In article <4c259cd4$0$1029$afc38c87@news.optusnet.com.au>, "Peter Webb" <webbfamily@DIESPAMDIEoptusnet.com.au> wrote:
> "Virgil" <Virgil@home.esc> wrote in message > news:Virgil-71EA75.23485925062010@bignews.usenetmonster.com... > > In article > > <b3413a4e-567b-4dfb-8037-21f14b826ede@g1g2000prg.googlegroups.com>, > > Newberry <newberryxy@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?
