Uncountability of the Rationals?
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From: Marshall on 14 Jun 2010 11:01 On Jun 14, 7:46 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jun 13, 4:55 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > On Jun 13, 4:37 am, Tony Orlow <t...(a)lightlink.com> wrote: > > > "complete relation", set of all pairs of members from each set, > > > For any sets S and T, we call that S X T. > > Sorry, CORRECTION: > > I meant (S u T) X (S u T). > > Hmm, it seems there should be a common term for that. Perhaps there > is, but it's not coming to my mind now. Maybe it is 'complete > relation', but I don't recall ever reading it. "Cross product" perhaps? Marshall
From: MoeBlee on 14 Jun 2010 11:07 On Jun 14, 10:01 am, Marshall <marshall.spi...(a)gmail.com> wrote: > On Jun 14, 7:46 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > On Jun 13, 4:55 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > > On Jun 13, 4:37 am, Tony Orlow <t...(a)lightlink.com> wrote: > > > > "complete relation", set of all pairs of members from each set, > > > > For any sets S and T, we call that S X T. > > > Sorry, CORRECTION: > > > I meant (S u T) X (S u T). > > > Hmm, it seems there should be a common term for that. Perhaps there > > is, but it's not coming to my mind now. Maybe it is 'complete > > relation', but I don't recall ever reading it. > > "Cross product" perhaps? 'cross product' is just another term for 'Cartesian product'. But here we mean not just the Cartesian product of S and T, but rather the Cartesian product of SuT and SuT. I.e., {<x y> | {x y} subset of SuT} = SuT X SuT. I don't recall that, given S and T, there is an ordinary terminology to say {<x y> | {x y} subset of Sut}. MoeBlee
From: Brian Chandler on 14 Jun 2010 11:52 David R Tribble wrote: > Brian Chandler wrote: > > Seems to me something has gone wrong here... > > Okay, before we all start talking at cross purposes here... We already are. > Virgil wrote: > >> Real functions do allow infinite quantities either as arguments or > >> values, so that you are making false statements. > > > > Brian Chandler wrote: > > Really? What does Virgil mean by "real functions"? One might guess > > (particular if one were as mathematically ignorant as Tony) that he > > means "real-valued functions". But to make V's statement true, I > > think, he would just have to mean "real functions" as opposed to > > "fake functions". Pretty confusing. > > Yes, okay, Virgil probably meant "actual" functions, as opposed > to functions (or "formulae") in Tony's sense. And yes, functions > can certainly have non-real domains and values. No, Virgil explicitly says that he means real-valued functions. But it turns out that when he wrote "do allow", this was actually a typo for "do not allow". > Tony Orlow wrote: > >> Don't correct me incorrectly. Tony managed to read "do allow" as meaning "do allow", and is rightly annoyed that Virgil is spouting nonsense -- albeit really a typo for non-nonsense. But UIMM, Virgil felt it appropriate to respond with one of his usual spews, without bothering to read his own words to see if they made sense. This is not how communication is supposed to work, chaps (and chapesses should there be such). Brian Chandler
From: Brian Chandler on 14 Jun 2010 12:04 FredJeffries wrote: > On Jun 8, 3:03 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > MoeBlee refers to extrapolating from the finite to > > the infinite when considering set size. Here are > > two equally intuitive notions that describe set > > size for finite sets: > > > > 1. Sets in bijection with each other have the same size. > > 2. The whole is strictly greater than the part. > > (paraphrased from Euclid) > > > > The problem is that for (Dedekind-) infinite sets, there > > is no example of set size that preserves both of these > > notions, since by definition there exists a bijection > > between a D-infinite set and one of its proper subsets. > > > > So the best we can do is choose one of these two equally > > good notions to preserve. Standard Cantorian cardinality > > rejects the second in favor of the first. But I see no > > reason that we can't reject the first in favor of the > > second -- and this would give us TO's Bigulosity. This is nonsense, of course. "Standard Cantorian cardinality" (SCC) doesn't "reject" anything -- but then TP is totally hung up on "standard theorists" (or whatever they're called this week) "rejecting" things. Anyway, yes, cardinality is a natural extension of the "size" comparison of finite sets which preserves the first notion. But the simplest measure that preserves the second notion is the subset relation: perfectly sound "standard theorists'" maths, but inevitably only a partial ordering. Bigulosity simply isn't a theory in any normal sense -- just a braindump of Tony's musings. > Which of the following is larger? > A = {1, 2, 3} > B = {1, 4, 5, 7} > I can use bijections to see that {1, 2} has smaller cardinality that > {1, 2, 3} without referring to the fact that the first is a subset of > the second. But I can't find a way to say which of {1, 2, 3} or {1, 4, > 5, 7} is larger with respect to your second notion without using a > bijection somewhere. Come on... for each of the two finite sets, write the names of the elements on evenly spaced points around a circle. Join points in the obvious way to form polygons. To compare the size of two sets, simply measure the internal angles of their respective polygons, and whichever has the larger angle is the larger set. Brian Chandler
From: Brian Chandler on 14 Jun 2010 12:15
Tony Orlow wrote: > On Jun 14, 1:21 am, Brian Chandler <imaginator...(a)despammed.com> > wrote: > > David R Tribble wrote: > > > Tony Orlow wrote: > > > > Actually, in Bigulosity, there are much larger countably infinite > > > > sets. The square roots of the naturals have a higher "density" than > > > > the naturals. > > > > > How is that? The two sets should be exactly the same size > > > (your size(), not just cardinality). Every natural has a square root, > > > and every natural square root has a corresponding natural. > > > > Can't you see? You're just using "cardinality" to point out a > > bijection between naturals and their square roots. Bigulosity Theory, > > or BT, simply rejects such arguments, in its search for the Answer. > > > > Consider any large number Q (perhaps a quillion): it is clear that in > > the range 1-Q there are Q naturals, but there are (roughly, at least) > > Q^2 real values x such that x^2 is a natural. Now appealing to ICI, > > PCT, QD, and any other TLAs to hand, declare Tav* to be our unit > > infinity, let Q tend to Tav, and lo! the answer we wanted. > > > > * Last letter of the Hebrew alphabet according to Wikipedia. > > > > It is true that BT, of which I am but a beginning student, leaves > > nagging doubts. Suppose, for example we thought about the bigulosity > > of the set NQ > > > > NQ = { {p,q} | p^2 = q e N} (set of pairs of naturals with their > > roots) > > Done. You have an omega X omega^2 matix. You omega^3 elements. Is this the answer, Dear Leader? "You omega^3 elements"? This sentence no verb? Explication possible? I would have thought that since their is one pair {p,q} for each natural q, that there might be omega of these pairs. Or possibly, since their is one pair {p, q} for each root of a natural p (of which, DL, you say there are omega^2), then there might be omega^2 of these pairs. But you tell us there are omega^3?? I see no matrix, by the way. I also wonder what the bigulosity is of the set of pairs {p, p} where p is a natural? Brian Chandler I > Next.... > > > > > But I suspect that our Leader will determine that this is a Trick Set, > > to which special rules apply. We may have to declare a new Unit > > Infinity, or Recalibrate the naturals, or something. But you'll have > > to wait for the answer till he comes back from lunch. > > > > Brian Chandler |