Uncountability of the Rationals?
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From: Jesse F. Hughes on 11 Jun 2010 07:48 Brian Chandler <imaginatorium(a)despammed.com> writes: > So if S = { x^2 | 1 <= x <= Bigun } the function f(x) is x^2. This is > a monotonically > increasing function: OK! Now "the inverse function" is (obviously!?) > sqrt(x), so the number of squares is > > sqrt(Bigun) - sqrt(1) + 1 = sqrt(Bigun) > > And so on. I mean, the number of primes, for example, is found > immediately using the inverse prime function, and um Lordy. I was kidding when I wrote that Tony can prove sqrt(aleph_1) > ln(aleph_1) earlier. I had plumb forgot that Tony really does talk about the square root of Bigun. -- "After years of arguing I realize that your intellects are too limited to fully grasp my work. [...] Still, no matter how child-like your minds are, [...] since you have language, [...] there's a chance that I'll be able to find something that your minds can handle." --JSH
From: Tony Orlow on 11 Jun 2010 08:15 On Jun 10, 2:42 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jun 10, 1:35 pm, Tony Orlow <t...(a)lightlink.com> wrote: > > > and I > > have to retreat > > You retreat every time you fail to define your terminology in a non- > circular way, every time you fail to answer the substantive questions > but instead blame the actually informed people here for what you claim > to be their obtuseness or dishonesty, every time you fail to answer > the substantive questions but instead resort to things like corny "yo' > mamma" talk. > > MoeBlee No, I tend to retreat when the criticism gets too personal and harsh and starts to seriously erode my motivation and self esteem. BTW, you mention a goal of set theory being to have a "size" for every set. 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. Uncountably infinite sets can have an absolute size in terms of zillions. Both kinds of infinity can be multiply tiered using infinite case induction on all the arithmetic functions used in bijections. Also, sometimes I retreat to rethink something, when an issue in my thinking is pointed out, which has happened several times, and which I don't regret. I welcome logical criticism. When it gets personal, I can get personal back, but I'm trying to behave myself. Are you? TOny
From: Tony Orlow on 11 Jun 2010 08:30 On Jun 10, 3:40 pm, Virgil <Vir...(a)home.esc> wrote: > In article > <0f90b6ed-0038-4424-9c4b-189bc7a1b...(a)d4g2000vbl.googlegroups.com>, > Tony Orlow <t...(a)lightlink.com> wrote: > > > > > > > On Jun 9, 10:10 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > > On Jun 7, 7:30 am, Tony Orlow <t...(a)lightlink.com> wrote: > > > > > On Jun 5, 11:36 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > > > > If the T-riffics are distinct from the H-riffics, then I > > > > > would like to learn more about the T-riffics before I > > > > > attempt to pass judgment. I don't mind learning more > > > > > about sets other than the classical real numbers (i.e., > > > > > standard R) and standard set theories. > > > > The T-riffic numbers are much like the adic numbers in that they > > > > express number strings of infinite length in a finite representation, > > > > and therefore are only capable of expressing "rational" numbers with > > > > respect to the scale we are addressing. The digital point of normal > > > > digital systems is retained to the right of the 1's digit at location > > > > 0, but in addition we can insert other digital points uncountably to > > > > the left or right of this middle point. Also, like classical digital > > > > systems, one can use any natural base above 1, so we may have binary, > > > > octal, decimal, or hexidecimal (or whatever) T-riffic number systems. > > > > The choice of extra digital points to the right or left of the > > > > classical digital point is based on the formulaic infinitude one want > > > > to express. In order to be consistent with normal digital systems, > > > > with the classical digital point at location 0, we can place other > > > > digital points to the right or left by specifying the digit location > > > > relative to that point. In a T-riffic of base x, for instance, one > > > > zillion would be expressed with a 1 to the left of a point at > > > > logx(zillion). Thus, where z is a zillion, one zillion in base two > > > > would be: > > > > 1.(log2(z))000...000.0. > > > > A zillionth can be expressed as: > > > > 0.000...001.(-log2(z))0 > > > > Thanks. So essentially it is a system of infinitesimals. > > > Hi Transfer. My pleasure. I'm glad you responded. I thought you were > > running off to steal my Field's Medal. J/K... > > > The T-riffics are a digital system capable of representing ordered > > infintiesimals as well as infinities. The "digital points" can be > > placed anywhere, though they aren't necessary unless the uncountably > > apart in the uncountable sequence of digit positions. > > > > Later on, Tribble points out that it is similar in > > > some ways to the AP-reals. A key difference between > > > the AP-reals and the T-riffics is that the former > > > is decimal, while the latter is binary. > > > No, the T-riffics can be expressed in any base, as can the H-rifiics. > > As a neo-taoist I tend to like binary. 0 and 1 are rather > > elemental. :) > > > > In searching for a way to make the T-riffics more > > > rigorous, we can start by letting z, the symbol for > > > "zillion," be a primitive and go from there. > > > Yes. > > E 0 > > E 1 > > 0<1 > > x<y -> E z: x<z<y > > > This leads to some level of infinites between counting numbers. > > Including the reals we can call this z, for a zillion, not to be > > confused with Z, the set of all integers. Then we marry count to > > measure for continuous space. > > > > > Now, central to my theory is infinite-case induction. You may dredge > > > > up a thread I started some years ago called "Infinite Induction and > > > > the Limits of Curves", in response to a challenge regarding infinite- > > > > case induction from Chas Brown. In any case, infinite-case induction > > > > is simply an extension of finite inductive proof to the infinite case, > > > > without reference to any limit ordinals or transfinite concepts. > > > > Thanks to all the critics and naysayers I was able to refine the rule > > > > so that it was consistent. Simply stated, any inequality which may be > > > > inductively proved to be true for any value greater than some > > > > particular finite value can be considered true for any positive > > > > infinite value, provided that the difference between the two > > > > expressions upon which the inequality is based does not have a limit > > > > of zero as the variable approaches infinity. Thus x+2<x*2<x^2<2^x<x^x > > > > for any x greater than 2. > > > > Ah, infinite-case induction. Once again, TO is hardly the > > > first person to desire that induction be extended to > > > infinite as well as finite cases. > > > We are not alone. It was pointed out that x^2<2^x for x>4 and not x>2, > > but omega>4, anyway, right? :) > > > > TO claims that his induction rule is consistent, but I > > > suspect that no one is convinced of its consistency. I've > > > tried to find a way to make something similar consistent, > > > but so far it hasn't worked. > > > :( > > > I believe every aleph is greater than any finite number, and as long > > as the difference establishing an inequality does not shrink to > > nothing, the difference persists, and order is established. Where does > > it break? > > > > We need something roughly like this: > > > > Infinite-Case Induction Schema: > > > > If phi is a formula that doesn't mention the primitive > > > symbol z, then as many closures of: > > > Wait: primitive symbol z is a zillion? > > > > (AxAy ((x<y & phi(x)) -> phi(y))) -> phi(z) > > > > as possible to avoid inconsistency are axioms. > > > > To make this rigorous, we need to replace "as many > > > closures as possible" with a rigorous rule to determine > > > which closures we are discussing. Otherwise, the schema > > > is called "oracular" and will be rejected.- Hide quoted text - > > > > - Show quoted text - > > > Allow me to retort (okay, not ala Pulp Fiction, but...). Consider f(x) > > and g(x) to be any two given arithmetic formulae on x. If: > > Are your "formulae" any different from functions, and if so, in what way? > And if so what are their domains and codomains? > No, they are exactly invertible functions, as are used in any bijection. IFR is simply an extension of bijection and an extension of natural density which derives more distinctions than either of those. > > > > (1) It can be classically inductively proven that f(x)>g(x) for every > > real value greater than some finite x (the base case), and > > Does that imply that these are real functions? > Yes, they can be any invertible arithmetic functions. Does that pose a problem for you? Probably, since I am applying them to infinite values. However, that's what infinite case induction is for. > > > > (2) The difference between the two terms which establishes the > > inequality does not have a limit of zero as x approaches infinity, > > then > > What is the "arithmetic" of the set in which those differences occur? > The arithmetic of the set? The difference between f(x) and g(x) is f(x)-g(x) for any given natural x. If the limit of f(x)-g(x) tends to zero as x tends to oo, then infinite case induction fails. Otherwise it holds. > > > > (3) If omega is greater than any finite counting number, or c or > > aleph_1 (whether they are the same or not), then the same rule also > > applies to such large numbers. > > > Please let me know which step seems vague. > > All of them. > > I hope I clarified some of that for you. Tony
From: Jesse F. Hughes on 11 Jun 2010 08:47 Tony Orlow <tony(a)lightlink.com> writes: >> > (1) It can be classically inductively proven that f(x)>g(x) for every >> > real value greater than some finite x (the base case), and >> >> Does that imply that these are real functions? >> > > Yes, they can be any invertible arithmetic functions. Does that pose a > problem for you? Probably, since I am applying them to infinite > values. However, that's what infinite case induction is for. No, your description of infinite case induction only gave certain sufficient conditions for concluding that f(omega) > g(omega). It does *not* give any reason to believe that a function f defined on natural (or real) numbers can be extended to infinite sets in some unique way. You are presuming that without stating it explicitly. What you really want, but are not stating, is something like this: For every (invertible?) function f:R -> R, there is a unique extension f':? -> ? (Ord -> Ord?) such that ... (what?). -- "I am a force of Nature. Time is a friend of mine, and We talk about things, here and there. And sometimes We muse a bit [...] and then We watch them go... in the meantime, Time and I, We play with some of them, at least for a little while." --- JSH and His pal, Time.
From: Tony Orlow on 11 Jun 2010 09:10
On Jun 10, 5:06 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jun 10, 2:19 pm, Tony Orlow <t...(a)lightlink.com> wrote: > > > > You retreat every time you fail to define your terminology in a non- > > > circular way, every time you fail to answer the substantive questions > > > but instead blame the actually informed people here for what you claim > > > to be their obtuseness or dishonesty, every time you fail to answer > > > the substantive questions but instead resort to things like corny "yo' > > > mamma" talk. > > Please cite an instance recently where I've used any circular logic. > > I said circular TERMINOLOGY. You have no primitives, so your > terminology is never grounded. It either goes circularly (and I showed > you instances of that a few years ago), or it just dangles at terms > themselves not defined not primitive. Yes, go back a few years. It's like arguing with one's wife. "What about that thing you said about my hair last summer???" > > Or, please, I'd love for you to list your exact primitives: constants, > predicate symbols, function symbols (or words that signal such > formalisms). > > A few years ago you made a pretense of doing so, but it was an > embarrassing mess. I even took a good amount of time to help you out. > It ended up with you getting impatient and childishly defensive as you > ordinarily do, so that nothing really was accomplished. > > MoeBlee Moe - You seem to think that coming up with a complete alternative to transfinite set theory from the ground up is a simple matter to be easily spewed out. Then when I try to tell you where it's going, you deride me for not explaining every detail, including how logic works. So, you'll just have to wait for the book. TOny |