Uncountability of the Rationals?
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From: David R Tribble on 4 Jun 2010 16:02 Tony Orlow wrote: > In Cantor's list of reals, for every digit added, the list doubles in > length. I assume you mean a list of finite-length binary fractions representing reals in [0,1). > In order to follow his logic, we need not consider any real > numbers which are not rational. In any such list of rationals, it is > always true that we can concoct another rational which is not on the > list. Oh come on, Tony. You know this isn't right. A list of finite-length rationals is either a finite list itself, being a list of reals of n bits or less and therefore containing no more than 2^n reals in total; or it's an infinite list of finte-length reals (i.e., all of them are binary fraction that end with a trailing list of repeating 000... digits). In the first case, it's not Cantor's list of reals, so it's moot. In the second case, the real that's not in the list is a sequence of binary digits that differs by at least one digit from every other (rational) real in the list. This diagonal number, by the way, is most likely not rational (unless it just so happens to end in a repeating binary digit sequence). > Irrational numbers need not even be considered. So, are the > rationals not to be considered uncountable, by Cantor's own logic? Well, first you have to have the correct description of Cantor's diagonal argument. You don't. > BTW, I understand that the rationals are considered countable because > of a rather artificial bijection with N, Or any other bijection with N. There are an infinite number of them to choose from. I personally like the one I independently re-discovered a few years ago (as a response to one of your posts, in fact, if I recall correctly), which is at: http://david.tribble.com/text/sequence1.html > but if Cantor's argument > really has anything to do with the reals, as opposed to the powerset > (which is what it's really all about), then why is it entirely > unnecessary to even consider irrational reals in his construction? Again, you have to have the right description of the list. Once you do, you can see that your conclusion is not warranted. -drt
From: MoeBlee on 4 Jun 2010 16:04 On Jun 4, 1:32 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jun 3, 7:34 am, Tony Orlow <t...(a)lightlink.com> wrote: > > and the simplistic assumption of equivalence of > > "size" based on bijection alone. > > I've presented to you at least a hundred times the answer (in > variation) that instead of the word 'size' we could use the word > 'zize'. Then your protest disappears. You've never dealt with that > point. P.S. I don't claim that such a "formalistic" rebuttal to you settles the question. That is to say, I don't begrudge that there may be a legitimate philosophic dispute over whether set theoretic cardinality does or does not capture our intuitive notion of size in regards infinite sets. (That set theoretic cardinality does capture our notion of size in finite cases I take to be ordinarily uncontroversial). However, we must at least address my "formalistic" rebuttal at least first, so to recognize that in the sense of set theory as a formal theory, whether we call call cardinality 'size' or 'zize'. That said, however, still I don't take the import of set theory to be to settle such questions except as they pertain to mathematical inquiry, in which sense the notion of cardinality does seem to be viable. This is so especially since at least I don't know of a challenging formalization of the notion of size in the infinite case that is any more intuitive than set theoretic cardinality, as at least set theoretic cardinality is motivated by the intuitive notion of one- to-one correspondence but also taken in the infinite case. More specifically, Orlow himself has not presented any theory that satisfies EITHER criteria: (1) formally coherent, (2) more intuitive than set theoretic cardinality, as Orlow's notions are even LESS intuitive than set theoretic cardinality. That he personally finds his own ruminations more intuitive is not sufficient philosophical motivation for patiently waiting for him to someday put his floating ideas into the form of a formal theory. MoeBlee
From: MoeBlee on 4 Jun 2010 16:09 On Jun 4, 2:50 pm, Tony Orlow <t...(a)lightlink.com> wrote: > You cannot disagree with the fact that the rationals are a dense set > whereas the naturals are sparse, WRONG, WRONG, WRONG. I've explained this to you several times, and you persist to ignore. A set is dense with respect to some ORDERING. There IS a dense ordering of the set of natural numbers, and there is an ordering of the set of rationals that is NOT dense. What you can say is that the set of natural numbers is not dense with regard to the usual ordering on the set of natural numbers and that the set of rational numbers is dense with regard to the usual ordering on the set of rational numbers. > with a countably infinite number of > rationals lying quantitatively between any two given naturals, can > you? Do you not see that in some sense there appear to be more > rationals than naturals? Yes, but some investigation shows that that is a superficial view, since we see also that we can order the naturals densely and order the rationals discreetly. MoeBlee
From: MoeBlee on 4 Jun 2010 16:13 On Jun 4, 2:58 pm, Tony Orlow <t...(a)lightlink.com> wrote: > On Jun 4, 1:42 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > > which we find in the logical literature? There is of course no > > suggestion the set of theorems of PA + Con(PA) is uncountable. > > This also sounds very fascinating, and I'd like to know more. It's not fascinating. Aatu is just reminding of the dry fact that in conversational mathematics, we do use the word 'more' in a sense not confined to cardinality. A more prosaic example: If I have a countably infinite set of symbols, but "add on" another symbol, then I say "now I have more symbols" though I don't mean that in the literal sense that the "new" set has greater cardinality. MoeBlee
From: David R Tribble on 4 Jun 2010 16:20
kunzmilan wrote: >> The uncountability of the reals is simply based on the fact, that >> there are more rational numbers than there are the natural numbers. > Tony Orlow wrote: > The rationals have the same cardinality as the naturals, though there > are surely more rationals per unit interval of the real line than > there are naturals. Yes, but it's a curious fact that there are the same number of rationals as naturals on the _entire_ real number line. kunzmilan wrote: >> The rational numbers are defined as the natural number i divided by >> the natural number j. Thus, rational numbers can be greater than 1, >> too. > Tony Orlow wrote: > One might think there were something like aleph_0^2 rationals, but > that's not standard theory. Actually, there are Aleph_0^2 rationals. And Aleph_0^2 = Aleph_0. -drt |