An uncountable countable set
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From: Virgil on 20 Aug 2006 15:27 In article <ec9u24$at1$1(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > Virgil wrote: > > Actually, I would bet against N being a member of a subset of N, myself. > > I'd bet against you. Does TO really wish to bet that N IS a member of a subset of N? > > Oh, Virgil, please try rereading this. S={{1},{1,2},{1,2,3},...}, > N={1,2,3,...}. It's a member of S. Since every member of S, at least as represented above, contains a largest member but TO's N does not contain a largest member, why is that N a member of that S? Is TO off on his "largest natural" kick again? > > > > > > >> Yes, and it's inductively provable likewise that the size of a successor > >> ordinal is one greater than its max element. That would make aleph_0 1 > >> greater than the max finite natural. > > > > There goes TO's delusion again. What ever makes TO think that a process > > which cannot not end must end? > > > > If it doesn't end what makes Virgil think he can tack a count on it? I can "tack on " bijections of endless sets by defining those bijections. And in standard mathematics that is one way of counting. > am not saying there IS a max finite ordinal. I am saying that that's > what the assumption of aleph_0 implies when one is allowed to apply > inductive proof to the infinite case. The only acceptable forms of induction in ZF or NBG are finite induction and transfinite induction, neither of which has TO shown he knows how to apply. > > > >> The only way around this is the > >> declaration of the limit ordinals, but that's just a philosophical > >> monkey wrench. > > > > it certainly throws a monkey wrench in TO's gearbox, but ZF, ZFC and NBG > > are unaffected by non-existence of TO's alleged maximum finite natural. > > It creates the problem. Not in ZF, ZFC or NBG, and problems occurring in the isolation of TOmania are irrelevant to mathematics. > Nowhere have I ever claimed there IS a max > finite natural. I am saying that's the implication of your theory, > unless you throw out induction when n is infinite. For those cases we can sometimes invoke transfinite induction, but it doesn't give any of the results that TO wants. > > >>> There is no equivalent model in which the ordinal of each element is > >>> equal to the element itself as that would require sets to be members of > >>> themselves, which is impossible for ordinals. > >> Don't confuse max with size. Relate them formulaically. > > > > Does TO mean that a maximum does not have to be larger than those things > > of which it is allegedly the maximum? > > > > If not, then what does TO mean by "max"? > > Max value in the set vs. size of the set. But TO is always going on about how for his version of the naturals they are the same. What does one do for sets which are not ordered, or ordered sets which have no max?, both of which are abundant.
From: Virgil on 20 Aug 2006 15:32 In article <eca295$h7i$2(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > David R Tribble wrote: > > Tony Orlow wrote: > >>> It would seem to me that any good theory > >>> of infinite sets could be applied to infinite sets of points, such as > >>> the reals in (0,1] or those in (0,2], and be able to draw conclusions > >>> such as that there is twice as many points and twice as much space in > >>> the second as in the first > > > > Once again, you are confusing 'cardinality' with 'volume' . The first > > deals with denumeration of sets, while the second deals with geometric > > distance. > > > > I'm not confusing them, I'm relating them. When, if ever, TO shows that he has even the smallest familiarity with measure theory only then will he have anything relevant to say about the relation between denumeration of sets and geometric measures of regions.
From: imaginatorium on 20 Aug 2006 15:51 Tony Orlow wrote: > imaginatorium(a)despammed.com wrote: > > Tony Orlow wrote: > >> Dik T. Winter wrote: > >>> In article <ec8h1q$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. > > > > Fine. When's the book coming out, then? > > I am working on an outline right now. When the boys are back in school, > I'll have time and peace to actually start putting it all together. > Thanks for asking. > > PS - you still owe me an actual answer to my question, or at least an > explanation as to why you don't accept my answer to yours. I thought we > had a deal, that didn't include pink elephants. I'm sorry, too many posts - you'll have to remind me exactly which question I'm supposed not to have answered. Is it isomorphic to the one about drive shafts in a dog? It would help for a start if you check for the presence of i-words and f-words, because if you can't rewrite it without using them, I can guarantee I will not be able to attach any clear meaning to your question. It isn't honest to pretend that the question is one using normal usage, because you will not interpret the answer to _that_ question in the way it was intended. > > > > >>> > Indeed, they don't form > >>> > a field. Tsk tsk. > > > > I think this is you - I wonder what you mean by a "field"? > > Dik said transfinite numbers didn't form a field. Look it up. Yes, I know what Dik (and all other mathematicians) mean by a field. Do you? > > > >>> 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. 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. Well, contrary to your expectations, the complex numbers do form a ring. I can't make any sense of your comment - what is a "2-D ring" for a start? Actually, I have never understood why the name "ring" is given to what mathematicians mean by the word. Can you explain? Brian Chandler http://imaginatorium.org
From: Virgil on 20 Aug 2006 16:13 In article <eca31m$itn$1(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > David, this is not a matter of proof given standard definitions, but a > redefinition of the principle of induction such that it applies to the > infinite case. When an inductive proof demonstrates that f(x)>g(x) for > all x greater than some y, all infinite values falls into that category. > So, it simply amounts to removing the restriction that induction only > applies to finite n. Once this restriction is lifted, then we can > "prove" that, for instance, for all n>2, n^2>2n, and therefore this also > holds for all infinite n. The implications of this assumption are a very > nice system of ordering infinities which goes far beyond what > cardinality can even hope to attain. Until TO comes up with an entire axiom system, like ZFC or NBG, in which his extended principle of induction is embedded so that we can examine that system for internal conflicts, I, for one, will not accept any principle, or its consequences, so obviously reliant on TO's erratic intuition.
From: Virgil on 20 Aug 2006 16:21
In article <eca564$m9h$1(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > The concept with the limit ordinals is that > they are the first after the last of something that doesn't end, the > next biggest thing. But, when all successor ordinals are exactly one > larger than their max element, then limit ordinals are something else > entirely. The declaration of omega as that thing right after all the > finites itself leads to silliness. That silliness is all in TO's head, where it has good company in the silliness of accepting TO's intuition as an arbiter of good sense. > > > > >> The only way around this is the declaration of the limit ordinals, > >> but that's just a philosophical monkey wrench. > > > > Ordinals and cardinals are necessities if we want to talk about > > set "order" and "size" in any kind of logical, well-defined way. > > Not limit ordinals and transfinite cardinalities Then one is stuck with only finite counts a la "Mueckenh". > > > > > What do you call the least ordinal that is greater than all finite > > ordinals? You don't have a name for it, do you? > > obviously, that's omega, if you entertain such a useless idea. I don't. But To entertains all those crazy suggestions his intuition makes without ever thinking about them. > > > > If you are going to keep taking this political approach to > > mathematics, condemning what's already established and functional > > but not providing any workable alternatives, you might want to > > consider running for office instead. > > > > It's not functional, and I have provided functional alternatives. It functions well enough for those who know how to use it, which TO does not, and TO's alternatives do not function outside the twilight zone of his TOmania. |