Uncountability of the Rationals?
in [Math]
|
Prev: Collatz conjecture
Next: Beginner-ish question
From: MoeBlee on 10 Jun 2010 12:17 On Jun 10, 7:17 am, Tony Orlow <t...(a)lightlink.com> wrote: > I present a box of tools. You present a rock. What you call a "rock" accomplishes a specific task, viz. ensuring that every set has a certain cardinality. What you call a "tool" does not accomplish that task, thus your "tool" and the "rock" are not comparable in that sense. / MoeBlee: Look at that airplane. It takess 300 people at a time from New York to L.A. in five hours. ToeKnee: No, wings are schlocky. MoeBlee: Okay, what's your invention? ToeKnee: Look, I can roll this ball across the tabletop without wings. MoeBlee: But the airplane carries 300 people at a time from New York to L.A. in five hours. Even though your ball may be nice to you because it doesn't have schlocky wings, it can't carry 300 people at a time from New York to L.A. All it carries is the dust on it for a distance of about five feet. So you can't compare the airplane, which accomplishes a certain purpose, with your ball, which does not accomplish that purpose. ToeKnee: You just don't get it. Sigh. ChanceFor PrinceOfPull: There goes MoeBlee again silencing alternative theories. / MoeBlee
From: David R Tribble on 10 Jun 2010 13:02 Tony Orlow wrote: >> In some cases Bigulosity cannot make a greater distinction than >> cardinality, ... > David R Tribble wrote: >> 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. > Tony Orlow wrote: > No, That's IFR in particular which is used on countable sets of reals > mapped from some segment of N. So then what is the *general* case for IFR? Tony Orlow wrote: >> ... and may have to simply classify a set as countably infinite. > David R Tribble wrote: >> What about all the sets with cardinality greater than Aleph_0? >> Are their bigulosities all the same? If so, how do you show this? > You didn't answer this question, so I'll ask it again: What are the bigulosities of sets with cardinality greater than Aleph_0? [insults snipped] David R Tribble wrote: >> Once you offer some logical rigor for your notions, people >> will be more open to discussing them. For now, all we have >> is imprecise descriptions and your assurances that they work >> and are truly wonderful. > Tony Orlow wrote: > And I'll I get from you are defensive insults. You'll get criticism of your lack of mathematical coherence and rigor, like you always do. I'll let you do all the name-calling and insulting. David R Tribble wrote: >> Could you show us an example or two where >> Natural density fails but your "IFR" works? > Tony Orlow wrote: > 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. Which is ...? What is the IFR result for comparing the sizes of set N and the square naturals? Natural density says it's 0, what does IFR say it is? David R Tribble wrote: >> 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. > Tony Orlow wrote: > I didn't say that. Where IFR is applicable it gives precise results, > but that is only for linearly ordered sets of real quantities. Does that mean that it does not given precise results for other types of sets?
From: David R Tribble on 10 Jun 2010 13:10 Tony Orlow wrote: >> I present a box of tools. You present a rock. > MoeBlee wrote: > What you call a "rock" accomplishes a specific task, viz. ensuring > that every set has a certain cardinality. > > What you call a "tool" does not accomplish that task, thus your "tool" > and the "rock" are not comparable in that sense. > > MoeBlee: Look at that airplane. It takess 300 people at a time from > New York to L.A. in five hours. > > ToeKnee: No, wings are schlocky. > > MoeBlee: Okay, what's your invention? > > ToeKnee: Look, I can roll this ball across the tabletop without wings. > > MoeBlee: But the airplane carries 300 people at a time from New York > to L.A. in five hours. Even though your ball may be nice to you > because it doesn't have schlocky wings, it can't carry 300 people at a > time from New York to L.A. All it carries is the dust on it for a > distance of about five feet. So you can't compare the airplane, which > accomplishes a certain purpose, with your ball, which does not > accomplish that purpose. > > ToeKnee: You just don't get it. Sigh. > > ChanceFor PrinceOfPull: There goes MoeBlee again silencing alternative > theories. It's obvious by asking all these questions of "proof" and "definition" and "coherence", that you're just a hater. Like me. And all the rest of the posters here who keep asking the same questions over and over. We're all just haters.
From: Tony Orlow on 10 Jun 2010 13:21 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: (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 (2) The difference between the two terms which establishes the inequality does not have a limit of zero as x approaches infinity, then (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. Thanx, Tony
From: Tony Orlow on 10 Jun 2010 13:23
On Jun 10, 12:17 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jun 10, 7:17 am, Tony Orlow <t...(a)lightlink.com> wrote: > > > I present a box of tools. You present a rock. > > What you call a "rock" accomplishes a specific task, viz. ensuring > that every set has a certain cardinality. > > What you call a "tool" does not accomplish that task, thus your "tool" > and the "rock" are not comparable in that sense. > > / > > MoeBlee: Look at that airplane. It takess 300 people at a time from > New York to L.A. in five hours. > > ToeKnee: No, wings are schlocky. > > MoeBlee: Okay, what's your invention? > > ToeKnee: Look, I can roll this ball across the tabletop without wings. > > MoeBlee: But the airplane carries 300 people at a time from New York > to L.A. in five hours. Even though your ball may be nice to you > because it doesn't have schlocky wings, it can't carry 300 people at a > time from New York to L.A. All it carries is the dust on it for a > distance of about five feet. So you can't compare the airplane, which > accomplishes a certain purpose, with your ball, which does not > accomplish that purpose. > > ToeKnee: You just don't get it. Sigh. > > ChanceFor PrinceOfPull: There goes MoeBlee again silencing alternative > theories. > > / > > MoeBlee Gimme some o' what's havin' ToeFaceKnee |