"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.
