"Virgil" <Virgil@home.esc> wrote in message news:Virgil-7885B6.13355726062010@bignews.usenetmonster.com... > 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.
Well, Cantor used the list explicitly in his construction.
