Uncountability of the Rationals?
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From: Virgil on 10 Jun 2010 15:33 In article <76dd99bf-5095-4c31-9e2f-bd0b49147fb5(a)x27g2000yqb.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > On Jun 9, 2:19�pm, David R Tribble <da...(a)tribble.com> wrote: > > David R Tribble wrote: > > >> You completely avoided answering my question, which was > > >> meant as a concrete example to answer the bigger question: > > >> �"is bigulosity general enough for to be useful for all sets?" > > > > Tony Orlow wrote: > > > In some cases Bigulosity cannot make a greater distinction than > > > cardinality, ... > > > > You mean that in *most* cases it cannot. Based on your description, > > bigulosity only applies to ordered sets of monotonically increasing > > numeric members. That's a very tiny subclass of all interesting > > sets. > > No, That's IFR in particular which is used on countable sets of reals > mapped from some segment of N. Which is, indeed, a very tiny subclass of interesting sets. > I have precisely stated what IFR and ICI are. If you cannot > understand, it's only due to deliberate ignorance. TO has often claimed a precision of statement which he has not actually provided, as, for example, his recent definition of "sequence", which, in effect, was the same as "totally ordered set". > > And I'll I get from you are defensive insults. And they were SUPPOSED to be offensive! > > > > > > IFR goes far beyond either of those already. (sigh) > > > > So you claim. Could you show us an example or two where > > Natural density fails but your "IFR" works? > > I just did. Natural density gives a measure of zero for any set mapped > from the naturals which does not use a linear mapping formula. Any > higher power mapping formula, such as the set of squares, results in a > measure of 0. In IFR, results agree with natural density for linear > formulas. For any other invertible mapping formula, which includes all > bijections, IFR gives a formulaic result in terms of omega. Granted, > the formulas don't make sense to you unless you accept the concept of > infinite-case induction, which has not been refuted as inconsistent in > any manner. In mathematics, one must present a positive argument before there is any need of refutation. > > > > > Precisely defining what exactly "IFR" is would help, too. > > If it really does give multiple results for the same sets (as > > you've said), then you can't go around claiming that it is > > mathematically sound. > > I didn't say that. Where IFR is applicable it gives precise results, > but that is only for linearly ordered sets of real quantities. So it only works on subsets of the reals with its standard ordering? Since that is a well known area, we would need considerable evidence that your techniques add anything before taking them seriously, and so far we have not seen any.
From: Tony Orlow on 10 Jun 2010 15:34 On Jun 10, 3:05 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Transfer Principle <lwal...(a)lausd.net> writes: > > To me, the difference between ZFC and non-ZFC theories > > is more analogous to the difference between a flight > > on American Airlines from JFK to LAX, and a flight on > > Virgin America from JFK to LAX. In this analogy, > > MoeBlee is forcing everyone to fly only on American > > Airlines and not Virgin or any other airline. > > No, he's not. Moe has *never* said that ZFC is the only acceptable > theory. > > > In another thread, he claims that he doesn't consider ZFC to be the > > best theory, but the fact that he compares ZFC to an airliner and > > other theories to rubber balls speaks for itself. > > The fact is that Tony, AP, etc., have *not* offered any coherent > mathematical theory at all *and you know it*. Thus, if Moe criticizes > their blatherings, then it is extraordinarily disingenuous to claim this > as evidence that he accepts only ZFC. > > -- > Jesse F. Hughes > > "Really, I'm not out to destroy Microsoft. That will just be a > completely unintentional side effect." -- Linus Torvalds "Disingenuous" means "lying". I believe Transfer's comment falls into the category of a best-guess interpretation of Moe's motives. You, Moe, Virgie, The Tribble and others seem completely closed to the concept of any improvement on the standard obfuscation. For, "none shall drive us from the Garden which Cantor has created for us". If it doesn't produce fruit, it's time to plant a new bed, or at least fertilize. TOny
From: Virgil on 10 Jun 2010 15:40 In article <0f90b6ed-0038-4424-9c4b-189bc7a1b7dd(a)d4g2000vbl.googlegroups.com>, Tony Orlow <tony(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? > > (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? > > (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? > > (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. > > Thanx, > > Tony
From: Virgil on 10 Jun 2010 15:43 In article <aed4ddae-8d91-4d6f-8ff0-2b232a48109e(a)q29g2000vba.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > 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 > > Please cite an instance recently where I've used any circular logic. > Thanks, > > Tony Tony's logic is nowhere as neat and orderly as circles are.
From: MoeBlee on 10 Jun 2010 17:06
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. 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 |