An uncountable countable set
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From: Tony Orlow on 15 Sep 2006 16:02 Virgil wrote: > In article <45085f4c(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> David R Tribble wrote: >>> Tony Orlow wrote: >>>>> If you remove an element, the proper subset should ALWAYS be smaller by >>>>> 1. That is the case for me. For a theory to claim a proper subset is the >>>>> same "size" as the proper superset is an immediate deal-breaker for me. >>> David R Tribble wrote: >>>>> If by "different size" you mean that you cannot pair up all the >>>>> elements from both sets, then you're going to have a difficult >>>>> time proving that for any infinite set. (You have never show this, >>>>> BTW.) >>>>> >>>>> If by "different size" you mean something other than some way of >>>>> denumerating (counting) the elements of the set (e.g., by assigning >>>>> them different natural indices), then you should use a different term, >>>>> because it's confusing. Oh, and you have to prove that it works >>>>> (you have never shown this, either). >>>>> >>>>> Start with a simple proof: Take the set of naturals, N = {0,1,2,3,...} >>>>> and remove one element to get set S = {1,2,3,...}. Now show that >>>>> the "T-size" of N is exactly one less than the T-size of S. In other >>>>> words, find a way to show that every counting of S versus every >>>>> counting of N always leaves one element of N (0) left over. >>> Tony Orlow wrote: >>>> Use IFR. >>> A.k.a. a bijection. You see where this is going? >>> >> Yes, bijection with measure. >> >>>> N maps to S using f(n)=n+1. The inverse of that function is >>>> g(x)=x-1. >>> Which proves that every n in N has an x in S. Where is that >>> leftover element that was removed from N? If N has more >>> elements than S, shouldn't N have a member that can't be >>> mapped to any member of S? >>> >> One can map all sorts of sets. N contains every element of S - map those >> first, to themselves. Now you have one left over. >> >>>> So, over the range of 0 to N, |S|=|N|-1. >>> Funny how you don't define what |X| is. You're using standard >>> symbols but obviously with a different meaning, since "|X|" means >>> "cardinality of X" when X is a set. Your IFR bijection proves that >>> |S| = |N|. >>> >> |X| means size of, like the absolute value of a real. >> >>>> Map N to the Evens E using f(n)=2n. The inverse function is >>>> g(x)=x/2, so over the range of N, >>>> the evens have |N|/2 elements. Isn't that intuitively satisfying? And >>>> gee, it works for finite sets accurately too!! >>> Same thing as above, it proves that every n has an x. Where are >>> the leftover unmapped elements of N that make S a smaller proper >>> subset of N? >>> >> All the odds. N contains all elements of E, plus the elements of O. IFR >> works to the level of accuracy of detecting a change of 0ne element out >> of an uncountable number. > > But in standard mathematics, it is trivial that any ordering of sets in > which every proper subset of a set is 'smaller' than its superset, is no > more that a partial ordering, and can never be a total ordering, so that > there must be sets which cannot be compared. > > So that TO's requirements are self-contradictory. You dingbat. I never said that the subset relation was the only method of ordering by any means. But, whatever method of ordering by size one uses should AT LEAST not violate that general principle, which the standard approach does unabashedly. IFR preserves this concept intact, and works for both finite and infinite sets mapped to the hypernaturals. :)
From: Tony Orlow on 15 Sep 2006 16:08 Virgil wrote: > In article <45098084(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Dik T. Winter wrote: >>> In article <4506d1ae(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> >>> writes: >>> > David R Tribble wrote: >>> ... >>> > > Start with a simple proof: Take the set of naturals, N = {0,1,2,3,...} >>> > > and remove one element to get set S = {1,2,3,...}. Now show that >>> > > the "T-size" of N is exactly one less than the T-size of S. In other >>> > > words, find a way to show that every counting of S versus every >>> > > counting of N always leaves one element of N (0) left over. >>> > >>> > Use IFR. N maps to S using f(n)=n+1. The inverse of that function is >>> > g(x)=x-1. So, over the range of 0 to N, |S|=|N|-1. Map N to the Evens E >>> > using f(n)=2n. The inverse function is g(x)=x/2, so over the range of N, >>> > the evens have |N|/2 elements. Isn't that intuitively satisfying? And >>> > gee, it works for finite sets accurately too!! >>> >>> How many elements has the set of primes? >> There is no well-known function that maps n to the nth prime, so IFR >> doesn't apply. Do you have an inverse function that specifies the nth >> prime for all neN? Didn't think so. > > Then there are sets whose size TO cannot measure, which makes his > "measure" less useful that cardinality, at least cardinality in ZFC and > NBG. Haha!! Good one. "There are the same number of primes as there are naturals, a proper superset." Good answer. It's almost as bad as, "There are an infinite number of rationals between any two naturals, but there are the same number of each." What a "theory"! It's mathematical creationism. Of course you snipped the actual answer, so as to make it look like I didn't answer, just because IFR doesn't directly apply.
From: Randy Poe on 15 Sep 2006 16:14 Tony Orlow wrote: > Haha!! Good one. "There are the same number of primes as there are > naturals, a proper superset." Good answer. Obviously you don't believe that. Here's an equivalent statement: If n is a natural, there is an n-th prime. Do you think that's false? Do you think that there is a natural n such that there is an (n-1)th prime, but no n-th prime? - Randy
From: Tony Orlow on 15 Sep 2006 16:19 Han de Bruijn wrote: > mueckenh(a)rz.fh-augsburg.de wrote: > >> David R Tribble schrieb: >> >>> Tony Orlow wrote: >>> >>>> Wolfgang's and my position is that N is unbounded but finite, >>> >>> "Unbounded but finite" is a contradiction, meaning "not finite but >>> finite". I'm sure you and Wolfgang think this double-think makes >>> sense, but the rest of us don't. >> >> Your position only reflects the miseducation in mathematics during the >> last decades. >> >> Actual or finished infinity is a contradicton. Surpassed infinity is a >> contradiction. >> >> Unbounded but finite is mathematical reality. Think of the set of all >> natural numbers which have been realized by writing down these numbers. >> Think of the set of known prime numbers. Think of the set of written >> novels. Think of the set of postings. >> >> These sets are unbounded because they can be extended without end. >> Nevertheless they are always finite. > > Sorry for jumping in so late. But VM is quite right, of course. We have > encoutered utterly absurd consequences of thinking otherwise, like the > mainstream "theorem" that the probability of a natural being a multiple > of 3 doesn't exist. While the obvious truth is that it is equal to 1/3 . > > This topic has been discussed at length in a thread called "Calculus XOR > Probability". Let Google be your friend, eventually. > > Han de Bruijn > Hi Han - How are you? Welcome to the thread. Calculus XOR Pobability was a great thread. I'm going to have to include a chapter in my book about it. It's quite true that probability over the naturals or any infinite set implies an infinitesimal probability for at least of of the possible alternatives. Personally, I rather like the notion of infinitesimals and specific infinities, though Wolfgang does not. But, I spport him in his assessment of the relationship between lement count and value in the naturals. They are equal. Have a nice day! :) Tony
From: Tony Orlow on 15 Sep 2006 16:28
Han de Bruijn wrote: > mueckenh(a)rz.fh-augsburg.de wrote: > >> Virgil schrieb: >> >>> Counting can be done by making tally marks or moving pebbles, for >>> example, entirely without numbers, though we have become so >>> sophisticated that we may have trouble realizing it. >>> >>> It is whether the tally marks or collected pebbles biject with the >>> objects counted which is the issue. >>> >>> And no number need ever be mentioned or used. >> >> Your tally marks and moving pebbles are numbers. Get more sophisticated >> in order to see it. > > Precisely! Mathematicians get confused by the idea of a "bijection", > which is an Equivalence Relation, which in turn is a "generalization" > of "common equality" (yes: the one in a = b). But the funny thing is > that EQUALITY HAS NEVER BEEN DEFINED. So there is actually nothing to > "generalize". Equivalence relations are a "generalization" of nothing. > > But, fortunately, reality is more simple than this. Every equality is > an equivalence relation. And every equivalence relation is an equality. > So the bijection between tally marks or collected pebbles with counted > objects means, indeed, that tally marks and moving pebbles ARE numbers. > > Han de Bruijn > I agree with that last statement, but would disagree that equality is not definable. It depends on difference, most basically, and where none is detected, two things can be said to be equal. This may be tricky with things like infinite sets, where there is no bound, so a generalization is made from the finite to the infinite. I agree that simple bijection, while sufficient for finite sets, falls short when it comes to infinite sets, and some notion of measure needs to be introduced for formulaically compare them. What do you say to that? Tony |