Uncountability of the Rationals?
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From: David R Tribble on 13 Jun 2010 14:52 David R Tribble wrote: >> 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. > Tony Orlow wrote: > Actually, in Bigulosity, there are much larger countably infinite > sets. The square roots of the naturals have a higher "density" than > the naturals. How is that? The two sets should be exactly the same size (your size(), not just cardinality). Every natural has a square root, and every natural square root has a corresponding natural. (I assume you're only dealing with the positive roots.) > In terms of natural density, they probably have a > measure of oo? The set of positive natural square roots is not a subset of the naturals, so it does not have a natural density. > Not sure. For me, there are omega^2 number of square > roots of naturals within the range of N. Are you sure about this? Shouldn't there be omega square roots, or 2*omega if you're also counting negative roots? Are you sure you're not talking about the square roots of R?
From: Virgil on 13 Jun 2010 15:01 In article <7aec0c0d-c177-46ab-8c47-460016762bab(a)s9g2000yqd.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > ICI contraidicts the model of the von Neumann ordinals, and extends > natural density through IFR. To no purpose, as far as anyone can see. The von Neumann ordinals, on the other hand, have proven to be quite useful to the theory.
From: Virgil on 13 Jun 2010 15:03 In article <bf8bd12c-7c3e-436d-aeb8-1e4eb6034947(a)d8g2000yqf.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > On Jun 11, 3:54�pm, Virgil <Vir...(a)home.esc> wrote: > > In article > > <cd83f80f-337f-40d5-8ae5-764b216c9...(a)j4g2000yqh.googlegroups.com>, > > �Tony Orlow <t...(a)lightlink.com> wrote: > > > > > On Jun 10, 4:57�pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > > To have one and to be one are pretty different. Anyway, yes, > > > > continence has never seemed your problem; the exact opposite. > > > > > Don't forget to wipe this week. > > > > I was always under the impression that it was the responsibility of he > > who was incontinent to wipe himself. > > Perhaps several times a day Several times a day does seem a bit incontinent.
From: David R Tribble on 13 Jun 2010 15:18 David R Tribble wrote: >> 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. > Tony Orlow wrote: > 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. Come on, Tony, you've been told all this many times before. A string cannot contain more than a countable number of digits (symbols from the source alphabet). A string is either finite in length, or (countably) infinite in length. You knew that, of course. The list of all finite-length digit strings is a countably infinite list. The other list of all (countably) infinite-length digit strings contains only (countably) infinite-length strings, and is an uncountably infinite list. Actually, no such list can exist, at least not in the form of a "list" (as per Cantor's Theorem). All these strings can be arranged as infinite paths in an infinite tree, though.
From: David R Tribble on 13 Jun 2010 15:42
Virgil wrote: >> Real functions do allow infinite quantities either as arguments or >> values, so that you are making false statements. > Tony Orlow wrote: > Don't correct me incorrectly. Standard real functions take real value > parameters and return a real result. Where they have return an > infinite value, they are declare "undefined" or to have some infinite > "limit" at that point, and where any parameters are of infinite > nature, the modern mathematical procedure is to refer to the "limit" > of the function as one or more variables approach oo or -oo. To lie by > accusing someone is lying is...lame. Virgil has the advantage of knowing more math than you, Tony. You should not pretend to be an expert when you're not. It makes you look silly and pretentious. A real-valued function, by definition, has a domain from which it takes its arguments, and a range (or codomain) of values it "returns" (i.e., maps those arguments to). The function maps each argument value from the domain to a single result value in the range. If the function has no mapping for a particular argument, then the function is not defined for that argument value, and that value is not within the domain of the function. Likewise, real functions take real arguments, which means that infinity (+/-oo) cannot be in the domain set of a function, and thus can never be an argument to a function. An example is the reciprocal function, f(x) = 1/x. The domain of f is R \ {0}, i.e., all real values except 0. The function does not "return infinity" for argument 0, rather, f is simply does not have any result for it. It would be more correct to say that f(0) is "nothing" or "the empty set". Likewise, the real square root function, s(x) = sqrt(x) for all x >= 0, has the domain [0,oo). The function is not defined for any argument outside its domain, such as -1, meaning that there is no value for s(-1). It's not an "undefined value" of the function, rather the function has no value to map that argument to. It's amazing what you can learn with a little study: http://en.wikipedia.org/wiki/Domain_of_a_function http://en.wikipedia.org/wiki/Codomain http://en.wikipedia.org/wiki/Analytic_function -drt |