"Tim Little" <tim@little-possums.net> wrote in message news:slrni1m25q.jrj.tim@soprano.little-possums.net... > On 2010-06-18, Virgil <Virgil@home.esc> wrote: >> Since you object to there being any bijection from N to any superset >> of S, you must equally be rejecting any surjection from N to S and >> rejecting any injection from S to N, since from any such bijection >> such a surjection is easily derived. > > Yes, Peter is very confused. >
No.
> >> So in what sense do you claim that the the set S of computable >> numbers is countable? >> >> It is certainly not in any sense that I am aware of. > > Heh, these two sentences would be great to quote out of context.
Hey, they are great even in context. The computable numbers definitely are countable, as google will verify. (Or can easily be proved by associating computable Reals with TMs that produce them, and there are only countable TMs).
> In > the context of Peter's premise the latter is true, but in ordinary > context of mathematics it is very obviously false. >
I made no premise.
I simply provided an example of a set which is:
(a) Countable, but (b) Cannot be listed explicitly.
Cantor's diagonal proof shows that Reals cannot be explicitly listed. This is *not* equivalent to the statement they are uncountable, as the example of the set of all computable numbers shows. These cannot be formed into a list, but are nevertheless still countable.
