Uncountability of the Rationals?
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From: Virgil on 14 Jun 2010 15:52 In article <4599cfae-a652-4034-a420-ac9db2ed294f(a)q12g2000yqj.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > On Jun 12, 3:47�pm, FredJeffries <fredjeffr...(a)gmail.com> wrote: > > On Jun 12, 12:42�pm, David R Tribble <da...(a)tribble.com> wrote: > > > > > Jesse F. Hughes wrote: > > > >> You [Tony] sometimes say that you > > > >> don't think omega is the smallest infinite ordinal, but I'm not sure > > > >> what you mean when you say that. > > > > > Tony Orlow wrote: > > > > I don't use the word "ordinal" in my arguments (sure, go find one > > > > mention from 1996 or whatever). I say there are a wide spectrum of > > > > countable and uncountable infiniites, given the right techniques. > > > > > What Tony means is that size(E) < size(N) for the sets N and > > > E = {0,2,4,6,...}. Specifically, he means that size(E) = size(N)/2, > > > where "size(S)" is his personal definition of "set size". > > > > No, he told me that Size(E) = Size(N)/2 + > > 1http://groups.google.com/group/sci.math/msg/df77005159c91c90 > > because of 0. > > > > At least, I THINK that's what he told me. > > > > > > > > > > > > > He also allows for things like sqrt(size(N)) and log(size(R)). > > > These "values" of his are "different infinities", all based on his > > > idea of a "unit infinity".- Hide quoted text - > > > > - Show quoted text - > > Actually, you're right. N should always start with 1 for IFR. I made > an error of 1 by including 0. If we include 0, then we may say E=N/2+1 > and O=N/2, because of that extra even element. Apologies. But the von Neumann naturals are quite as good as any others, and they, or something like them, are coming to be the preferred form in mathematics, so when everyone else has 0 in their naturals, poor TO will be left out in the cold.
From: Virgil on 14 Jun 2010 16:24 In article <03571022-ccc2-4995-aeb1-8cbd8e2690bf(a)g19g2000yqc.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > On Jun 13, 3:42�pm, David R Tribble <da...(a)tribble.com> wrote: > > 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. Real functions do NOT allow infinite quantities either as arguments or values, so that you are making false statements. I have at least once corrected that statement. > > It looks like he may also be able to tell who's talking, which gives > him an advantage over you. HE'S the one saying real functions can take > infinite argument or return infinite values. I find it revealing that the only time TO considers me to be right is when I have clearly erred. al-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. Correct! > > > > 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". > > Tell it to Virgil. I concur that f(x) has no value, i.e., is not defined, at x = 0. > > > > > 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. > > No kidding. Thanks for the kindergarten lesson. Show it to Virgil. David is right re this, and TO should learn from him. > > > > > 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
From: Virgil on 14 Jun 2010 16:34 In article <0090bd76-afa4-4925-93df-98b00889e1b6(a)x21g2000yqa.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > On Jun 13, 3:56�pm, David R Tribble <da...(a)tribble.com> wrote: > > Jesse F. Hughes wrote: > > >> I know what the function n |-> n^2 means for infinite values. �I'm > > >> asking you know about the function sqrt. �What is sqrt(omega)? �What > > >> properties does it satisfy? > > > > Tony Orlow wrote: > > > (x>1 -> sqrt(x)<x) ^ (omega>1) -> sqrt(omega)<omega > > > It's the size of a smaller countably infinite set, the set of squares > > > of naturals. > > > > All this does is state a property of something you call "sqrt(omega)". > > Yes, it's less than omega, Bigulosity-wise due to ICI. > > > > > You left out the parts that > > a) actually define what sqrt(omega) is, and > > Suffice it to say it's countably infinite but less than omega Thus equal and less than sumultaneoulsy? > > > b) show that it actually exists. > > It exists as "actually" as omega, which only exists virtually. I've > made that clear oodles of times. The consistence of cardinality has been established. The consistency of "bigulosity" has not been established. And until it is, the various "existences" it would require are, at best, still hypothetical and conditional, rather than actual. > > > > > Oh, and you also neglected to mention if "x", "1", and "omega" are > > reals, naturals, ordinals, cardinals, bigulosities, or whatever. > > sqrt is a real function for x>0. And for a large and growing portion of the world, 0 is a member of N. > "Omega" is the size of N starting at > 1, which isn't an absolute number, but an expression of potential > infinity. "Omega" is the ordinality of N, not its cardinality. Aleph_0 is its cardinality starting from 0 or from 1.. > > > > > (You did state that "sqrt(omega)" is a "set size", but did not state > > what a "set size" is.) > > Bigulosity according to IFR and ICI. Cardinality is a perfecty adequate "set size", and TO has yet to show the need for any other. > > > > > You've been asked this before: > > Are you saying that sqrt(omega) * sqrt(omega) = omega? > > Yes, if such an operation is desired. For instance, I would say the > set all pairs (x,y) where x and y are both squares of naturals would > have a Bigulosity of omega. > > > If so, what do you mean, exactly, by "omega" and "*"? > > omega is the size of N starting at 1, and * is normal multiplication > extended through ICI to establish order among infinities. GIGO! > > Tony
From: Virgil on 14 Jun 2010 16:45 In article <66c0fd40-129d-4e57-9b90-02bb8314ca45(a)i31g2000yqm.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > f(x)-g(x) > 0 for x>n|neN > lim(x->oo: f(x)-g(x)) >0 That second line does not follow from the first line. E.g., f(x) = x + 1/x, g(x) = x > Therefore > A x>n f(x)>g(x) > And since > omega>n (and all infinite x>n) > Then > f(omega)>g(omega) > > > > > (1) There is no such x's in the domain if the domain is w+. So we must > > have the domain of f to be ordinal-greater than w+. And I take it '<' > > is ordinal-greater here, not cardinal-greater. > > No ordinals. They have nothing to do with Bigulosity. > > > > > (2) Anyway, with the domain thus modified, of course there is such an > > a, and you don't need the antecedent of your statement for that. > > > > > > "number of elements" > > > > > Count. Member of N+. "Zize" of a set. Bigulosity. Basta? > > > > N+ is the set of all positive natural numbers? Or (I forgot from last > > time) you mean 'N+' to be w+ (i.e., the wu{w})? > > N+ is, as others have used it, the set of naturals and hypernaturals, > the infinite extension of w. > > > > > So either "count" means a positive natural number, or count means a > > countable ordinal, or something else? And that's all Biglosity is? > > Bigulosity is a better "set size" in the infinite case than > cardinality. Claimed but never proven. Until TO, or someone else, can provide some useful , or at least interesting, results via "Bigulosoity" which cardinality can not, "Bigulosoity" is of no use at all. > > > > > > "value range" > > > > > a<x ^ x<=b = (a,b] > > > > I.e, {x | x in the range of f and f in (a b]} I guess? > > > > Be careful, just as a matter of keeping variables tidy, previously you > > were using 'x' for members of the domain. > > > > Isn't the value range under consideration a subset of the domain? 'x' > is used for all sorts of things. Clarity suggests using different variables for members of different sets. For example. x^2 + x^2 = x^2 does not convey circles anywhere nearly as well as x^2 + y^2 = r^2.
From: Jesse F. Hughes on 14 Jun 2010 16:49
David R Tribble <david(a)tribble.com> writes: > Serves me right for trying to think like Tony. > > Oh well, this is easily solved! The problem is that TA1 is too > general, so we need to limit its scope. Instead of any property > P, let's just consider arithmetic properties and operators. > So, perhaps something like this: > > Tony's Axiom [TA2] > Given arithmetic function f(x) = y, where x and y are reals, > then f(w) = u, where w and u are infinite ordinals. > > Yeah, I know this really isn't any better when taken any further, > but Tony really needs to put his money where his mouth is and > just go ahead and write his assumptions down already. He can't > proceed with any of his other statements about bigulosity, > ICI, IFR, or whatever until he starts with an axiom or two. > And he knows it. It seems to me that he's stated his assumption, but he doesn't get quite how much it assumes. Given any real valued functions f and g, if lim (f - g) > 0, then f(x) > g(x) where x is any infinite number. As a consequence of this, I guess, it follows that f and g are defined on infinite numbers, though we don't know anything about their values aside from the fact that f(x) > g(x). -- Jesse F. Hughes "Why do the dirty villains always have to tie your hands *behind* ya?" "That's what makes them villains." --Adventures by Morse (old radio show) |