Uncountability of the Rationals?
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From: MoeBlee on 4 Jun 2010 18:29 On Jun 4, 3:28 pm, Tony Orlow <t...(a)lightlink.com> wrote: > On Jun 4, 2:37 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > On Jun 4, 11:16 am, Tony Orlow <t...(a)lightlink.com> wrote: > > > > other > > > measures of sets can also exist, consistent with the axioms, but > > > inconsistent with cardinality. > > > It is not ordinarily disputed that definitions may be given > > stipulatively. So what? > > "stipulative - no dictionary results" - dictionary.com Try 'stipulative definition' on the Internet. > > > > How does TO define 'sequence' without reference to something like the > > > > SET of naturals as indices? > > > > It is a set wherein every element is either before or after (not > > > immediately) every other element. That's one possible definition, > > > Yes, the objection is that it's more general than the ordinary > > definition, thus loses the special sense obtained from the ordinary > > definition. But again, if you don't like the ordinary definition of > > 'sequence', then just imagine everywhere 'sequence' appears in set > > theory discussion that the word is 'zequence'. > Do you not remember the thread not long ago where you methamaticians > started arguing amonst yourselves? What is the "ordinary" definition, > again? I just gave it to you. So what that some people discovered that there are different uses? > Never mind. It's a sidebar, as always with you. Because getting straight the subject matter on which you shoot your big mouth off is always a side matter with you. > Take a deep breath. Deep advice. > Love, Someone should write a song or a poem on that subject one of these days. MoeBlee
From: Ludovicus on 4 Jun 2010 20:09 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. No. The Farey Series of order n have 3*(n/pi)^2 terms. When n ---- infinte, we have all the rationals in the unit interval, but the formula shows that that quantity is countable. That is, it have the same power of the naturals. Ludovicus
From: Jesse F. Hughes on 4 Jun 2010 21:31 David R Tribble <david(a)tribble.com> writes: > Aatu Koskensilta wrote: >> The supply of morons in this world is inexhaustible. >> So naturally you object also to the statement >> PA + Con(PA) proves more true statements about naturals than PA. > > I don't follow this. > > If the number of true (constructible) statements proved by PA and > the number of true statements proved by PA + Con(PA) are both > countable, does it not follow that those two sets have the same > cardinality? Aatu's point is simply that, in some contexts, we say that this set has more elements than that set even though they have the same cardinality. Sometimes, we use the word "more" to refer to the superset relation. -- Jesse F. Hughes "I think the problem for some of you is that you think you are very smart. I AM very smart. I am smarter on a scale you cannot really comprehend and there is the problem." -- James S. Harris
From: Tony Orlow on 5 Jun 2010 01:44 On Jun 4, 4:09 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > 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. That's usually what I mean. I'll be more specific if not. > > > 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. Perhaps simple bijection as a proof of equinumerosity is superficial. That's also a possibility. :) Peace, TOny > > MoeBlee
From: Tony Orlow on 5 Jun 2010 01:45
On Jun 4, 4:04 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > 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). I really appreciate that philosophical concession, and freely admit that cardinality captures the essence of set size with respect to finite sets. No controversy there. > 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'. Okay, phrase missing, but more or less understood. Proceed.... > > 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 Well, sure, MoeBlee. I'm not sure how easy you think it is to concoct an original mathematical theory regarding infinity on one's own. In some sense I have been working on this for thirty years. The past several have been a revival, both of interest in understanding the standard theory, and in improving on it. There will remain questions unanswered when I'm done, I'm sure, but the continuum hypothesis is not one of them. Poll: would you rather see a collection of all axioms and theorem derivations, or a more explanatory approach? I'm thinking the formal axiom set and development is more of an appendix. Best Regards, ToeKnee |