An uncountable countable set
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From: Tony Orlow on 20 Aug 2006 23:09 Dik T. Winter wrote: > In article <ec9qh6$762$1(a)ruby.cit.cornell.edu> aeo6(a)cornnell.edu writes: > > Dik T. Winter wrote: > > > In article <ec8h1q$prm$1(a)ruby.cit.cornell.edu> aeo6(a)cornnell.edu writes: > > > ... > > > > Yes, that's a shame, isn't it? It would seem to me that any good theory > > > > of infinite sets could be applied to infinite sets of points, such as > > > > the reals in (0,1] or those in (0,2], and be able to draw conclusions > > > > such as that there is twice as many points and twice as much space in > > > > the second as in the first, rather than coming to to useless conclusion > > > > that the infinite sets are equal in size. > > > > > > You are looking for measure theory. > > > > I looked at measure theory a little. Lebesgue measure doesn't come close > > to what I'm concocting. I am trying to integrate set theory with > > combinatorics, set density, Lebesgue measure, topology, etc, and all it > > takes is a housecleaning and a few new simple rules. > > O. > > > > > Transfinite cardinalities > > > > don't really measure anything, though, do they? > > > > > > They measure something, but not what you want them to measure. > > > > As far as I can see, they "measure" with a fluid yardstick, based on > > non-logical arbitrary axioms. If cardinalities measured set size at all > > accurately, then they would always reflect the addition or removal of > > elements with a change in value. It's actually really not hard to > > achieve that. > > Arbitrary axioms? They are soundly based on equivalence relations and > bijections. What are the non-logical axioms used in that? > > > > > Indeed, they don't form > > > > a field. Tsk tsk. > > > > > > There is not something inherently wrong with a collection of objects > > > not forming a field or a division ring, or whatever. However, if that > > > is the case you must be prepared to find that division is not generally > > > possible. Take for instance the Sedenions, an extension of the Cayley > > > numbers (or Octonians). In the Sedenians general division is not > > > possible because there are zero divisors. > > > > That is an area I don't know too much about except that it's an > > extension of the complex numbers. > > That shows. If you do not know what I am talking about, do not demean > it with your "Tsk tsk.". I'm sorry Dik. I didn't mean to demean what you were saying, but only to point out the shortcomings of the present system. I didn't mean to be rude, but a tad humorous. I suppose my efforts were in vain, as usual. Tsk tsk. (that's to me) > > > Complex numbers are really not single > > quantities, at least in terms of linear order, and so I wouldn't expect > > them to form a ring, but perhaps more of a 2-D ring, or toroidal > > surface, in that sense. > > They not only form a ring; they form a field. The Quaternions form a > division ring. The Cayley numbers do not form a ring (associativity > is not guaranteed), but division can be defined. In the Sedenions > division can not be defined. Okay. Like I said, I don't know too much about the extensions to the complex numbers. I'd have to do some research into that area, which indeed interests me, before I could engage in that discussion. Sorry. > > > However, when we are talking about raw counts of elements in a set, that > > lies somewhere on the infinite real number line, and it is quite > > possible to formulaically compare and order infinite values along this > > line. Don't you think? > > I was not talking about comparisons, I was talking about division. Perhaps, but the root discussion is relative size of infinite sets, and how to distinguish them. Besides, the unit infinity and infinitesimal solve the problem of division. > > > The projectively extended reals, in the end, form a circle and a ring. > > Not a ring in the mathematical sense. > > > With the addition of infinitesimals and specific infinite quantities, > > even division by 0 can be handled, probably even for complex numbers, > > Division by 0 can not be handled in a ring. According to MathWorld, indeed, there is an exception for 0 in the optional Multiplicative Inverse condition. Only required due to the aversion to infinitesimals. Alas! Why cannot we tie this all together? http://mathworld.wolfram.com/Ring.html :) Tony
From: Dik T. Winter on 20 Aug 2006 23:05 In article <ecb69r$88o$1(a)ruby.cit.cornell.edu> aeo6(a)cornnell.edu writes: > Dik T. Winter wrote: .... > > > That's the von Neumann ordinals. I am talking about {}, {1}, {1,2}, ... > > > If by convention one says the empty set has max=0, then the empty set > > > can be included this way. Otherwise, just leave it out and start with {1}. > > > > But you make conclusions about an ordinal number. > > I make conclusions about the ordinal number system, yes. No. You start with naturals and are telling about sets of naturals, and from that you conclude things about ordinal numbers. > > I question your "inductively provable". When you consider the von > > Neumann ordinals I can state that the value of an ordinal is larger than > > each of its elements. > > When you prove this inductively, how much larger is the size of the set > than its maximum element? Is it always 1 larger than the max element? Unstated in general. But *if* there is a largest element, it will be 1 larger. > If > so, then how does the set size at some point jump to infinity, while all > elements remain finite? Just because the set of all finite ordinals does not have a largest element? > Is there a largest finite, and beyond it a > point, beyond which everything is infinite? No, there is no largest finite. Because if there were, adding 1 to it would give you a larger finite. > That's a very magical point, > indeed! O, how I'd like to bring my unicorn there for a picnic! :D Cute. But that point does not exist. > > I can use quite the same induction here as you are > > using and come to the conclusion that the ordinal of the set of all naturals > > is larger than each natural. > > With the vN ordinals, 1 greater, implying that aleph_0 is 1 greater than > some finite. Nope. Won't work. Nope, you do not understand. > > Consider the following. The ordinal number of each initial segment of > > ordinal numbers is larger than each of the ordinal numbers in that > > segment. That works for finite and for infinite ordinals. > > Sure, but it's vague, and doesn't provide fine distinctions in size like > my Inverse Function Rule. Possibly. I quit reading you quite a long time ago, until you invaded this thread clearly not knowing what the discussion was about. Binary representations. Sheesh. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Tony Orlow on 20 Aug 2006 23:14 Dik T. Winter wrote: > In article <ec9pqd$6iu$1(a)ruby.cit.cornell.edu> aeo6(a)cornnell.edu writes: > > Dik T. Winter wrote: > > > In article <ec8gln$oc9$1(a)ruby.cit.cornell.edu> aeo6(a)cornnell.edu writes: > > > > Dik T. Winter wrote: > > > ... > > > > > What is the wrongness of the definition? Let me refrase: > > > > > Let L be the list of all natural numbers > > > > > Let K be the sequence of digits such that for each p in L, K[p] = 1. > > > > > What is *wrong* with that definition? > > > > > > > > Sorry to interject, but could you tell me how long a sequence K is? > > > > > > Can you not determine it? I would think it follows immediately from the > > > definition. > > > > > > > If > > > > it's finite, then you only have a finite list L, and if it's countably > > > > infinite, then you have an uncountable list of naturals? > > > > > > What is your problem? From the definition I would say that the size > > > of the sequence of digits in K is equal to the size of the sequence > > > of natural numbers in L. > > > > Here is my problem. Given normal combinatorics, there are 2^n unique > > bit strings of length n. > > You have not followed the discussion and so you are telling nonsense. > The discussion was about unary representations, so there is exactly > one unique strings of length n. Re-read the complete discussion and > after that come back again. I'm sorry. There were a number of things going on in the discussion, and I recently came back, and you're right. I thought these were two different topics, but this is the same as the unary half-square enumaration. Apologies. It must have been talk of Virgil's tree that confused me with its binary wiles. ;) But, it's still an interesting question, how many bits are required to enumerate the reals. Eh? There ain't any such number. Heh. :) Smiles, Tony
From: Tony Orlow on 20 Aug 2006 23:25 Dik T. Winter wrote: > In article <ec8if0$rtd$1(a)ruby.cit.cornell.edu> aeo6(a)cornnell.edu writes: > > Dik T. Winter wrote: > ... > > > > Tony Orlow wrote: > > > > > A circle is a regular polygon with an infinite number of infinitesimal > > > > > sides. The formula for the angle of the vertices of a regular > > > > > polygon of n sides is 2*pi*(1/2-1/n). The limit of this formula > > > > > as n->oo is pi. > > > > > > Tony, I see you did agree with Wolfgang Mueckenheim in this thread. But > > > I should warn you. According to WM neither pi nor that limit do exist. > > > > > > (Gives me a deja vu about JSH agreeing with EEE. While the latter proved > > > that FLT was false and the former proved that it was right.) > > > > Oh, how I love being compared to JSH! > > You misunderstand. What struck me was that again two people in violent > disagreement with each other are complimenting each other. I am not sure how Wolfgang feels about me, but I don't feel like I am in violent opposition to him. I don't feel in violent opposition to anyone here, but think maybe some are in that with me. I hope WM doesn't feel that way. I think Virgil likes feeling that way, but I worry for his heart. > > > Indeed, I agree with WM's logic concerning the identity relationship > > between element count and value in the naturals. He's quite correct in > > that regard. > > Well, you and he are not. The logic is flawed. How so, precisely? > > > I'm a devoted post-Cantorian delver into > > infinity. While he takes the argument put forth as proving that the set > > of naturals is finite, he does so with the assumption that infinite > > naturals cannot exist. > > And his proof is not a proof. When any positive infinite value is considered greater than any finite value, and you prove inductively the p(x) is true for x > finite y, well, yes, it's a proof. Proofs depend on starting assumptions, rules and facts. > > > For my part, I agree that the set of finite > > naturals is finite, though unbounded, > > In that case you are not using standard mathematical terminology. I > have no idea what a finite but unbounded set is. Yes, that's contradiction in terms given the Dedekind definition of an infinite set. WM's point is very subtle. I know that because I have raised it a bunch of times over the last year+, as I think has he. But, where there is a constant finite distance on a line between points, and no point is infinitely distant from any other, there is a finite range, and only a finite number of such disjoint intervals can occupy that space. When countable infinity is defined as a 1-D space where everything is within a finite distance of each other, it's not really an infinite space. > > > but that there exists an infinite > > set of naturals, which includes infinite values. > > That is alright with me, only, do not call them naturals, because that is > extremely confusing. Yes, I think "hypernaturals" as a superset of the naturals is best, though that name has been used, and may carry some unwanted ssumptions with it. Still, for now, "hypernaturals"? Dik - have you given much thought to infinite values? :) Tony
From: Virgil on 20 Aug 2006 23:28
In article <ecb0gm$rie$1(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > Virgil wrote: > > In article <ec9s3c$8qv$1(a)ruby.cit.cornell.edu>, > > Tony Orlow <aeo6(a)cornell.edu> wrote: > > > >>> Then these infinite numbers would not deserve the name "natural number". > >> Why is that? If they are whole numbers, each with successor, does that > >> not fit the bill? > > > > What is your definition of "whole number", TO, that allows it to be > > applied to objects not members of the minimal of the inductive sets > > required by the axiom of infinity? > > A point on the real line exactly reachable with an endpoint by placing > unit intervals end-to-end starting from 0 (using either a "countable" or > "uncountable" number of intervals). No wonder TO has so much trouble with arithmetic. > > > > > One of TO's faults is that he will not allow any definition to mean > > only what it states, but always has to suppose it to mean something else. > > This is a very anti-mathematical anti-logical attitude, and explains > > much of TO's difficulty in comprehending things logical and things > > mathematical. > > > > More vacuous dismissal. Whatever. Those aspiring to be mathematicians are supposed to dismiss the vacuous. > > > In ZF, ZFC andr NBG, a set, S, is called an inductive set if and only if > > (1) {} is a member of S, and > > (2) Whenever x is a member of S, then union {x,{x}} is a member of S. > > The axioms of infinity in those systems declare that there exists > > inductive sets, and other axioms guarantee that there must be a minimal > > inductive set which is called the set of natural numbers and its > > members, and only its members, are natural numbers in those systems. > > Very untrue. Those are the von Neumann ordinals, but that is not the > general definition of an inductive set. They are the only sort of inductive set whose existence is guaranteed by the axiom of infinity(also called the inductive axiom). http://en.wikipedia.org/wiki/Axiom_of_infinity There is a set N, such that the empty set is in N and such that whenever x is a member of N, the set formed by taking the union of x with its singleton {x} is also a member of N. Such a set is sometimes called an inductive set. > > > > If TO wants to call anything else natural numbers, he has put himself > > outside the pale. > > Into the tan, indeed. Out from the cave. Over the rainbow, if you will. > Why not? Why object to infinite induction as an alternative to > tranfinitology when it's no less reasonable? Why hide behind Georg? There is a perfectly legitimate version of transfinite induction, see: http://en.wikipedia.org/wiki/Transfinite_induction or http://en.wikipedia.org/wiki/Three_forms_of_mathematical_induction but it does not support TO's "extended induction" or "infinite induction" claims. |