Uncountability of the Rationals?
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From: Aatu Koskensilta on 15 Jun 2010 03:51 Tony Orlow <tony(a)lightlink.com> writes: > On Jun 11, 6:56�pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > >> The formulas you mentioned don't entail that there is such an ordinal >> number, true. But we do prove from our AXIOMS that there exists such >> an ordinal number. > > Uh huh. How so? You can find the details in any set theory text. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 15 Jun 2010 04:03 Tony Orlow <tony(a)lightlink.com> writes: > What does it mean to say omega is larger than any n in N? Which axiom > defines what this means for omega? A definition defines what it means for omega. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 15 Jun 2010 04:15 Tony Orlow <tony(a)lightlink.com> writes: > Hahaha. Our computational technology is based on George Boole's work, > which preceded the flowering of set theory, and he wasn't even a > mathematician. George Boole wasn't a mathematician? He was a professor of mathematics, so I take it this is just an expression of an unusually low opinion of his work. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Tony Orlow on 15 Jun 2010 11:09 On Jun 14, 4:49 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > David R Tribble <da...(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)- Hide quoted text - > > - Show quoted text -
From: Tony Orlow on 15 Jun 2010 11:09
On Jun 14, 4:45 pm, Virgil <Vir...(a)home.esc> wrote: > In article > <66c0fd40-129d-4e57-9b90-02bb8314c...(a)i31g2000yqm.googlegroups.com>, > Tony Orlow <t...(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. No, there is an "and", a '^', between them. They, together with the fact that any infinite set size is greater than any natural number, are stated axiomatically to imply that the same inequality between functions applied to infinite set sizes maintain their integrity in the absence of a limit of no difference in the positive infinite case. > > E.g., f(x) = x + 1/x, g(x) = x Yes, this is like the example of {1/n: neN}, a compactified monotonically decreasing set. That was pretty well handled. Even, as I thought about it, my initial example of [5,60] worked, since it would be floor(1/60) - ceiling(1/5)+1 = 0-1+1 = 0, the number of inverses of natural n in that range. That is, this would be like that if you were talking about sets defined using f and g. But you are talking about infinite case induction. So, please remember when considering it the stipulation that f(x)-g(x) does not have a limit of 0 as x->oo. > > > > > > > Therefore > > A x>n f(x)>g(x) > > And since > > omega>n (and all infinite x>n) > > Then > > f(omega)>g(omega) No comment? > > > > (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. > It's more general than natural dnsity, that's for sure. > > > > > > > "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.- Hide quoted text - I didn't use x in two different senses in one statement, did I? No, I was talking about two different things, like when I use functions f and g both in ICI and IFR. > > - Show quoted text - Tony |