Uncountability of the Rationals?
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From: Brian Chandler on 14 Jun 2010 01:21 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) 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
From: Brian Chandler on 14 Jun 2010 01:27 Virgil wrote: > In article > <b27319dd-1f53-49e5-a876-753b6301f008(a)g39g2000pri.googlegroups.com>, > Brian Chandler <imaginatorium(a)despammed.com> wrote: > > > Seems to me something has gone wrong here... > > > > David R Tribble wrote: > > > Virgil wrote: > > > >> Real functions do allow infinite quantities either as arguments or > > > >> values, so that you are making false statements. > > > > 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. > > In standard mathematical usage a "real function" is one whose domain and > codomain are both subsets of the set of real numbers. > And that standard usage meaning is what I mean by the phrase. OK. So what do you mean by saying "Real functions do allow infinite quantities either as arguments or values, ..." (Ah! Perhaps I see. You meant "Real functions do not allow infinite quantities either as arguments or values, ...", which is only one word different. In the infinite limit, which is where sci.math is going eventually, I suppose this is no difference at all.) Brian Chandler
From: David R Tribble on 14 Jun 2010 01:32 Brian Chandler wrote: > Seems to me something has gone wrong here... Okay, before we all start talking at cross purposes here... 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. Tony (and I) are talking about real-valued functions. Tony Orlow wrote: >> Don't correct me incorrectly. Standard real functions take real value >> parameters and return a real result. Where they have return an >> infinite value, they are declare "undefined" or to have some infinite >> "limit" at that point, and where any parameters are of infinite >> nature, the modern mathematical procedure is to refer to the "limit" >> of the function as one or more variables approach oo or -oo. To lie by >> accusing someone is lying is...lame. > David R Tribble wrote: >> A real-valued function, by definition, has a domain from which it >> takes its arguments, and a range (or codomain) of values it "returns" >> (i.e., maps those arguments to). The function maps each argument >> value from the domain to a single result value in the range. >> >> If the function has no mapping for a particular argument, then the >> function is not defined for that argument value, and that value is >> not within the domain of the function. >> >> Likewise, real functions take real arguments, which means that >> infinity (+/-oo) cannot be in the domain set of a function, and thus >> can never be an argument to a function. > Brian Chandler wrote: > Right. Well see above. Isn't that just a more carefully written > version of what Tony said? No. Tony says that real-valued functions can return infinite values. I said that they can't. It's a subtle distinction, but Tony is saying that whenever a function returns an infinite value for a particular argument, "standard" mathematics declares that function to be "undefined" for that value. He states it like it's a cop-out by standard mathematics. Remember, Tony is trying to posit that real functions are allowed to operate on non-real infinite values (assuming a few criteria are met) just as if those infinite value acted like real values. The real function f(x) = x+1 works on all real x, but Tony claims that it also works on infinite x in exactly the same way. It's like he's trying to eliminate the notion of an "undefined" function value and just go ahead and replace it with "infinite result". David R Tribble wrote: >> An example is the reciprocal function, f(x) = 1/x. The domain of f >> is R \ {0}, i.e., all real values except 0. The function does not >> "return infinity" for argument 0, rather, f is simply does not have >> any result for it. It would be more correct to say that f(0) is >> "nothing" or "the empty set". > Brian Chandler wrote: > I certainly don't think it would be correct to say that the reciprocal > of zero is the empty set. Okay, but I think it makes sense to say that there are no values of the reciprocal function for argument zero, or that there is no reciprocal of zero. Which to me sounds a lot like saying that the set of values for recip(0) is empty. We don't disagree, we just might have slightly different notions of what "undefined" means for functions as sets. (And I never claimed to be an expert.)
From: FredJeffries on 14 Jun 2010 01:53 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. Which of the following is larger? A = {1, 2, 3} B = {1, 4, 5, 7} > > I'd love to believe that there can be a completely > rigorous theory applicable to the sciences in which the > first notion is rejected for the second. I see no reason > for the lack of symmetry in that there's a sensible set > size that maintains the first property but none that > maintains the second property. 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. So, even for finite sets, I don't think that the situation is as symmetric as you would have us believe. > > We know all about standard cardinality. I want to know > how to develop a set size that preserves the second > property of finite set size. How about this one: for a set A, first well-order P(A). For any two subsets a, b of A if a is a subset of b (or b of a) we know which is larger. If not, if the cardinality of the set difference a - b is greater (less) than that of b - a, then a is larger (smaller) than b. Otherwise, use the well ordering from step 1: whichever is the first element of {a, b} according to that well-order is the smaller.
From: Virgil on 14 Jun 2010 02:06
In article <4e59f788-b0dd-4a2d-af66-c08b54b6bded(a)z15g2000prh.googlegroups.com>, Brian Chandler <imaginatorium(a)despammed.com> wrote: > Virgil wrote: > > In article > > <b27319dd-1f53-49e5-a876-753b6301f008(a)g39g2000pri.googlegroups.com>, > > Brian Chandler <imaginatorium(a)despammed.com> wrote: > > > > > Seems to me something has gone wrong here... > > > > > > David R Tribble wrote: > > > > Virgil wrote: > > > > >> Real functions do allow infinite quantities either as arguments or > > > > >> values, so that you are making false statements. > > > > > > 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. > > > > In standard mathematical usage a "real function" is one whose domain and > > codomain are both subsets of the set of real numbers. > > And that standard usage meaning is what I mean by the phrase. > > OK. So what do you mean by saying "Real functions do allow infinite > quantities either as arguments or > values, ..." If I left the "not" out, it was just poor prof reading. > > (Ah! Perhaps I see. You meant "Real functions do not allow infinite > quantities either as arguments or > values, ...", which is only one word different. In the infinite limit, > which is where sci.math is going eventually, I suppose this is no > difference at all.) > > Brian Chandler |