Uncountability of the Rationals?
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From: Tony Orlow on 7 Jun 2010 10:30 On Jun 5, 11:36 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > On Jun 5, 10:04 am, Virgil <Vir...(a)home.esc> wrote: > > > In article > > <796c4157-f47d-4aba-b056-8ddc38d46...(a)c10g2000yqi.googlegroups.com>, > > Tony Orlow <t...(a)lightlink.com> wrote: > > > Surely you remember the T-Riffics? > > Does Tony Orlow really want to maintain that ANY part of his idiotic > > "T-Riffics" was ->generally accepted<- ? > > Hold on a minute. Earlier, TO and Tribble were discussing > something called the H-riffics. Now Virgil is referring > to something called the T-riffics. > > If by "T-riffics" Virgil is actually referring to the > "H-riffics" as mentioned by Tribble, then for once, I > actually agree with Virgil. For according to Tribble, the > H-riffics lack a value for 3. Even _I_ can't accept a > theory in which one can't even prove the existence of 3 > (especially if it does prove the existence of 4, which, > being a power of two, does exist in this theory). > > 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. Hi Transfer - You deserve an explanation of these number systems I developed. You are likely one of the few that might actually appreciate them without unreasonable derision (reasonable criticism is more than welcome), and they are getting rather mangled in the various misinterpretations that others are dredging from their databanks. So here's a quick synopsis of their goals and methods. I'll post now about the T-riffics, as they appeared first, and post afterwards about the H-riffics, if that's okay with you. The T-riffic numbers were developed to numerically represent infinite and infinitesimal numbers so that arithmetic could be performed on them which may produce other infinities, infinitesimals, or even finite numbers as a result. In addition to the finite unit One, there is an infinite unit, Big'Un, and its multiplicative inverse, the infinitesimal Lil'Un. I have recently decided to coopt the vague term "zillion" in place of Big'Un, and a "zillionth" in place of Lil'Un. The idea is that there are a zillion points in any half-open real unit interval, such as (0,1], each of those points occupying one zillionth of that interval. Thus, in the interval [3,5] there would be two zillion and one points. I had first imagined these points as each being a separate real number, but it was immediately apparent that any point within any countable distance form any given point would be the same standard location and represent the same standard real number. However, that countable distance in zillionths represents a precisely measured infinitesimal difference between two points. Notice that I conceded that these are not exactly points in the classical sense, but are more like Newton's fluxions, having infinitesimal length and therefore being line segments on the infinitesimal scale. 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 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. However 1-1/x<1 does not hold in the infinite case, since the difference of 1/x has a limit of 0 as x increases without bound. By virtue of this rule we can say that one zillion is greater than 2 zillion, or that omega^2 is greater than omega, if any such number omega exists and is greater than 2. So, we may place multiple digital points within a T-riffic number, denoted each by some function on z (a zillion). For instance, a zillion plus the square root of a zillion in base 2 might be epressed as: 1.(log2(z)).000...001.(log2(z)/2)000...000.0 Perhaps I should stop here for questions. This post is tiring me out a little, and I'm probably not the only one. But, that's a good thing. As I tell my kinds, when you get tired, either physically or mentally, after you rest, you're stringer than before. Have a a nice day :), Tony
From: William Hughes on 7 Jun 2010 10:52 On Jun 7, 11:30 am, Tony Orlow <t...(a)lightlink.com> wrote: [on the origin of the T-riffics] You look at dividing the [0,1] interval into a large (finite) number of small parts. This works well, and words such as "first", "next" and "last" make sense. So you have a "first" interval after 0, a last interval before 1 and a next interval after 0.5. So you think, instead of divinding [0,1] into a large number of small parts, I am going to divide it into an infinite number of infinitessimal parts. However, when you do this, "first", "last" and "next" go away. The T-riffics are your attempt to have your cake and eat it. However, you can't have an infinite number of reals and still have "first", "last" and "next". - William Hughes
From: David R Tribble on 7 Jun 2010 11:34 Transfer Principle wrote: > Hold on a minute. Earlier, TO and Tribble were discussing > something called the H-riffics. Yep. I don't wish to re-start any discussions of Tony's pet projects, but here is a brief thumbnail sketch, from memory, of his "H-riffic" sequence and "T-riffic" number system. Tony's H-riffics were his attempt to well-order the reals. It's basically a sequence of real values, starting with 1, then doubling the number of values at each level (a binary tree forking). If x is an H-riffic, then so are 2^x and 2^-x. He used to have a discription posted somewhere on the Internet, but I can't find it. See this discussion for a partial explanation: http://mathforum.org/kb/plaintext.jspa?messageID=5298645 What was (is) lost on Tony is that: 1. this is a countable set, so it can't possibly contain all the reals. 2. this omits vast uncountable subsets of the reals. Specifically, any real that is a multiple of an integer power of natural k where k is not 2. For example, all of the multiples of powers of 3, including 3 and 1/3. Apparently Tony is still quite convinced that his sequence of reals denumerates and thereby well-orders all of the reals, in spite of all evidence to the contrary. > Now Virgil is referring to something called the T-riffics. The T-riffics are Tony's attempt at representing his "unit infinity" (Big'un) and infinitesimals (Lil'un) in digital form. IIRC, it used a notation like 123...789:000...001, where each group of infinite digits between the ':' delimiters represented the digits of a number at some level of infinity. The rightmost group were normal finite numbers, so 0:123 was the standard number 123. Groups added in the left represented higher levels or powers of unit infinity, so that 1:0 was equal to his Big'Un. Likewise, groups added on the right were levels of infinitesimals, powers of Lil'un. The problem with his notation, of course, was in making it consistent with standard arithmetic and even self-consistent. Numbering the digits within each group, for example, proved to be problematic for Tony. Likewise, defining how addition, multiplication, and comparison/ordering were beyond Tony. However, he keeps claiming that he's putting the finishing touches on his system, and it will be ready Any Day Now. > But the fact that TO uses the word "bigulosity" strongly > implies that he's using a theory _other_than_ ZFC. Except that Tony believes that his ideas of Bigulosity and unit infinities work within the standard system of arithmetic, (i.e., within ZFC). He's trying to extend/correct the "accepted but misguided" ideas of infinities and set "size" within the standard theory.
From: Tony Orlow on 7 Jun 2010 11:36 On Jun 5, 11:36 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > On Jun 5, 10:04 am, Virgil <Vir...(a)home.esc> wrote: > > > In article > > <796c4157-f47d-4aba-b056-8ddc38d46...(a)c10g2000yqi.googlegroups.com>, > > Tony Orlow <t...(a)lightlink.com> wrote: > > > Surely you remember the T-Riffics? > > Does Tony Orlow really want to maintain that ANY part of his idiotic > > "T-Riffics" was ->generally accepted<- ? > > Hold on a minute. Earlier, TO and Tribble were discussing > something called the H-riffics. Now Virgil is referring > to something called the T-riffics. > > If by "T-riffics" Virgil is actually referring to the > "H-riffics" as mentioned by Tribble, then for once, I > actually agree with Virgil. For according to Tribble, the > H-riffics lack a value for 3. Even _I_ can't accept a > theory in which one can't even prove the existence of 3 > (especially if it does prove the existence of 4, which, > being a power of two, does exist in this theory). > > 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. Hi again Transfer - I hope you enjoyed the description of the T-riffics. I guess I'll see what questions you have about that. In the meantime, I took a break and will now try to describe the H-riffics. As you might imagine, I rather liked the T-riffics when I thought of them, but don't like the H-riffics so much. ;) I introduced the H-riffic number system in a thread I started called, "Well Ordering the Reals", as a possible enumeration of the real numbers which might be considered a well order on them, since apparently the axiom of choice implies one must exist, and yet, identifying one appears to be an elusive goal for the mathematical community. I wanted to see exactly what criteria were used to discount it, and ultimately ended up deciding that if the H-riffics didn't represent a well ordering of the reals, then nothing did, and it didn't really matter. Anyway...... The H-riffics of a given base b are based on the idea that: 1 is a number if x is a number then both b^x and b^-x are both numbers. So, for instance, given b=2, we have the follwing numbers: 1 2,1/2 4,1/4,sqrt(2),sqrt(2)/2 16,1/16,... etc. If we denote each number by a string of 0's and 1's each representing the expontiation by x and -x, respectively, certainly 3 cannot be expressed in any finite string using H-riffic base 2, but I think is represented as an infinite string, as someone years ago confirmed. I cannot remember if he said it was 001001001... but it was something like that. In any case, it becomes strange at this point, because after this infinite string that equals 3 lie more numbers: 3 8,1/8 256,1/256,8th-root(2),1/8th-root(2) etc. So, the infinite string continues after some uncountably distant point? That's what it seems to me. It's related to my "zillion" somehow, but not at all sure how. I dunno what to do with it, particularly. I guess it's not a well order? They're certainly somewhat horrific to try to calculate, a superset of tetrations apparently. If you can think of a use for them, please do let me know. Thanks, Tony
From: Tony Orlow on 7 Jun 2010 11:49
On Jun 6, 5:16 am, Brian Chandler <imaginator...(a)despammed.com> wrote: > Transfer Principle wrote: > > On Jun 5, 9:59 am, Virgil <Vir...(a)home.esc> wrote: > > > In article > > > <db5cbe4b-a8b8-4b6a-ae47-05fa05dd6...(a)i28g2000yqa.googlegroups.com>, > > > Tony Orlow <t...(a)lightlink.com> wrote: > > > > Yes, that's where I apply N=S^L. > > > Which is a wrong now as when first dropped on an unsuspecting world. > > > IIRC, TO's statement N=S^L means that the number of > > strings of length L from a language of size S is > > equal to N. > > > I disagree with Virgil that it's "wrong." > > No, of course the actual meaning of "N=S^L" is correct. The problem is > that when Tony says "apply N=S^L" he refers to one of his "proofs" > that an infinite set of natural numbers must include at least one > number which is itself "infinite" (though he never really defines what > this means). I will try to reconstruct the argument, which goes > something like: Oh, here we go..... The argument that the truly infinite set of naturals must include an infinite natural has nothing particular to do with N=S^L. That's a simple matter of the nth element of any initial segment of N being equal to n and existing for any segment of size n or greater. > > (Tony 'knows' that the set P of all 'natural numbers' must include > "infinite naturals", but can't just state it.) Been there done that. Your failure to comprehend is not necessairly due to my inability to express myself. > > So, consider "N=S^L". The number of strings (over alphabet size S) of > length L is N. Yes. > > Well, the number of strings (over alphabet size S) of maximum length L > is N. (Not exactly true, but close. Failiing to distinguish "finite > strings of no fixed maximum length" from "possibly infinite strings" > is at the heart of this argumentation technique.) Good imaginatorializing. > > But for there to be an infinite number of strings ("set N = oo"), > since S is constant, we have "L=oo". (Confusion and non-sequitur) (On your part, without a doubt) Because N=S^L, the only way N can be infinite is if either S or L is infinite. With an infinite alphabet we have an infinite number of words of any given length, except 0 of course. Further, N can only be uncountable, in my opinion, if either S or L or both are uncountable. That's where we disagree, I believe. > > Therefore a set of strings which is infinite must include some > infinitely long strings. Ergo, the set of "natural numbers" includes > some "infinite natural numbers". Somewhat related, but really not. At least I left some kind of impression. > > Well, I guess this is a "non-standard theory"... > > Brian Chandler :) Tony |