Uncountability of the Rationals?
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From: David R Tribble on 12 Jun 2010 20:19 Jesse F. Hughes wrote: >> I never overrule the random sig picker. > Marshall wrote: >> -- >> "People make mistakes. Better to live today and learn the truth, than >> to be one of those poor saps who died deluded, thinking they knew >> certain things that they just didn't. Thinking they had proofs that >> they didn't." --James S. Harris, almost too sad for a .sig > David R Tribble wrote: >> Wow. Your sig picker must be psychic, or perhaps artificially intelligent. > Marshall wrote: > It demonstrates its relevance via the same mechanism that > Herc uses when pointing to random pages in a book. Tiny alien gnomes living in its spleen?
From: Jesse F. Hughes on 13 Jun 2010 00:24 David R Tribble <david(a)tribble.com> writes: > Tony Orlow writes: >>> PS - If you wouldn't mind laying off, the JSH references get a tad tiring. >> > > Jesse F. Hughes wrote: >> I never overrule the random sig picker. >> >> -- >> "People make mistakes. Better to live today and learn the truth, than >> to be one of those poor saps who died deluded, thinking they knew >> certain things that they just didn't. Thinking they had proofs that >> they didn't." --James S. Harris, almost too sad for a .sig > > Wow. Your sig picker must be psychic, or perhaps artificially > intelligent. It really is random. Or else I have Herc-like powers and the Gods themselves are commenting on our threads. -- "Witty adolescent banter relies highly on the use of 'whatever.' Anyone out of high school forced to watch more than an hour of 'Laguna Beach' might possibly feel the urge to beat themselves about the head with a large stick." -- NY Times on an MTV reality show
From: Tony Orlow on 13 Jun 2010 06:42 On Jun 11, 12:45 pm, David R Tribble <da...(a)tribble.com> wrote: > Tony Orlow wrote: > > I would like you consider a slightly modified goal: To exactly order > > sets according to size, and to distinguish between those that have an > > absolute size and those that do not. For me, no countably infinite set > > has an absolute size, but only one relative to other countably > > infinite sets, especially the standard N, starting at 0. > > Wouldn't it be easier to say that some chosen countably infinite > set has a standard "absolute" size, and then that all other such > sets have a size that is some fraction of that size? Thus every > set then has an "absolute" size, based on the standard set size. No, it is easier to establish some absolute unit infinity for the number of points in the unit line segment, and wed the uncountable to measure, while treating the uncountably infinite as more of an undetermined size, parametrically compared over iterations. > > For convenience, you'd want to choose the "largest" countably > infinite set for the standard "absolute" set size. This would > probably be N, since it contains all of the countable naturals. > (Another logical choice would be Z.) Given this, you could then > say that every countably infinite set does indeed have an > "absolute" set size. Actually, in Bigulosity, there are much larger countably infinite sets. The square roots of the naturals have a higher "density" than the naturals. In terms of natural density, they probably have a measure of oo? Not sure. For me, there are omega^2 number of square roots of naturals within the range of N. However N, starting from 1 (not 0 as I mistakenly said before) is the standard countable infinity, compared to which all others are measured, according to IFR. > > Of course I've glossed over all the important details, but that's > really your job to provide. I think I've responded appropriately. Peace, Tony
From: Tony Orlow on 13 Jun 2010 07:02 On Jun 11, 1:43 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jun 11, 8:20 am, Tony Orlow <t...(a)lightlink.com> wrote: > > > The inverse function rule. If a set of reals is bijected with a > > segment of the naturals > > I.e., for a countable subset of R ('R' being the set of real numbers - > the real numbers as we ordinarily understand them?) > > > using an invertible formula, > > What is an "invertible formula"? And the terms you use to answer that > question, do they have defintions too? Does ANYTHING you say end with > axioms and primitives or AT LEAST some language that is informal (thus > not necessarily in sequence back to primitives) but COMMON enough that > other people may grasp your notions without resort to special PERSONAL > terminology? > Consider, to begin with, a monotonically increasing function, which of necessity has a monotonically increasing inverse function. That should be a good starting point. > > then that inverse > > formula may be used to calculate the number of elements in the set > > within any given value range. > > "may be used to calculate" (do you mean there is a recursive function, > or is your notion of 'calculate' something different from Church- > Turing?) No, I mean with a simple function which may be proved inductively to be greater or less than some other function of the same variable, for a number greater than any finite number. I mean, if f(x)>g(x) for all x>a, for some finite a, then the same applies to a positive infinite a (the size of an infinite set) as long as f(x)-g(x) doesn't have a limite of 0 as x->oo. > > "number of elements" Count. Member of N+. "Zize" of a set. Bigulosity. Basta? > > "value range" a<x ^ x<=b = (a,b] > > > Where a<b, > > I take it '<' is the standard ordering on R. Yes. a<b ^ b<c -> a<c ~(a<b ^ b<a) ~(a<b v b<a) = (a=b) > > > and where S is a set mapped > > from the naturals using f(n)=x, > > I take it S is a subset of R. For the purposes of IFR, yes. I am sure extensions are possible, but I should establish this first. > > f is the bijection you mentioned earlier? Um, yes? f: N->S is half the bijection, along with g: S->N in the other direction, S being a subset of R. > > > and where there exists an inverse > > function g(x)=n such that f(g(x))=g(f(x))=x, > > I.e., where g is the inverse of f. Yes. > > > the number of elements in > > the set S within the range [a,b] is given by floor(g(b))-ceiling(g(a)) > > +1. Try it on any finite set. Now imagine applying it to the range > > [0,omega]. > > As far as I can tell, this is not well defined, since it is RELATIVE > to f (g being the inverse of f). So what we would have is "number of > elements of [a b] PER f" is [...]. But, even worse (to the extent I > understand what you're trying to say) there IS NO bijection between [a > b] and any segment of w, perforce no bijection between S and a segment > of w. In the words of Scooby Doo, "Rrhuuhrrr?" Please try a simple example and see if it works. I've described the technique for the finite case. Apply it. Play with it. Open ourself up to new definitions of "set size". > > Would you please restate this whole thing in standard terminology, or > if mixed with your own terminology, please define to either standard > terminology or to primitives. > > MoeBlee I think that should be clear. Love, Tony
From: Tony Orlow on 13 Jun 2010 07:08
On Jun 11, 2:47 pm, David R Tribble <da...(a)tribble.com> wrote: > Tony Orlow wrote: > >> I was wondering what thoughts you [Walker] had on the countably infinitely long > >> complete list of digital strings, if anything. > > David R Tribble wrote: > >> That would be the countable list of all finite-length digital strings, > >> wouldn't it? Or would it be the uncountable list of all infinite- > >> length > >> digital strings? > > Tony Orlow wrote: > > Um, excuse me, but is the list of all finite digital strings not > > ultimately countably infinite in width? If it is finite, then please > > state this finite width. > > If you mean what is the number of digits needed to list > all possible finite-length digital strings, the answer is omega. > So yes, the "width" of the list is countably infinite. > > > If it is countably infinite, then how does it > > differ from your uncountably long list? > > Well, the list of all finite-length digit strings contains > only finite-length strings, and is a countably infinite list. > The list of all infinite-length digit strings contains > only infinite-length strings, and is an uncountably infinite > list. I'm sorry, Tribbs, but are you implying that these strings are longer than countably infinite, or are you sying the lists are equally wide, OR, are you saying there are different "kinds" of countably infinite widths? Please expliculationalize. :) TOny |