Uncountability of the Rationals?
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From: Tony Orlow on 15 Jun 2010 11:16 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. That's actually a nice succinct way to state it, however I generally include the condition that finite-case induction applies to f-g as well, to make clear that it is an extension of the finite form of proof to the infinite case. > > 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). No, that's correct. At best you can say one set is greater then the other, and you can quantify that difference as the ratio of the sizes functionally defined over [1,w]. There is some math that can be done, but it's nothing all that super-special. The special part is that it provides for a countably large spectrum of countably infinite sets which satisfy normal intuitions, such as the proper subset always being smaller, as well as more general notions. That seems like a little bit of an advance to me. > > -- > 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 - > Tony
From: Tony Orlow on 15 Jun 2010 11:21 On Jun 14, 7:41 pm, Virgil <Vir...(a)home.esc> wrote: > In article > <357d9d42-f0ac-4fdc-bf9a-8092183ca...(a)3g2000vbg.googlegroups.com>, > Tony Orlow <t...(a)lightlink.com> wrote: > > > On Jun 14, 3:49 pm, Virgil <Vir...(a)home.esc> wrote: > > > In article > > > <49f89dd7-4c7f-434d-af8c-391194e15...(a)a30g2000yqn.googlegroups.com>, > > > Tony Orlow <t...(a)lightlink.com> wrote: > > > > > Nope. They simply say that IF N has a size, it cannot be a member of > > > > N. > > > > Note that for the von Neumann naturals, the 'size' of a natural is never > > > a member of that natural. So why should it be any different for the set > > > of all of them? > > > How many elements in the null set? > > Is the size of the set with no elements an ELEMENT of that set? > > If so that would make it not empty after all! > > Read more carefully, TO! Okay, but the size of the set is equal to the number represented by the von Neumann ordinal, if they start at 0. If one wants it to start at 1, I suppose one could associate ech set with the deepest nesting of brackets in the set, which starts with one level. No, they are not elements of the sets, but measures thereof. TOny
From: Tony Orlow on 15 Jun 2010 11:28 On Jun 14, 7:46 pm, Virgil <Vir...(a)home.esc> wrote: > In article > <23e0c345-10ff-4172-8a1e-0c230c170...(a)f8g2000vbl.googlegroups.com>, > Tony Orlow <t...(a)lightlink.com> wrote: > > > On Jun 14, 3:23 pm, David R Tribble <da...(a)tribble.com> wrote: > > > Brian Chandler wrote: > > > >> I certainly don't think it would be correct to say that the reciprocal > > > >> of zero is the empty set. > > > > Tony Orlow wrote: > > > > In standard mathematics it is "undefined", both positively AND > > > > negatively infinite. It doesn't exist. The set of solutions at 0 is > > > > empty, ot perhaps a pair of hyperreals. > > What is undefined or does not exist is neither positively nor negatively > infinite. 1/x does not have a value, or a solution, at x = 0. > I believe there are formal systems where this value may be considered +/- oo, as they are equal in many respects. There are a few versions of the number circle, which can be generalized to more than one dimension, as toruses and their extensions. After all, for y=1/x, y approaches the same limit as x->oo or x->-oo. > > > > > A hyperreal can be returned by a real-valued function? > > > Only possibly where it is undefined. Don't ask me for a method. I > > said, "perhaps". > > > Tony > > It is certainly not standard for a function all of whose values all > reals to have a value which is not a real, whether hyperreal or unreal > in some other way. Given that the infinite value represents a counting number greater than any given natural, and that two functions have a difference without a limit of 0 as the independent variable increases from some natural n without bound, that difference remains in the infinite case. There is no reason to think it disappears, and the difference is functionally quantifiable. Love TOny
From: Jesse F. Hughes on 15 Jun 2010 14:17 Tony Orlow <tony(a)lightlink.com> writes: > 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. > > That's actually a nice succinct way to state it, however I generally > include the condition that finite-case induction applies to f-g as > well, to make clear that it is an extension of the finite form of > proof to the infinite case. > >> >> 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). > > No, that's correct. At best you can say one set is greater then the > other, and you can quantify that difference as the ratio of the sizes > functionally defined over [1,w]. There is some math that can be done, > but it's nothing all that super-special. The special part is that it > provides for a countably large spectrum of countably infinite sets > which satisfy normal intuitions, such as the proper subset always > being smaller, as well as more general notions. That seems like a > little bit of an advance to me. That little bit of math is utterly unclear. For instance, you claim that sqrt(w) * sqrt(w) = w, but that does *not* follow from the above principle, *even if* we assume that the above principle entails that sqrt(w) is defined. The above principle just allows us to conclude that sqrt(w) < w. -- "You are beneath contempt because you betray mathematics itself, and spit upon the truth, spit upon decency, and spit upon the intelligence of the world. You betrayed the world, and now it's time for the world to notice." -- James S. Harris awaits Justice for crimes against Math.
From: Virgil on 15 Jun 2010 15:37
In article <3bee359c-97d1-4bdc-ac57-7086e9249303(a)e5g2000yqn.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > 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. "Integrity"? Until you have proved that the extensions you are forcing on certain real functions are always consistent with each other, you don't have any integrity to maintain. |