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From: Virgil on 27 Oct 2005 15:18 In article <MPG.1dcab744b1856f9198a57e(a)newsstand.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > Randy Poe said: > > Since f:N->E is both injective and surjective, it is bijective. > Okay, thanks, how about a proof of bijection between the natural > numbers and the binary strings? Which set of naturals and which set of binary strings? In TOmatics there are at least two of each. For the standard bijection between the Dedekind infinite set of finite naturals and the Dedekind infinite set of finite binary strings, map each finite natural to its finite binary string representation and voila! Since the Dedekind infinite set of finite natural is the only set of naturals outside of TOmatics, that bijectin will have to suffice, at least everywhere outside chaos.
From: Virgil on 27 Oct 2005 15:38 In article <MPG.1dcab8fe3f850ff98a580(a)newsstand.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > > I wonder if you've noticed that (with the possible exception of > > other cranks) you are the only one who goes about "declaring" this > > and that, or "applying" values to things you want to "measure" .. > LOL that's a good one. You declare aleph_0 to be some smallest > infinity, applying this "value" to a set with no size so as to > measure it. We *define* Aleph_0 to mean the cardinality of the Dedekind infinite set of finite naturals, N. It can easily be proven that there is an injection from N to any Dedekind infinite set, which *proves* that, by the Cantor definition of order of cardinalities, aleph_0 is as small as any Dedekind infinite cardinality can be. All of this is quite simple logically, which is no doubt why TO cannot grasp it. > Please! Don't make me wet my pants laughing. When and how you micturate, as long as you do in only in TOmatica, is of no interest outside TOmaica.
From: Virgil on 27 Oct 2005 15:46 In article <MPG.1dcaba9ffcd3c6998a581(a)newsstand.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > Robert J. Kolker said: > > Tony Orlow wrote: > > > > >> > > > > > > Perhaps. Cardinality is better than nothing. And yet, I think > > > that noting that the size really is not a number with an absolute > > > value but needs to have some value applied to it in order to > > > measure it, > > > > You continually confuse "how many" with "how much". > Pah! Bob, I equate "how many" with "how much" when there is a 1-1 > correspondence between element value and element count, as is the > case with the naturals. I am not confused. That sounds like a cry for help! And TO certainly could use some. > > > The former is denominated by cardinal numbers. That later by some > > form of measure. Google <measure theory>. The simplest sorts of > > measures are length, area, volume etc. and generalizations thereof. > > > > Mathematicians do not confuse these two kinds of quantities, but > > you do, therefore you are not a mathematician. Your inability to > > learn from your betters indicates strongly that you will never > > learn to be one. > You have no clue what I'm doing here, do you Bob? One might guess trolling, since it is certainly neither mathematics nor logic. The whole point is > to bring these various disparate measures that don't get along, and > turn them into a cohesive team that can discriminate all types of > sets accurately. So how many pounds tall are you, TO? > > > > Before you criticize well established mathematical theories you > > should learn what they are first. You have not done this and you > > show no indication of ever doing so. > I know what they are, but they don't make sense. That TO claims they do not make sense to TO has several possible interpretations, one of which is that TO is too stupid to comprehend, another is that he is merely trolling. My own opinion is that it is a mixture of these two.
From: Virgil on 27 Oct 2005 15:54 In article <MPG.1dcabbdb7bb59b1d98a582(a)newsstand.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > Virgil said: > > There is no such thing as 'the' size of the set, there are many > > sizes, depending on which proprties one is looking at. But only the > > cardinality size can be applied to every set. all other types of > > size are limited in what sets they can compare. > Do you have one tool in your tool drawer? Is it a hammer, a > screwdriver, or a wrench? Probably just a rock. Cardinality works for all sets. TO's "sizes" do not. Better having one tool that always works than any number of broken tools. > > > > > > I hope that's okay, even if it sounds like babbling hogwash. Does > > > deep hogwash run still? Hmmm.... > > > > If TO is content to wallow in his hogwash, ... > Come wallow, O Virgil, for the sun is hot, and the hogwash is > refreshing! Cool thy meaty rump in the mud, as you bask in the > afternoon breeze. :D The difficulty is that I cannot cross the gap between into TOmatica, and TO's pond of hogwash lies in TOmatica, and lies, and lies.
From: Randy Poe on 27 Oct 2005 15:57
Tony Orlow wrote: > Randy Poe said: > > Since f:N->E is both injective and surjective, it is bijective. > Okay, thanks, how about a proof of bijection between the natural numbers and > the binary strings? Which naturals, ours or yours? Which binary strings? How about the one I claim elsewhere exists, the bijection between *N (or its binary representation), and P(N)? I like to pin things down: Defn: An "infinite bit string" is a map s:N->{0,1} so that every natural number n in N is associated with either 0 or 1. The "n-th" bit is s(n), also written s_n. So I've got exactly one bit for every n in N. When talking about "infinite strings" I want to be specific about just what labelling set I'm using to label my bits. (Being ambiguous about that is exactly where the fallacy in your "bijection" lies). Do we have a definition of *N separate from the bit strings? I'm not sure. So let me just say that *N will be the set of all infinite bit strings under the above definition of infinite bit string. Consider the obvious map to subsets: f:*N->P(N), where f(s) = {n in N: s_n = 1}. So s_n = 1 if and only if n is in f(s). There is also the obvious inverse g:P(N)->*N. Let w be any element of P(N), i.e. w \subset N. Then g(w) is an infinite string with g(w)_n = 1 if n in w, 0 otherwise. Claim: f is surjective. Proof: g(w) exists and is a valid infinite bit string for any w in P(N), and f(g(w)) = w. Thus, for any w in P(N), there exists s in *N such that f(s) = w, and f is surjective. Claim: f is injective. Proof: Consider s and s' such that f(s) = f(s'). That is, for every n in N, (n in f(s)) if and only if (n in f(s')). But then (s_n = 1) if and only (s'_n = 1). So s = s'. Since f(s) = f(s') => s = s', f is injective. f is a map from *N to P(N) which is surjective and injective. Therefore f is a bijection from *N to P(N). - Randy |