The right way to question the uncountability of the reals
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From: George Greene on 14 Jun 2010 17:03 On Jun 12, 9:23 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > 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: Yes, we have seen that one before. It is not even set theoretic. What was going to be relevant to all the talk in the threads was some infinitary SET theoretic proof,something that somehow exploited some property OF INFINITE SETS. But THE POINT is, Cantor's theorem about the different sizes SIMPLY HAS NOTHING TO DO with infinite sets. Infinity is crucially exploited in the "other" proof you mention, but that is just proof that that is an inferior proof. Infinity is NOT ACTUALLY RELEVANT to the question!
From: George Greene on 14 Jun 2010 17:05 On Jun 13, 4:46 pm, Lee Davidson <l...(a)meta5.com> wrote: > 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. That's only "the" issue "here" IF you're using THAT proof. If you're using a set-theoretic proof then infinity is NOT even relevant AT ALL.
From: Rupert on 15 Jun 2010 03:03 On Jun 14, 6:25 am, Lee Davidson <l...(a)meta5.com> wrote: > 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. This seems more to do with the question of how we would go about verifying that there exists a physical continuum than with mathematics.
From: Rupert on 15 Jun 2010 03:04
On Jun 14, 6:38 am, Lee Davidson <l...(a)meta5.com> wrote: > 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. It is probably best if you read the exposition in Chapter 2 of "Subsystems of Second Order Arithmetic". |