Why can no one in sci.math understand my simple point?
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From: Virgil on 27 Jun 2010 16:43 In article <4c274337$0$12922$afc38c87(a)news.optusnet.com.au>, "Peter Webb" <webbfamily(a)DIESPAMDIEoptusnet.com.au> wrote: > "Virgil" <Virgil(a)home.esc> wrote in message > news:Virgil-4CE082.01003126062010(a)bignews.usenetmonster.com... > > In article <4c259cd4$0$1029$afc38c87(a)news.optusnet.com.au>, > > "Peter Webb" <webbfamily(a)DIESPAMDIEoptusnet.com.au> wrote: > > > >> "Virgil" <Virgil(a)home.esc> wrote in message > >> news:Virgil-71EA75.23485925062010(a)bignews.usenetmonster.com... > >> > In article > >> > <b3413a4e-567b-4dfb-8037-21f14b826ede(a)g1g2000prg.googlegroups.com>, > >> > Newberry <newberryxy(a)gmail.com> wrote: > >> > > >> >> > > No. (3) is not true, as it is based on a false premise (that the > >> >> > > computable > >> >> > > Reals can be listed). > >> > > >> > How is countability any different from listability for an infinite set? > >> > > >> > Does not countability of an infinite set S imply a surjections from N > >> > to S? And then does not such a surjection imply a listing? > >> > >> It implies a listing must exist, but does not provide such a listing. > >> > >> The computable Reals are countable, but you cannot form them into a list > >> of > >> all computable Reals (and nothing else) where each item on the list can > >> be > >> computed. > >> > >> In order to list a set, it has to be recursively enumerable. Being > >> countable > >> is not sufficient. > > > > Both countability and listability appear to be the case if and only if > > a listing exists, but neither requires specifying that listing. Is that > > not so? > > No. > > If we take "listable" to mean we can (ummm) make a list of exactly those > elements and no other, then this is not correct. A set can be countable but > not listable. > > AFAIK, "listable" is not a formally defined mathematical term. The formal > term in mathematics which is closest to the intuitive idea of being able to > explicitly list the members of a set is that it is "recursively enumerable". > Not "countable". Being countable is necessary but not sufficient. Then how can you prove a set to be countable? As far as I am aware there is only one way, by creating, or at least proving the existence of, a list of its members. I do not consent to your definition of "listable". > > This is the problem I have with the standard presentation of Cantor's > proof - it starts with a "list", and then proves there is an item missing > from the list. Proving that no list can be prepared proves only that the > Reals are not recursively enumerable, not the stronger condition they are > uncountable. > > I have no problem with the very similar proof of Cantor's that the power set > has larger cardinality than the set itself, which suffices to prove the > Reals are uncountable. It doesn't talk about "lists".
From: Virgil on 27 Jun 2010 16:46 In article <deb0b948-84ed-4177-93f1-fccc3968608a(a)y11g2000yqm.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 27 Jun., 05:13, Virgil <Vir...(a)home.esc> wrote: > > > > In order to do so, I posed the question: Does the list consisting of > > > A0, A1, A2, A3, ... contain its antidiagonal or not? > > > > No list contains its antidiagonal, nevertheless, any list together with > > its antidiagonal can be listed, and �in a large variety of ways. > > But it is impossible to list the countable set that I described. Except that several people have describes precisely how to do so! > > That is the point! And it is obvious that it cannot be circumvented. Not to anyone who has described precisely how to do so.
From: Virgil on 27 Jun 2010 17:00 In article <8a924020-947b-43cf-8ca9-d011c05b1b43(a)s9g2000yqd.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 27 Jun., 13:01, "Mike Terry" > <news.dead.person.sto...(a)darjeeling.plus.com> wrote: > > > > In order to do so, I posed the question: Does the list consisting of > > > A0, A1, A2, A3, ... contain its antidiagonal or not? > > > > > You said no. Therefore your assertion "which CAN be listed" is plainly > > > wrong. > > > > The fact that a list does not contain its antidiagonal does not mean the > > list cannot be listed!. > > But that is not the question! Please read carefully. The question is > whether there is a countable set that cannot be listed. This set is > given by the original list and all its possible antidiagonals. That presumes that the set of all its possible "antidiagonals", i.e., nonmembers, is countable, which begs the question. If one has a particular rule for creating an antidiagonal of a listing and then inserting it in that listing, that all the elements of the original list and all constructed elements can be listed in onew list quite easily, and several such methods have been shown here, which WM has carefully ignored. > > > > My final word on this: > > > > The set you have constructed: �{A0, A1, A2,...} is: > > > > a) �countable > > b) �can be listed, e.g. (A0, A1, A2,...) > > c) �of course the list (A0, A1, A2,...) has an antidiagonal Aw, > > � � which is not in {A0, A1, A2,...). �(This is obviously > > � � irrelevent to (a) and (b)). > > And your (a) and (b) ist obviously irrelevant for the present > discussion. > > > > So you are wrong. > > No. You simply cannot understand the meaning of a process which cannot > end (hence the results of which cannot be put in a complete list) > unless there is a list containing its antidiagonal. The set of naturals does not end, but can be "listed", so that "not ending" is not a valid objection. Any set which can be recursively defined can obviously be listed, just like the naturals, and the set that WM objects to being listable IS recursively defined. > But as I have > given the description in clear words, I don't want to repeat it. > Probably it would not support your understanding either. What is "clear" to WM is often not clear to anyone else, and what is clear to everyone else is often not clear to WM. WM's "mathematics" and the mathematics of the rest of the world have limited overlap. > > You may google under "supertask" to better inform you. According to Zeno, and WM, moving from point A to point B would be one of those impossible supertasks. But those of us not aware of its impossibility in their worlds do it anyway.
From: Owen Jacobson on 27 Jun 2010 22:30 On 2010-06-27 08:24:15 -0400, Peter Webb said: > AFAIK, "listable" is not a formally defined mathematical term. This could be because every time someone presents you with a clear, concise definition you don't happen to like, you stop replying to them[0]. The definitions you've been presented with numerous times *just in this thread* are all variations on "a list of elements of some set S is a surjective function L from N (the natural numbers) to S." You're free to involve recursive enumerability in your definition if you like, but be prepared for misunderstandings. -o [0] See for example <2010062122134453246-angrybaldguy(a)gmailcom>.
From: Peter Webb on 27 Jun 2010 22:59
"Owen Jacobson" <angrybaldguy(a)gmail.com> wrote in message news:2010062722302771524-angrybaldguy(a)gmailcom... > On 2010-06-27 08:24:15 -0400, Peter Webb said: > >> AFAIK, "listable" is not a formally defined mathematical term. > > This could be because every time someone presents you with a clear, > concise definition you don't happen to like, you stop replying to them[0]. > As I have said before, AFAIK there is no accepted definition of "listable". I provided a definition. Others want to define it differently. > The definitions you've been presented with numerous times *just in this > thread* are all variations on "a list of elements of some set S is a > surjective function L from N (the natural numbers) to S." The definitions I have seen are all equivalent to "countable". These are not good definitions for three reasons. Firstly, we already have a perfectly good word which means "there exists a surjection from N to the set" which everybody knows, and it is "countable". Secondly, the definition I proposed for "listable" is far more in accord with common usage. Just because you can enumerate all items sold in a supermarket does not neccesarily mean you can form a shopping list; a shopping list is not just a list every item sold in supermarkets, it is a specific list of exactly those items you need. Thirdly, I already provided a definition of "listable" which is equivalent to being recursively enumerable. If people try and redefine terms to mean something different, then there are going to be misunderstandings. > You're free to involve recursive enumerability in your definition if you > like, but be prepared for misunderstandings. You are addressing your criticism to the wrong person. "Listable" is not (AFAIK) a technical term with a well defined meaning. I provided a meaning. It is other people who want to redefine the term to simply mean "countable". They should use the term "countable" if they mean "countable". My definition of "listable" is not the same as countable. They are different concepts. For example, computable Reals are countable but not listable. Re-defining a term to mean something different is clearly going to cause problems. > > -o > > [0] See for example <2010062122134453246-angrybaldguy(a)gmailcom>. > |