Uncountability of the Rationals?
in [Math]
|
Prev: Collatz conjecture
Next: Beginner-ish question
From: MoeBlee on 7 Jun 2010 17:29 On Jun 5, 1:06 pm, Tony Orlow <t...(a)lightlink.com> wrote: > or is less dense in the natural quantitative order There is no general definition that we have (let alone that you have provided) of "the natural quantitative order". MoeBlee
From: MoeBlee on 7 Jun 2010 17:56 On Jun 7, 7:59 am, Tony Orlow <t...(a)lightlink.com> wrote: > On Jun 6, 5:31 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > On Jun 5, 9:34 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > > > On Jun 4, 3:16 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > As far as I know, neither Galileo nor Aristotle proposed what could > > > fairly be called a set theory. > > True, but they explored infinite sequences, Aristotle making the > distinction between the two general types, and Galileo exploring > bijection as a means of caomparison, and then conclusing that the > sizes of infinite sets simply don't exist or cannot be calculated. (1) That doesn't address my point about their role in your analogy. (2) I'll take your word for it that that was Galileo's conclusion. > the way, I only agree partly with Galileo's conclusion, but fully > agree with the distinction between actual/uncountable and potential/ > countable infinities. Whose distinction is that? It's not Aristotle's. As far as I know, Aristotle's disctinction was between actual and potentential infinity, as he endorsed the latter notion over the former. > > By the way, I should have mentioned that Orlow mentioned also Cantor > > and Dedekind as counterparts of Jesus in Orlow's analogy. I don't see > > how that would make sense in terms of his analogy, since Cantor and > > Dedekind proposed uncountability, contrary to Orlow's own analogy. > > If you really think that I don't believe in the distinction between > the countable and uncountable then I don't know who you think you were > talking to or what you think I said. You're reading into my remarks what is not in them. I didn't suggest in the least that you decline a distinction between countable and uncountable. > Additionally, it is not ZFC that > I object to, but the extension of the theory with the model or > cardinality as "set size" for the infinite case. Simple bijection does > not equate to "equinumerosity" in my book. This is where AGAIN you just skip recognizing my arguments and explanations (SEVERAL of them over years) about this. It seems you don't even know what I'm talking about, no matter how carefully, simply, or, in contrast, fully explain. I surmise you just skim right past. > That's "equicardinality" > for the mathematically rigorous, no? You're welcome to make up the word. In the meantime, 'equinumerous', 'equipotent', 'equipollent' are commone enough. Then, with cardinals incorporated, we can also say 'having equal cardinality', etc. > The analogy has to do with what Hilbert et al did with ZFC, and what > Constantine did with Christianity. I got that more or less. But the Jesus mention in the analogy was then left dangling. Not a big deal; I just wanted to see where you'd land with it. > > > > Anyway, so everything Galileo and/or Aristotle said was true and wise? > > > > So _nothing_ Galileo and/or Aristotle said was true > > > and wise? > > > Of course Galileo and Aristotle were wise in certain ways. So what? > > So were Abraham and Buddha, who each contributed to the wisdom of > Jesus. That's what. That's a fine non sequitur. > > > > > ToeKnee > > > > BoerEeng. > > > > "Boring." This, of course, represents the reaction > > > that MoeBlee would have if TO really were to post a > > > rigorous theory applicable to sciences based on his > > > own non-ZFC notions. MoeBlee would be "bored." > > > No it doesn't. STOP PUTTING WORDS IN MY MOUTH. > > > If Orlow ever came up with a rigorous theory I'd be amazed and > > fascinated. STOP PUTTING WORDS IN MY MOUTH. > > Don't misrepresent my position and it won't come back to bite you. What are you talking about? I've not intentionally misrepresented you, and when I have inadvertently, I've been diligent to correct myself. That is in stark contrast with Transfer Principle's posting about me. > > The point I made by saying "BoerEeng" is only that Orlow has worn out > > whatever amusement there ever was by signing himself with the > > affectation "ToeKnee" in response to my posts signed "MoeBlee". > > My ApoloGees. hyuck hyuck. The funny thing is that it bothers you at > all. Not as funny as that you're so pleased with your repetitiveness. > FootLeg 'FootMouth' would suit you better. MoeBlee
From: Virgil on 7 Jun 2010 19:48 In article <b4a074fc-10ea-4e1e-bc2d-e50846202c8b(a)w31g2000yqb.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > On Jun 7, 3:59�pm, Virgil <Vir...(a)home.esc> wrote: > > In article > > <9b865fc2-1836-4aa3-9f5f-016f04300...(a)u26g2000yqu.googlegroups.com>, > > �Tony Orlow <t...(a)lightlink.com> wrote: > > > > > The argument that the truly infinite set of naturals must include an > > > infinite natural has nothing particular to do with N=S^L. That's a > > > simple matter of the nth element of any initial segment of N being > > > equal to n and existing for any segment of size n or greater. > > > > One can have initial segments of infinite well ordered sets which have a > > last member and initial segments which do NOT have a last member, and > > the two types are quite distinct. TO attempts to overlook the difference > > and conflate the two types. > > > > In particular, for infinite well-ordered sets there is no necessity for > > their "initial sets" all to have largest members, and some of those > > initial sets necessarily don't have largest members! > > > > Consider, for example, the Cartesian product of the naturals with the > > naturals, NxN = {(m,n} : m in N and n in N} with the well-order > > (a,b) < c,d) > > iff �a < c or [ a=c and b < d ] > > > > Then for any n in N, the set of (a,b) for which a <= n is an initial > > segment with no largest element, so there are as many such initial > > segments without a largest as there are initial segments altogether. > > Look at what you quoted from me above. Where do I say anything about a > last element? This is typical deliberate misinterpretation and a red > herring. Thanks for the waste of time. > > Tony Tony has often argued that the set of naturals, by being itself infinite, must contain an infinite member. If anyone has wasted time with that sort of nonsense, it has been TONY.
From: David R Tribble on 7 Jun 2010 19:48 Tony Orlow wrote: > True, but they explored infinite sequences, Aristotle making the > distinction between the two general types, and Galileo exploring > bijection as a means of caomparison, and then conclusing that the > sizes of infinite sets simply don't exist or cannot be calculated. Not exactly. He never claimed that they don't exist. He did conclude that calculating with infinity does not follow the same rules as with finite numbers. Here's a nice short summarization of Galileo's observations about infinity: http://www.firstscience.com/home/articles/big-theories/a-brief-history-of-infinity_1223.html -drt
From: David R Tribble on 7 Jun 2010 19:54
MoeBlee wrote: > (1) For only occasional sets do we have a "usual" ordering. In > general, there is no definition provided (especially by you [Tony]) for "the > usual ordering of a set". So your analysis in that respect doesn't > even get off the ground. Indeed. Here are a few ways to write the set of naturals, all of them being exactly the same set: { 0, 1, 2, 3, 4, 5, 6, ... } { 0, 2, 1, 4, 3, 6, 5, ... } { 0, 2, 4, 6, ..., 1, 3, 5, 7, ... } { 0, 1, 3, 4, 6, 7, 9, 10, ..., 2, 5, 8, 11, ... } { ..., 5, 4, 3, 2, 1, 0 } { 0, ..., 4, 3, 2, 1 } Tony must realize that a set is simply a collection, specifically an *unordered* collection. Just because all of its members are natural numbers (or reals, or rationals, or whatever) does not automatically embue it with an ordering. -drt |