An uncountable countable set
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From: Virgil on 20 Aug 2006 10:00 In article <ec9no7$44k$1(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > Virgil wrote: > > In article <ec8gln$oc9$1(a)ruby.cit.cornell.edu>, > > Tony Orlow <aeo6(a)cornell.edu> wrote: > > > >> Dik T. Winter wrote: > >>> 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? > >> Hi Dik - > >> > >> Sorry to interject, but could you tell me how long a sequence K is? 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? > > > > How does having a countable list of naturals require an uncountable list > > of naturals? > > Is K a list of natural numbers? Pay attention. K is the sequence of > digits required to represent the naturals in binary. Since it is, at worst, merely a union of countably many finite sets, it is, at worst, countably infinite. At least in any axiomatically justifiable set theory. Perhaps if TO were ever able to produce an axiom system in which to consideer his claims ...
From: mueckenh on 20 Aug 2006 10:01 Virgil schrieb: > The sequence of edges leading up to any edge is finite. So a finite > sequence of 0's and 1's, 0 for left child, 1 for right child, uniquely > identifies each edge, but it takes an endless sequence of 0's and 1's to > identify any endless path. > > Thus ? do you know what you are talking about? > in infinite binary trees, the set of edges is countable but the set > of endless paths is not. Regards, WM
From: Virgil on 20 Aug 2006 10:08 In article <ec9nrs$44k$2(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > Virgil wrote: > > In article <ec8h1q$prm$1(a)ruby.cit.cornell.edu>, > > Tony Orlow <aeo6(a)cornell.edu> wrote: > > > >> It would seem to me that any good theory > >> of infinite sets could be applied to infinite sets of points, such as > >> the reals in (0,1] or those in (0,2], and be able to draw conclusions > >> such as that there is twice as many points and twice as much space in > >> the second as in the first > > > > That would require each point to occupy enough space that only a finite > > number of them could fit into any finite interval. > > WRONG! > > It requires that some relationship be set up between a particular > infinity of points and a finite length. What you suggest would be a > finite set of finite elements, not any kind of infinite set. If TO insists that infinitely many points can be compressed into a finite space, then one can just as easily compress twice as many points into the same space. If one takes the points of (0,2] and places each point from x at position x/2 in (0,1], one has compressed "twice as many" points into (0,1] as were originally there. Thus there cannot be any fixed proportionality between the "number of points" in an interval and its length for intervals of positive length.
From: Tony Orlow on 20 Aug 2006 10:10 Dik T. Winter wrote: > In article <ec8h1q$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. > > > 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. > > > 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. 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. Perhaps the sedenions form the surface of an 8-D hypertorus in terms of being closed under various arithmetic operations. 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? The projectively extended reals, in the end, form a circle and a ring. With the addition of infinitesimals and specific infinite quantities, even division by 0 can be handled, probably even for complex numbers, quaternions, sedenions, etc. But, I haven't delved into that yet. :) TO
From: Tony Orlow on 20 Aug 2006 10:24
Dik T. Winter wrote: > In article <ec8hlf$qq8$1(a)ruby.cit.cornell.edu> aeo6(a)cornnell.edu writes: > > Dik T. Winter wrote: > ... > > > Eh? It holds for every finite number but does not hold for an infinite > > > number of finite numbers? What do you mean with that? If there are > > > infinitely many finite numbers, I would say that as it holds for > > > every finite number, it also holds for infinitely many finite numbers. > > > > If the formula applies to an infinite number of finites, then does the > > sum of the aleph_0 finite naturals equal (aleph_0^2+aleph_0)/2? > > I think there is a misunderstanding. The formula > sum{i = 1 .. n} i = n * (n + 1) / 2 > holds for every finite number n, so it holds for infinitely many finite > numers n (as there are infinitely many finite numbers n). But we can > not switch to an infinite sum (that is something different). Well, we can. If we turn to what we know about infinite series, we can apply notions such as, if every term in series A is greater than its corresponding term in series B, then the sum is obviously greater. You claim there are infinitely many n e N, aleph_0 of them. So, if that formula represents the sum of the first x terms, and you plug in aleph_0 for x to include all of them, then you get that result. > > > In > > standard theory, would this not equal aleph_0, and if so, does it make > > sense that sum(n=1->aleph_0: 1) = sum(n=1->aleph_0: n), when n>1 for all > > n>1? The scond would appear to be clearly a larger sum. > > No that does not make sense. sum{n = 1 .. oo} 1 is not defined, the same > holds for sum{n = 1 .. oo} n. If you want to use them you have to > provide a definition for them. All that needs doing is declaring a unit oo and allowing it to be used formulaically. sum{n = 1 .. x} 1=x, so sum{n = 1 .. oo} 1=oo. sum{n = 1 ... x} n=(x^2-x)/2, so sum{n = 1 .. oo} n=(oo^2-oo)/2. It was perhaps a year ago that three of us independently said the sum is (|N|^2-|N|)/2. It's not a problem dealing with infinite values, once you declare an infinite unit. "Diverges" doesn't specifically describe the value of the sum. :) > > > > With ordering you lose, so not the last. {1, 11, 21, ...} are an example > > > of a set that contains 10% of the natural numbers (whatever that may mean). > > > > I think that what WM is pointing out here is that aleph_0 does not > > behave like a normal number when it cannot be divided to determine a > > midpoint of the set. This is a problem when viewed quantitatively. > > He may be trying to point that out, but not in a way that I do understand. > Moreover, I have stated over and over again that aleph_0 does not behave > like a normal number. It is just people like WM and you that wish that > if behaves like a normal number, but it does not do so. I see no problem > with that. You can not find a midpoint in the ordered set of natural > numbers using a measure derived from the standard measure of the reals. True. WM and I and others want numbers to behave like numbers, and cardinalities and ordinals simply do not. We find them useless and a digression from the study of quantity and representation which we see as being the foundations of math. It irks people like us when set theory claims to be the foundation of math, and yet makes all sorts of exceptions and new rules for infinite values. There are good reasons why this is a constant issue in this forum. It's some kind of logical construction, but for folks like us, it's really not related to mathematics. WM can correct me if I misrepresent his position, but I think that bothers him the way it does me (and in other ways). |