)lausd.net> wrote: > >> On Jun 15, 2:03�pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > >> > >> > Tony Orlow <t...(a)lightlink.com> writes: > >> > > IF it has an algebraic bijection with N+ it can be compared therewith. > >> > What's "algebraic bijection" mean? > >> > >> I assume TO means a bijection f: N+ -> S such that f is an > >> algebraic function (i.e., a polynomial, rational, radical). > > > > Actually, I meant a general formulaic relation, not necessarily > > algebraic, but with an inverse function that can be determined through > > algebra, not necessarily restricted to polynomials, but also including > > exponents and logs, etc. > > That's not very explicit. TO tends to avoid being explicit whenever possible.
From: Virgil on 16 Jun 2010 15:26 In article <3d816782-ee7e-4974-9202-399716bac746(a)18g2000vbi.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > > > > > If you prefer, for set H: > > > > > 0eH > > > xeH -> -(2^x)eH ^ (2^x)eH > > > > > For all xeR except for x=0 exists parent element yeR:y=log2(abs(x)). Such a claim requires proof, and even if true, I doubt that TO could prove it in his own. > > > > So, 1/3 is in R because log_2(1/3) is in R, and that's in R because > > log_2(log_2(1/3)) is in R and so on? > > > > Somehow, I don't get it. > > It sounds like you get it. R is closed under this pair of operations, > except that 0 has no parent. That is a claim that requires proof, as it is no at all obvious, for example, that any finite number of iterations of { 0eH \/ (xeH -> -(2^x)eH /\ (2^x)eH)} will produce 3 or 1/3
From: David R Tribble on 16 Jun 2010 16:52 Brian Chandler wrote: > I do speak Japanese, and I sometimes dream > things in Japanese, which make sense at least as far as dreams ever > make sense. But I don't speak German, other than highly fragmentary > bits of youth hostel stuff: however, I recall a memorable dream in > which I was with some German speakers, who were indeed speaking > German, and just like in real life I could sort of semi-understand > parts of it. How did my brain generate this German for me to semi- > understand? The same thing happens to me occasionally. An article I read a few years ago suggested that our minds have pretty good "intelligence simulators", which explains why when people you know show up in your dreams they tend to act and say things exactly like they would in real life; you brain is doing an impressive job of modeling their real behavior. Modeling people speaking in foreign tongues is yet another impressive feat.
From: David R Tribble on 16 Jun 2010 17:03
Tony Orlow wrote: >> Tav us a bad name, but whether you are talking about omega or a >> zillion, the equivalence relation holds. > Brian Chandler wrote: > Why is "Tav" a bad name? In any conversation with you, I am certainly > not talking about the "real" omega, because you don't understand what > it is. If there is "declaring of unit infinities" to be done, I shall > do it in the Hebrew alphabet. (Irrelevant anecdote: I remember Conway > using 'Beth', and commenting that the Hebrew alphabet is underused...) FWIW, "Beth" is already used to name infinite cardinals: http://en.wikipedia.org/wiki/Beth_number Essentially, cardinal Beth_0 = Aleph_0, and Beth_(n+1) = 2^Beth_n. Which makes them convenient to use as cardinals if you assume CH is true, since Beth_1 = c. Personally, I think "Tav" is a good name for a large "set size" distinct from the existing Alephs and Beths. Certainly better than "bigulosity" and "zillion". I used "eta" for my suprareal "large number" primitives, mostly because the Greek "eta" looks kind of like an "n". |