Why can no one in sci.math understand my simple point?
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From: Peter Webb on 21 Jun 2010 02:33 "Owen Jacobson" <angrybaldguy(a)gmail.com> wrote in message news:2010062100324493978-angrybaldguy(a)gmailcom... > On 2010-06-19 22:31:56 -0400, Peter Webb said: > >> Cantor's original proof requires the list to be provided in advance > > No, it does not. It's an existence proof showing that: > > IF a list of reals L(x) (where L : N -> R) exists, THEN a real A_L exists > such that for every natural number n, L(n) ≠ A_L. > > Or, equivalently, showing that: > > For every function L from N to R, there exists a real r not in the image > of L. > > Being a constructive proof, it's usually *presented* as if it were an > algorithm or a procedure, but it's only for ease of comprehension. Umm, I am talking about Cantor's proof as it is generally presented, and in that case it is an explicit algorithm. > Unfortunately, that seems to have backfired on you. > So you are saying that Cantor's proof in its usual form is incorrect, as it pretends to include an algorithmic process for computing the anti-diagonal? > -o > > Incidentally, "countable" and "listable" mean the same thing in > conventional set theories: an infinite set S is countable if and only if > there exists an surjective function f from N to S. Since a "list" of > elements of S is most easily formalized as a surjective function from N to > S, denying that S is listable is equivalent to denying that it is > countable. > The existence of a surjective function from N to S proves it is countable. The ability to prepare a list of exactly the elements proves the list is recursively enumerable. If you cannot form the list, you have proved that it is not recursively enumerable. This is a somewhat weaker condition that it being uncountable.
From: Virgil on 21 Jun 2010 02:48 In article <4c1f04e8$0$14086$afc38c87(a)news.optusnet.com.au>, "Peter Webb" <webbfamily(a)DIESPAMDIEoptusnet.com.au> wrote: > "Tim Little" <tim(a)little-possums.net> wrote in message > news:slrni1tmi9.jrj.tim(a)soprano.little-possums.net... > > On 2010-06-21, Peter Webb <webbfamily(a)DIESPAMDIEoptusnet.com.au> wrote: > >> "Tim Little" <tim(a)little-possums.net> wrote in message > >>> On 2010-06-21, Peter Webb <webbfamily(a)DIESPAMDIEoptusnet.com.au> wrote: > >>>> No, I only need a finite algorithm which will calculate it to n > >>>> decimal places with finite input. > >>> > >>> You have previously stated that you can do this with "about three > >>> lines of Java code". > >>> > >>> Here's your Java method signature: int antidiagonalDigit(BigInteger n). > >>> > >>> Here are the first twenty digits of the first twenty members of my > >>> list: > >>> > >>> 93525532854532512838 > >>> 32127313472944276266 > >>> 70595916184994935423 > >>> 20733652719572401688 > >>> 30472031767118774150 > >>> 47190325821263633948 > >>> 74236814853458351851 > >>> 58521903865615844550 > >>> 91701104659863267390 > >>> 39510921669610680229 > >>> 19656091025330568974 > >>> 49591084533660072011 > >>> 81683520683673830124 > >>> 93720622611168810054 > >>> 50284245443806050152 > >>> 11702670934156383526 > >>> 58534679962278312978 > >>> 68478827933494896765 > >>> 83579375050000862329 > >>> 50241229144880453593 > >>> > >>> Now let's see your Java code. > >> > >> Here is my p-code. > > > > As I suspected, your claim that you could actually do it was pure > > puffery, and you probably know as little about programming as you do > > about mathematics. > > > > > >> Take the nth digit of the nth item. > > > > The n'th item of *what*? > > The list. Of computable Reals. That is supposed to include all Computable > Reals. > > > The list is not a parameter, it is not a > > global variable, it is completely out of scope. You fail already. > > > > > > Actually, Cantor's proof starts with a purported list of all Reals, and then > shows that at least one Real is missing. Technically, it isn't Cantor's proof at all, as the only real Cantor diagonal proof is not about reals but about binary sequences. > > It assumes that a list exists, then shows a Real is missing. If you didn't > supply the list in advance, he wouldn't be able to construct a Real missing > from the list. In fact, if you don't say what is on the list, it you can't > prove soimething is missing from it. There is no Real which is missing from > every list, but every list misses at least one Real. > > You do believe Cantor's original proof that the Reals cannot be listed, > don't you? > > > > - Tim
From: Tim Little on 21 Jun 2010 03:35 On 2010-06-21, Virgil <Virgil(a)home.esc> wrote: > Technically, it isn't Cantor's proof at all, as the only real Cantor > diagonal proof is not about reals but about binary sequences. Indeed. In fact it would be a lot easier if instead of reals, we talked about sets of natural numbers, and instead of computable reals, we talked about recursively enumerable sets of natural numbers. The diagonal proof for the uncountability of P(N) is very much simpler, and so there are fewer distractions from the core idea. It is also much closer to Cantor's original proof. This might help to prevent some of the extraneous confusions Peter has about computable reals, and would help focus on his *central* confusions. - Tim
From: WM on 21 Jun 2010 05:49 On 20 Jun., 21:55, Virgil <Vir...(a)home.esc> wrote: > > The reals in a certain Cantor-list are countable. And if you form the > > anti-diagonal of that list and add it (for instance at first position) > > to the list, this new list is also countable. Again construct the > > anti-diagonal and add it to the list. Continue. The situation remains > > the same for all anti-diagonals you might want to construct. Therefore > > all reals constructed in that way are countable. Nevertheless it is > > impossible to put all of them in one list, because there would be > > another resulting anti-diagonal. > > > Conclusion: It is impossible to obtain a bijection of all these reals > > with N although all "these" reals are countable with no doubt. > > Equal cardinality of two sets requires, by definition, a bijection > between them. > > The naturals are, by definition of COUNTABLE cardinality so that any set > of equally countable cardinality must have a bijection with the > naturals. But the set of all diagonals constructed according to the scheme given above is countable but cannot be listed. > > Cantor has shown that the set of all reals cannot biject with the > naturals, ergo, the set of all reals is NOT of countable cardinality. The same proof shows that some countable sets are of not countable cardinality. But you prefer not to think about that? Regards, WM
From: WM on 21 Jun 2010 05:51
On 20 Jun., 22:18, Virgil <Vir...(a)home.esc> wrote: > In article > <f92c169d-ee85-40c2-aa82-c8bdf06f7...(a)j4g2000yqh.googlegroups.com>, > > WM <mueck...(a)rz.fh-augsburg.de> wrote: > > On 20 Jun., 17:51, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > > > > Cantor was this utterly insane freak who chose not to accept Newberry's > > > word for it, and instead *prove* that there was no list of all real > > > numbers. Obviously, his proof is nonsense, because, after all, Newberry > > > said there was no list. > > > His proof is nonsense because it proves that a countable set, namely > > the set of all reals of a Cantor-list and all diagonal numbers to be > > constructed from it by a given rule an to be added to this list, > > cannot be listed, hence, that this indisputably countable set is > > uncountable. > > That is a deliberate misrepresentation of the so called "diagonal proof". But this proof can be applied to this countable set and shows its uncountability. Regards, WM |