Uncountability of the Rationals?
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From: Jesse F. Hughes on 18 Jun 2010 11:25 Tony Orlow <tony(a)lightlink.com> writes: > Well, wait a minute. I don't object to the set N or N+ or omega, as a > construction. So, as far as I am concerned, the axiom of infinity can > stay. I just wouldn't pretend that the set transitively closed under > membership is necessarily the greatest model of the natural numbers. Considering that there are, ooh, just *oodles* of sets transitively closed under membership, your conclusion seems perfectly reasonable. To take one example, the empty set is not the greatest model of the natural numbers. You honestly don't know how omega is defined, do you? The set omega is, by definition, the least (according to subset relation) set x such that 0 in x & (A y)(y in x -> y u {y} in x). (Ind) So, omega (w) is the unique set satisfying Ind(w) & (A x)(Ind(x) -> w c x). It's true (of course) that w *like every other ordinal* is a transitive set, but that's not its definition. Hope that helps. -- Jesse F. Hughes "Basically there are two angry groups. I am a harsh force of one. Against me is a society of mathematicians. So far it's been a draw." -- JSH gives another display of keen insight.
From: Brian Chandler on 18 Jun 2010 13:23 Tony Orlow wrote: > On Jun 17, 10:17 am, Brian Chandler <imaginator...(a)despammed.com> > wrote: > > Tony Orlow wrote: > > > On Jun 16, 7:45 pm, David R Tribble <da...(a)tribble.com> wrote: > > > > Tony Orlow wrote: > > > > >> Sometimes there are logical arguments themselves which don't appeal to > > > > >> everyone equally. > > > > How can that be? If the "logical argument" is actually invalid or > > nonsensical it wouldn't appeal to me, but then I wouldn't call it a > > logical argument. If the logical argument is valid why should anyone > > care whether you like it? > > Does "the moon is made of cheese" imply that "pigs can fly like > birds"? Is this a real question? These are rather speculative claims about the real world -- not mathematical statements which could be absolutely true or false. Although we know a lot about the moon, can we be absolutely certain it wasn't the result of a giant explosion in a primordial pizza factory. And "Pigs can fly like birds" is an oddly worded claim. Pigs certainly can fly like some birds, penguins and dodos, for instance. In one simple-minded approach, we can simply take both statements as false, and since false implies anything, yes, the implication is valid. How am I doing? Are you going to put this in the book? (Have you _actually_ started it yet?) > What is the value of 0^0? Crumbs. I dunno. > That's really the only logical loophole, in both Boolean and > quantitative expressions. It is? If I said 0^0 = 1, which axiom would be violated? (Ha ha, can't answer, can you... nyaaghhh nyaaghhh) > > > > >> What I have been told is that the size of omega must > > > > >> be larger than every natural since no natural is large enough to > > > > >> express it, and so it is some infinite number, aleph_0. However, that > > > > >> logical argument, as I pointed out, really just proves that aleph_0 > > > > >> cannot be finite. Along with the argument that any initial segment of N > > > > >> + of size x contains an xth element whose value is x, which would > > > > >> imply that aleph_0 or omega is a member of N+, we arrive at a > > > > >> contradiction implying that aleph_0 cannot actually exist. Which axiom > > > > >> contradicts this logic? > > > > Invalid? Nonsense? Hard to decide. Anyway this is not what you have > > been told, it is your mangled misunderstanding of what some misguided > > soul whiled away a few hours trying to drum into your head. > > So, no axiom violated. Good. If you say so. Will this bit be in the book too? > > > If the set is an initial segment of N+ with no largest member it has > > > no size. I have said this about a zillion times (or at least tav). So just tell me again: what does "initial segment" mean? It can't mean subset of N+ which stops somewhere, because that somewhere would be the largest member. And if it has no largest member, it's still going strong at the right hand end (the one that isn't there, I think), so in what sense is it "initial"?/ > > I think (hope) "N+" means something recognisable like the naturals. > > The positive naturals, as opposed to my misuse at one point when I > meant *N. That's been established. > > > These go on and on ("to the right") without ever ending. To me > > "initial segment" means a proper subset of the naturals that "stops > > somewhere", which is equivalent to saying "has a largest element". Can > > you explain what an initial segment with no largest element would look > > like? > > Like N+. Say, the initial segment of size aleph_0, or omega, or tav, > or whatever you want to call it, as if it exists more than virtually. Oh, golly, an answer. So your example of an initial subset of N+ is... N+. Is that right? Oh, no, the *initial segment* is, golly, *an* initial segment of size [two maths terms, one bit of Orlovia: I can call "it" what I like, but I don't know what "it" is] *of* "Orlovian N +". Hmm. But glory be! Now I've been told what "initial segment" means. But I'm glad it's not me writing the book. > > Hereabouts, you seem to have been saying that if a set doesn't have a > > last member to "be its size" this means the set "has no size"? Please > > confirm: is this the same as "has no bigulosity"? Do you have a > > definition of "has a size" for a set? > > I mean no absolute Bigulosity, but only one in terms of tav, or > whatever. Aha. So Bigulosity comes in absolute and relative varieties. Perhaps you could make Chapter 2 about "Different kinds of Bigulosity"? Brian Chandler
From: Virgil on 18 Jun 2010 15:14 In article <d91af8ca-3475-4d5d-9711-2a53b7446543(a)j8g2000yqd.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > You miss the point. I never said there was no deifnition. I said the > definition does not "floow from the axioms" in the sense that it is > not a theorem of the axioms. Does TO know of any definitions which ARE "theorems of the axioms"? Does he even know what it would mean for a definition to be a "theorems of the axioms"? > It may be a statement "consistent" in > that there is verifiably no contradiction, however, that does not mean > that deifnition is entailed by the axioms in any way. I rather think > my definitions for IFR, IC, and the combination are considerably more > succint and therefore matheatically beautiful. Just my opinion.... Such "beauty" lies only in the eyes of the creator, and lies and lies and lies.
From: Virgil on 18 Jun 2010 15:23 In article <490f4893-77a4-417c-9524-f4e1f4f6cbf3(a)k39g2000yqd.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > I was going to gloss over your drivel, because I don't want to get > back into your standard misconceptions of the infinite tree again. > Anti-Cantorian or no, Meuckenheim is correct in his arguments about > the infinite tree The standard view of the complete infinite binary tree is no more "wrong" than the standard view of the complete infinite unary tree, more commonly called the set of natural numbers, which WM finds equally wrong. And in WM's view both cardinality for nonfinite sets and therefore also Bigulosity are abominations. So if WM were right about infinite trees, TO's "Bigulosity" would have to be wrong.
From: David R Tribble on 18 Jun 2010 15:30
Ross A. Finlayson wrote: > Tony are your H-riffics the same as the H numbers? If you use the H > numbers to derive that fact or Tribble's Eta for example, that has to > go in the definition. Here by H numbers I mean the constructive > hyperreals (constructable). Tony's H-riffics have nothing to do with hyperreals or surreals. They are just reals, of the form ±2 ^ (±2 ^ (±2 ^ ...))) for some finite number of exponentiations. When arranged as a binary tree, each node is an H-riffic, the top node is 0; for node h, the left child node is 2^-h and the right child is 2^h. Thus the H-riffics are the countable nodes in the tree. Note that most reals are not nodes in the tree at all, specifically all reals of the form mk^n, for all integers m, k, and n, and k /= 2. Examples include 3, 1/3, 2/5, sqrt(7), etc. My suprareal eta (eta_1, actually) is a primitive real-like number that extends the reals, having the property (as an axiom) that x < eta_1, for all r in R. My suprareals form a hierarchy of disconnected sets, where eta_(n) < eta_(n+1), for all n in Z; eta_0 = 1. Each suprareal on eta_1 is a polynomial of the form h = x_0 + x_1*eta_1^1 + x_2*eta_1^2 + ... x_n*eta_1^n, where x_i are reals. Larger suprareals are built up from smaller suprareals, so g = h_0 + h_1*eta_2^1 + h_2*eta_2^2 + ... h_n*eta_2^2, where h_i are suprareals on eta_1, and so on for larger eta_n values. My suprareals resemble the hyperreals, though I'm not expert enough to say if they are the same (isomorphic) or not. They appear to be the inverses (almost) of the infinitesimals of the Levi-Civita field. http://david.tribble.com/text/hnumbers.html http://en.wikipedia.org/wiki/Levi-Civita_field While my suprareals have nothing in common with Tony's H-riffics, it is instructive to compare my suprareals to Tony's T-riffics, primarily to see how "numbers larger than reals" can be defined axiomatically. Which of course seems beyond Tony. -drt |