On Jun 26, 12:35 pm, Virgil <Vir...@home.esc> wrote: > In article > <8ced201d-fce9-447f-9fa9-b972f2014...@t5g2000prd.googlegroups.com>, > > > > > > Newberry <newberr...@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.email@example.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.
I was replying to this: "It implies a listing must exist, but does not provide such a listing."
"It", in this context is the statement "all computable reals are countable."
If an antidiagonal existed it would prove that there was no such list.
> > > > > > 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.- Hide quoted text - > > - Show quoted text -