The right way to question the uncountability of the reals
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From: Lee Davidson on 13 Jun 2010 16:25 On Jun 11, 12:38 am, Rupert <rupertmccal...(a)yahoo.com> wrote: > On Jun 11, 1:20 pm, Lee Davidson <l...(a)meta5.com> wrote: > > > > > > > Its likely that some of you have had thoughts like mine before, so I > > am directing this to those of you who havent taken to time to think > > of things from the perspective I will develop here. > > > Ive seen some threads in sci.logic and sci.math initiated by posters > > that were confident that they have found flaws in Cantors diagonal > > argument etc. I have to admit that I couldnt make that much sense out > > of their postings, but perhaps I didnt study these postings closely > > enough. I have just so much time. > > > At any rate, all proofs of the uncountability of the reals that I know > > of make use of some machinery of set theory (e.g. ZFC) and/or assume > > the reals form a complete metric space. > > > If we define the reals using ZFC as, e.g., Dedekind cuts, we can of > > course prove that the reals form a complete metric space and all > > proofs of uncountability carry through. > > > However, suppose we begin with what I would like to call the idealized > > empirical continuum. This is the sort of thing you draw pictures of in > > 10th grade plane geometry, you know, points, lines etc. in two > > dimensions. > > > Consider an idealized one dimensional continuum, that is, an infinite > > straight line. > > > When you say that this forms a complete metric space, intuitively, > > this means that the line is solid, that is, has no gaps. > > > Now, empirically, in order to investigate whether this line is really > > solid, equip yourself with microscopes, 2x, 4x, 8x, 16x, etc., through > > all the powers of 2. > > > Now suppose, for example, that you have reason to believe that there > > are points on this line corresponding to all the rationals. (Dont ask > > me how you know this.) > > > Now try to use your microscopes to determine whether there is a point > > on the line corresponding to the square root of 2. Suppose you can > > make an infinite number of observations using your microscopes, in an > > attempt to home in on the square root of 2. What do you see in your > > microscopic observations? This line always looks solid. > > > But does this tell you there is a point there, corresponding to the > > square root of 2? No. All this tells you is at most that, between any > > two points on the line, the line looks (and maybe is) solid. That is, > > that you have a dense ordering. > > > In other words, empirically if you can perform infinitely many > > microscopic observations all you can verify is that the line is a > > dense ordering. > > > But that it is a dense ordering does not imply, logically, that it is > > a complete metric space. > > > But the situation is even worse than I have depicted. You cant even > > prove, empirically, that points on the line corresponding to all > > rationals exist. Though you might be able to base this on some > > considerations that seem self-evident from your spatial intuition > > e.g., translation invariance, isotropy, reflection invariance > > whatever. I wont go into any details here; Im trying to think about > > whats reasonable here. > > > The general point here: sure, you can show me a Cauchy sequence. But > > how to you know that the non-existence of a point to which that > > sequence converges constitutes a gap? > > > Anyway, I suspect that you need to bring in the machinery of set > > theory in a significant way here. But those who disbelieve in the > > ontology of set theory and want to base real number theory on our > > spatial intuition alone have some reason for skepticism. > > > Me, I dont really have any trouble with the ontology of set theory. > > But, from a foundational point of view, I see a problem here. > > > By the way, you can certainly get the square root of 2, and lots else > > besides, if you can build up an empirical theory, in my sense, of > > ruler and compass construction. But, of course, this still leaves you > > with, at most, a proof of a countable two dimensional continuum. > > > But where Im stuck here is: how do you know that non-parallel lines > > intersect in a common point? If you cant know this, you cant even > > have ruler and compass construction. > > I don't think I understand what question you're asking. > > Are you asking a question about physical reality?- Hide quoted text - > > - Show quoted text - Rupert, I was trying to be as brief as possible in my post, so did not explain my assumptions completely. I'm considering an idealized world that contains ideal infinite straight lines such as the ancient Greek geometers imagined, and considering one as a number line. I pick out two arbitrary points on that line and label them as 0 and 1. Now, equipped with all my "microscopes" and associated measuring apparatus, I am asking what we who live in that world can know about that number line. And, yes, this is a question about physical reality in that world, though you can apply my considerations to this actual world. My ultimate point is that empirical observation of this "physical" number line might well tell us that it is a dense ordering of something (and let us assume points) but we can't prove that there is a point corresponding to the square root of 2, or, in fact, corresponding to any other real number except our arbitrary 0 and 1. And I am allowing for infinitely many microsopic observations and measurements in progressively increasing precision. And this proof still eludes us. So the continuum is infinitely divisible -- densely ordered -- just as the Greeks thought. I mean, in this idealized world. However, the points in the continuum might well be enumerable. Since real number theory was, I think, meant to give us an explanation of the continuum, possibly that explanation has failed. However, real number theory can still give us a "model" of the real continuum.
From: Lee Davidson on 13 Jun 2010 16:30 On Jun 11, 2:39 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jun 10, 10:20 pm, Lee Davidson <l...(a)meta5.com> wrote: > > > At any rate, all proofs of the uncountability of the reals that I know > > of make use of some machinery of set theory (e.g. ZFC) and/or assume > > the reals form a complete metric space. > > Usually, I find that the reals are characterized as the carrier set of > a complete ordered field (and all complete ordered fields are > isomorphic). Is this isomorphism true in all cardinalities? I've forgotten my model theory on this. In any case, you've given the algebraic characterization. However, I'm thinking of the characterization in terms of Dedekind cuts or Cauchy sequences. > > And we don't need full ZFC to prove that any such carrier set is > uncountable. Just a portion of Z set theory and an intuitionistically > acceptable portion of the logic suffice. True, full ZFC is not necessary. And ZF (i.e., using replacement) isn't either. > > As to tjhe line you mentioned. If I'm not mistaken, it is usually an > axiom of ordinary geometry (the ruler postulate) that every line is > 1-1 with the set of real numbers. I don't know that there is a way to > physically "examine" to confirm such a thing. That's my point. I think infinite divisibility is determinable, using my intinity of microscopes. But to establish that ruler postulate, we need to do more work. > > MoeBlee > > > > > > > If we define the reals using ZFC as, e.g., Dedekind cuts, we can of > > course prove that the reals form a complete metric space and all > > proofs of uncountability carry through. > > > However, suppose we begin with what I would like to call the idealized > > empirical continuum. This is the sort of thing you draw pictures of in > > 10th grade plane geometry, you know, points, lines etc. in two > > dimensions. > > > Consider an idealized one dimensional continuum, that is, an infinite > > straight line. > > > When you say that this forms a complete metric space, intuitively, > > this means that the line is solid, that is, has no gaps. > > > Now, empirically, in order to investigate whether this line is really > > solid, equip yourself with microscopes, 2x, 4x, 8x, 16x, etc., through > > all the powers of 2. > > > Now suppose, for example, that you have reason to believe that there > > are points on this line corresponding to all the rationals. (Dont ask > > me how you know this.) > > > Now try to use your microscopes to determine whether there is a point > > on the line corresponding to the square root of 2. Suppose you can > > make an infinite number of observations using your microscopes, in an > > attempt to home in on the square root of 2. What do you see in your > > microscopic observations? This line always looks solid. > > > But does this tell you there is a point there, corresponding to the > > square root of 2? No. All this tells you is at most that, between any > > two points on the line, the line looks (and maybe is) solid. That is, > > that you have a dense ordering. > > > In other words, empirically if you can perform infinitely many > > microscopic observations all you can verify is that the line is a > > dense ordering. > > > But that it is a dense ordering does not imply, logically, that it is > > a complete metric space. > > > But the situation is even worse than I have depicted. You cant even > > prove, empirically, that points on the line corresponding to all > > rationals exist. Though you might be able to base this on some > > considerations that seem self-evident from your spatial intuition > > e.g., translation invariance, isotropy, reflection invariance > > whatever. I wont go into any details here; Im trying to think about > > whats reasonable here. > > > The general point here: sure, you can show me a Cauchy sequence. But > > how to you know that the non-existence of a point to which that > > sequence converges constitutes a gap? > > > Anyway, I suspect that you need to bring in the machinery of set > > theory in a significant way here. But those who disbelieve in the > > ontology of set theory and want to base real number theory on our > > spatial intuition alone have some reason for skepticism. > > > Me, I dont really have any trouble with the ontology of set theory. > > But, from a foundational point of view, I see a problem here. > > > By the way, you can certainly get the square root of 2, and lots else > > besides, if you can build up an empirical theory, in my sense, of > > ruler and compass construction. But, of course, this still leaves you > > with, at most, a proof of a countable two dimensional continuum. > > > But where Im stuck here is: how do you know that non-parallel lines > > intersect in a common point? If you cant know this, you cant even > > have ruler and compass construction.- Hide quoted text - > > - Show quoted text -
From: Lee Davidson on 13 Jun 2010 16:34 On Jun 11, 5:40 pm, George Greene <gree...(a)email.unc.edu> wrote: > On Jun 10, 11:20 pm, Lee Davidson <l...(a)meta5.com> wrote: > > > At any rate, all proofs of the uncountability of the reals that I know > > of make use of some machinery of set theory (e.g. ZFC) > > OK. > > > and/or assume the reals form a complete metric space. > > Please. You can't MIX the two of these! Yes I can. Cantor's first uncountability proof develops an increasing sequence of a_i that is bounded above. He assumes this converges. Completeness is needed here, because such sequences in the rationals do not necessary converge to rationals. > > > If we define the reals using ZFC as, e.g., Dedekind cuts, we can of > > course prove that the reals form a complete metric space and all > > proofs of uncountability carry through. > > You have completely jumped the shark here. > The ZFC proof of the uncountability of the reals neither > KNOWS NOR CARES ANYTHING WHATSOEVER > about THE REALS! The ZFC proof of Cantor's theorem is a proof about > ALL SETS!! > It says that EVERY set has the property of not being as big as its > powerset! > EVERY set! Infinite sets in general and the reals in particular ARE > NOT SPECIAL > in this regard! I understand that the ZFC proof is absolutely general: power sets always have higher cardinality than sets of which they are power sets. If you try to apply this to the reals, you need a representation of the reals. And this is all absolutely correct, within ZFC, in fact, within Z. But I am talking about the continuum such as the ancient Greeks imagined, and wonder how we can apply proofs within Z to that concept, without bringing in the heavy machinery of set theory.
From: Lee Davidson on 13 Jun 2010 16:38 On Jun 11, 5:41 pm, Rupert <rupertmccal...(a)yahoo.com> wrote: > On Jun 12, 4:39 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > > There is a weak theory called RCA_0, which is conservative over PRA, > discussed in "Subsystems of Second Order Arithmetic", in which it can > be proved that no countable sequence of real numbers contains all real > numbers. > This sounds interesting, but I plead ignorance about this. Give more detail if you'd like.
From: Lee Davidson on 13 Jun 2010 16:46
On Jun 12, 9:23 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > George Greene says... > > > > >"The reals" is not even the right entity to talk about. > >Any old infinite set will do, if infinity is where you want to take > >this. > >But it simply shouldn't be. It's called CANTOR's theorem and Cantor > >was talking about SET THEORY, so you HAVE to begin WITH A SET THEORY > >and NOT with "the reals". > > Well, Cantor gave two different proofs of the uncountability of the > reals. One of them generalizes to arbitrary sets, while the other > one is very specific to the reals. > > The one that is specific to the reals goes like this: > > Let r_0, r_1, ... be an infinite list of reals. Then there > is a real r that is not equal to any real on the list. > Proof: > > We assume that the sequence r_n is dense, and has no largest > element (otherwise, it's trivial to find a real that is not > on the list). > > Define two sequences of reals a_n and b_n as follows: Let a_0 = r_0. > Let b_0 = the first real r_k in the original sequence such that > r_k > a_0. > > For n > 0, let a_{n+1} = the first real r_k such that a_n < r_k < b_n. > Let b_{n+1} = the first real r_k such that a_{n+1} < r_k < b_n. > > Then we have the following inequalities: > > a_0 < a_1 < a_2 < ... < b_2 < b_1 < b_0 > > a_n is an increasing sequence of reals bounded from above, > and b_n is a decreasing sequence of reals bounded from below, > and there is no overlap between the sequences. > > So let a_lim be the limit of a_n. Let b_lim be the limit of b_n. > Finally, let r = any number such that a_lim <= r <= b_lim. > > It can be demonstrated easily that r is unequal to any r_k. > > This proof uses essential properties of the reals, such as the > fact that they are totally ordered, and that every > bounded sequence of increasing (or decreasing) reals has a limit. > > -- > Daryl McCullough > Ithaca, NY Last reply post today. Thanks to all. Let me just add, briefly, that, in that idealized possible world containing a "number line" it looks to me like Dedekind cuts are absolutely respectible entities -- initial segments of the number line with no maximum elements (that's my definition) -- and in fact they aren't countable. However, these aren't points on the line, and all the points might well be enumerable. Ironically, Dedekind cuts are enough to enable us to model the real line, even if they might falsify it -- ontologically speaking -- so far as its structure in terms of points. Of course, this presupposes a point ontology -- that lines are "composed of points." We could equally well suggest that points don't exist, and lines are composed of line segments. Given enough assumptions about our ideal possible world, we can derive real number theory that way as well. But I submit that the continuum might remain rather puzzling. In any case, in a point ontoloty, it's the existence of that limit, as a point on the line, that Daryl referred to in his last paragraph, that is the issue here. |