Uncountability of the Rationals?
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From: Virgil on 11 Jun 2010 20:21 In article <cf9298b4-73ea-499f-aabd-85b618e81347(a)u26g2000yqu.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > > But it's good to be clear that your methods do not provide that every > > set has a "size". So in that sense the demand on your theory is much > > less than on ZFC, which was my original point. > > > > > > In some ways less, and in others more. It's a little Heisenbergian in > that respect. That there is a large amount of uncertainty built into any of TO's "creations" is trivially obvious. But that amount is certainly much greater than Heisenbergian.
From: Pol Lux on 11 Jun 2010 23:51 On Jun 10, 5:44 am, mstem...(a)walkabout.empros.com (Michael Stemper) wrote: > In article <c36fedbe-da79-4372-a21e-6e6fb14be...(a)h13g2000yqm.googlegroups..com>, MoeBlee <jazzm...(a)hotmail.com> writes: > > >On Jun 8, 5:03=A0pm, Transfer Principle <lwal...(a)lausd.net> wrote: > >> So the best we can do is choose one of these two equally > >> good notions to preserve. Standard Cantorian cardinality > >> rejects the second in favor of the first. But I see no > >> reason that we can't reject the first in favor of the > >> second -- and this would give us TO's Bigulosity. > > >Fine. Except "bigulosity" awaits a coherent definition. > > It's a perfectly cromulent term! > > -- > Michael F. Stemper > #include <Standard_Disclaimer> > This message contains at least 95% recycled bytes. Is anybody actually reading threads that are that long? I can't give it a good grade for fun, but I can give it a B+ for literary creativity. Fun grade: 3/10. Could do better. And also, it's been established countless times by sci.math that the rationals are uncountable, that the reals don't exist, nor do the negative numbers, and so on and so on. Let's break new ground, people!
From: FredJeffries on 12 Jun 2010 10:13 On Jun 8, 3:03 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > MoeBlee refers to extrapolating from the finite to > the infinite when considering set size. Here are > two equally intuitive notions that describe set > size for finite sets: > > 1. Sets in bijection with each other have the same size. > 2. The whole is strictly greater than the part. > (paraphrased from Euclid) > > The problem is that for (Dedekind-) infinite sets, there > is no example of set size that preserves both of these > notions, since by definition there exists a bijection > between a D-infinite set and one of its proper subsets. > > So the best we can do is choose one of these two equally > good notions to preserve. Standard Cantorian cardinality > rejects the second in favor of the first. But I see no > reason that we can't reject the first in favor of the > second -- and this would give us TO's Bigulosity. > > I'd love to believe that there can be a completely > rigorous theory applicable to the sciences in which the > first notion is rejected for the second. I see no reason > for the lack of symmetry in that there's a sensible set > size that maintains the first property but none that > maintains the second property. > > We know all about standard cardinality. I want to know > how to develop a set size that preserves the second > property of finite set size. The closest such notion I've seen discussed was Fred Katz's http://arxiv.org/abs/math/0106100 a while back which (if I remember correctly) only worked for the Natural Numbers and was dependent on the axiom of choice for an ultrafilter. Instead of trying to construct such a set size notion from the ground up, why don't you assume its existence as an axiom and see what you can prove from it: If there is some set sizing function on the powerset of a set A with the properties you like then will A be forced to have certain features? Perhaps you could use your postulated set sizing method to produce a well-ordering or non-principle ultrafilter or measure or ... on a given set. Anyhow, you might figure out how your axiom relates to others like the axiom of choice, continuum hypothesis, ... and/or find out just how feasible such a size notion would be or maybe even derive a contradiction. Just a suggestion
From: David R Tribble on 12 Jun 2010 13:39 David R Tribble wrote: >> Now explain how this function retains its arithmetic meaning >> when extended to non-real (infinite) x. Or even how it can >> be extended in this way at all. > Tony Orlow wrote: > Okay. You would use this function, say, to map the naturals starting > at 0 with the naturals starting at 1. For every element x of the first > set exists an element y in the second such that y=x+1, right? Now, the > inverse function is x=y-1, correct? In the infinite case (barring > limit ordinals, which do not figure into my theory) this difference > does not decrease to 0, but stays at a constant value of 1. Using IFR, > then, we can say that set y has 1 fewer elements than set x, which is > what we would intuitively desire, since it is missing element '0'. For > me, x+1>x for all x, not just for the finite. Okay, so you start with a function f(x) defined over all real x, determine that it and its inverse function g(x) meet certain requirements, then declare by fiat that those functions and relations also hold for non-finite x arguments. Your exact phrase is: "For me, [f(x) holds] for all x, not just for the finite [x]". Do you see the critical steps you omitted? Hint: what does "for me" mean?
From: MoeBlee on 12 Jun 2010 13:48
On Jun 11, 12:52 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > card({xeS | x e [a b]}) > = > (max(range(g restricted to [a b])) - min(range(g restricted to [a b]))) > +1 > > Yes, that is just a very roundabout way of restating the cardinality > of some finite subset of a closed real interval. P.S. Of course this assumes that {xeS | x e [a b]} is finite. Anyway, I'll wait for Orlow to re-state what exactly else (his "infinite case" he adds to this formulation. I strongly suspect it will involve some kind of subtraction w-n, for natural numbers n. But then it would be up to him to DEFINE such an operation. MoeBlee |