Uncountability of the Rationals?
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From: Virgil on 7 Jun 2010 15:40 In article <319308fd-b002-431f-95b8-36df0c4569ec(a)k39g2000yqd.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > 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. It becomes immediately obvious that TO's "numbers" are TOtally incompatible with real real numbers, so one cannot have both on any number line. Most of us prefer to keep the real reals so of necessity round-file TO's. > > 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: Virgil on 7 Jun 2010 15:59 In article <9b865fc2-1836-4aa3-9f5f-016f04300e78(a)u26g2000yqu.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > 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. One can have initial segments of infinite well ordered sets which have a last member and initial segments which do NOT have a last member, and the two types are quite distinct. TO attempts to overlook the difference and conflate the two types. In particular, for infinite well-ordered sets there is no necessity for their "initial sets" all to have largest members, and some of those initial sets necessarily don't have largest members! Consider, for example, the Cartesian product of the naturals with the naturals, NxN = {(m,n} : m in N and n in N} with the well-order (a,b) < c,d) iff a < c or [ a=c and b < d ] Then for any n in N, the set of (a,b) for which a <= n is an initial segment with no largest element, so there are as many such initial segments without a largest as there are initial segments altogether.
From: Virgil on 7 Jun 2010 16:03 In article <7e9d6608-87c1-4871-a034-5af3dba9409c(a)z10g2000yqb.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > On Jun 7, 12:02�pm, David R Tribble <da...(a)tribble.com> wrote: > > Tony's ideas share a lot in common with those of > > Archimedes Plutonium (of which thousands of his posts > > can be found here in sci.math). I.e., the sequence of > > naturals (eventually) contains infinite naturals, infinite sets > > must contain at least one infinite member, all numbers > > must have digital representations, and a few other wacky > > ideas. > > Oh, David, don't even go there. Archimedes thinks the universe is the > nucleus of a plutonium atom in a bigger universe, for god knows what > reason. I have never claimed the standard naturals include infinite > values, nor denied that the set of points in [0,1] is infinite in size > but contains no infinite values, nor claimed that there exists any > finite digital representation of any transcendental number nor a last > digit thereof, or any such wacky ideas. Your disingenuity is most > unbecoming, and I would appreciate it if you not spin fabrications > about what you believe to be my dysfunction. The splinter you perceive > in my eye is but a reflection of the log in yours. Actually, TO's expositions are frequently sufficiently tangled to lead one to suppose that anything possible within them, so that David's concluding that TO shares a lot with AP is not entirely unfounded.
From: David R Tribble on 7 Jun 2010 16:16 David R Tribble wrote: >> Tony's ideas share a lot in common with those of >> Archimedes Plutonium (of which thousands of his posts >> can be found here in sci.math). I.e., the sequence of >> naturals (eventually) contains infinite naturals, infinite sets >> must contain at least one infinite member, all numbers >> must have digital representations, and a few other wacky >> ideas. > Tony Orlow wrote: > Oh, David, don't even go there. Archimedes thinks the universe is the > nucleus of a plutonium atom in a bigger universe, for god knows what > reason. I have never claimed the standard naturals include infinite > values, nor denied that the set of points in [0,1] is infinite in size > but contains no infinite values, nor claimed that there exists any > finite digital representation of any transcendental number nor a last > digit thereof, or any such wacky ideas. Your disingenuity is most > unbecoming, and I would appreciate it if you not spin fabrications > about what you believe to be my dysfunction. The splinter you perceive > in my eye is but a reflection of the log in yours. You will recall, though, your attempts to define relational ordering for naturals inductively "without the restrictions of finiteness" or some such wordage. (Probably in the "Calculus XOR Probability" thread from a few years ago.) Your most recent comments reflect that you still hold to those ideas: | 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. As I recall, responses to your attempts pointed out that there are no inherent "restrictions on finiteness" in the definition of the naturals, and yet no natural ever actually "becomes" infinite. I find it interesting that Archimedes Plutonium was very annoyed when I pointed out to him that his new digital notation for infinite numbers (his "AP-adics" and "AP-reals) looked almost identical to your (independently created) "T-riffic" notation. He of course claimed first discovery of any such idea. David R Tribble wrote: >> To contrast Tony's ideas with mathematical ideas possessed >> of something resembling logical rigor (and to toot my own horn), >> have a look-see at my humble attempts to extend the reals: >> http://david.tribble.com/text/hnumbers.html > Tony Orlow wrote: > Indeed, humble, as you really do nothing particularly spectacular. I never claimed it was. I certainly have never claimed anything of mine as having the (potential) importance that you have implied for some of your ideas, anyway. David R Tribble wrote: >> My "suprareals" share some characteristics with Cantor >> normal form and also with the surreals. Turns out that >> they are actually a re-invention of sorts of the Levi-Civita field. >> See: >> http://en.wikipedia.org/wiki/Levi-Civita_field >> http://en.wikipedia.org/wiki/Cantor_normal_form > Tony Orlow wrote: > I've reinvented lots of things. That's fun, but not ground-breaking. Again, I never claimed they were. Most of it is simply intellectual exercise/entertainment for me.
From: Jesse F. Hughes on 7 Jun 2010 16:23
David R Tribble <david(a)tribble.com> writes: > Tony's ideas share a lot in common with those of > Archimedes Plutonium (of which thousands of his posts > can be found here in sci.math). I.e., the sequence of > naturals (eventually) contains infinite naturals, infinite sets > must contain at least one infinite member, all numbers > must have digital representations, and a few other wacky > ideas. What's so wacky about all numbers having digital representations? Maybe you meant *finite digital representations? -- "That's one of the more fascinating things about my research: simple methods, mostly elementary, succinct and, if I do say so my self, well-written expositions, but no normal human can follow at all, and even the best mathematical minds get lost quickly." -- James S Harris |