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From: Daryl McCullough on 31 Oct 2005 17:03 Tony Orlow says... >No element is ever identified, in any infinite bijection, >as mapping to the last element of an endless set. Well, obviously if a set doesn't have a largest element, then nothing maps to the largest element. Unfortunately, a power set *does* have a largest element, namely the entire set. So if you claim to have a bijection between some set A and P(A), then you must have something that maps to the largest element of P(A). >To expect this of the power set bijection is consistently >inconsistent It's part of the definition of "bijection" that if you have a bijection between A and B, then *every* element of B must be associated with a unique element of A, and vice versa. As has been pointed out to you, there *is* a bijection between the P(N), the set of all subsets of the (finite) naturals, and N -> {0,1}, the set of all bit strings over the finite naturals: If s is a bit string, then define f(s) = { n | s_n = 1 }. That is a bijection between the set of all bit strings over the finite naturals and the set of all subsets of the naturals. Note that we can define an ordering on the bit strings: s1 > s2 if s1 is not equal to s2, and for all n, s1_n >= s2_n So there is a natural "largest" bit string: max = that bit string consisting of all 1s and there is also a largest set of finite naturals: N = the set of all finite naturals. The mapping f described above maps the largest bit string to the largest set of finite naturals. So there is no problem getting a bijection between some set and a power set. I can tell you exactly which bit string maps to the entire set. -- Daryl McCullough Ithaca, NY
From: Virgil on 31 Oct 2005 17:22 In article <MPG.1dd01f9793455ef598a5aa(a)newsstand.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > Virgil said: > > In article <MPG.1dcaba9ffcd3c6998a581(a)newsstand.cit.cornell.edu>, > > Tony Orlow <aeo6(a)cornell.edu> wrote: > > > > > > 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? > If I told you I was 6'5" and weighed 47 pounds, would you believe me? How > about > 3'11" and 475 pounds? Is there no relation whatsoever? Chances are, you are a > fat set of cells, at least the top portion of the set. The issue of different measures for different qualities seems to be one that TO cannot bring himself to deal with. Cardinality of sets is a valid measure of what it measures. TO's alleged "value range" is not a valid measure of anything except when it coincides with "diameter".
From: Virgil on 31 Oct 2005 17:24 In article <MPG.1dd02013441ca13698a5ab(a)newsstand.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > Virgil said: > > 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. > So, yes, it's a rock. And when you break it, then you have TWO tools in your > cave. Good, Virgil. Very good. While TO has nothing in his TOmatics cave but delusions.
From: Daryl McCullough on 31 Oct 2005 17:06 Virgil says... >> Look again at the bijection I offered. Your mapping between *N and >> P(N) is not valid in Bigulosity Theory. > >On the contrary, in "Bigulosity Theory" anything, and its negation, is >valid. Actually, Tony's mathematics works like this: a formal proof doesn't show that something that something is true, it only shows that it is true unless there is some reason not to believe it. So Tony accepts the axiom of mathematical induction, but he does not accept all the theorems that are provable using that axiom. You have to look at theorems on a case-by-case basis to decide whether to believe them. -- Daryl McCullough
From: Robert Low on 31 Oct 2005 17:29
Daryl McCullough wrote: > So Tony accepts > the axiom of mathematical induction, but he does not accept all > the theorems that are provable using that axiom. You have to look > at theorems on a case-by-case basis to decide whether to believe them. Well, if you can't use your common sense to reject the theorems that are obviously wrong, what's the point of having any taste or discrimination at all? |