In article <8ced201d-fce9-447f-9fa9-b972f2014d8e@t5g2000prd.googlegroups.com>, Newberry <newberryxy@gmail.com> wrote:
> On Jun 25, 11:22 pm, "Peter Webb" > <webbfam...@DIESPAMDIEoptusnet.com.au> wrote: > > "Virgil" <Vir...@home.esc> wrote in message > > > > news:Virgil-71EA75.23485925062010@bignews.usenetmonster.com... > > > > > In article > > > <b3413a4e-567b-4dfb-8037-21f14b826...@g1g2000prg.googlegroups.com>, > > > Newberry <newberr...@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. > > Does such a listing have an anti-diagonal?
Every list of reals, or of infinite binary sequences, has an antidiagonal, in fact, has at least as many antidiagonals as elements. > > > 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.
In order explicitely to construct such a list, perhaps, but such lists may exist even when not explicitely constructable. And for any countable set, they do.
