In article <firstname.lastname@example.org>, WM <email@example.com> wrote:
> On 4 Jan., 22:38, Virgil <vir...@ligriv.com> wrote: > > > According to standard mathematics, a set is "countable" if and only if > > there is surjection from N to that set. > > If and only if? Do you agree that a subset of a countable set is > countable?
Since a surjection from N to a superset of S trivially implies a surjection to S, it also implies that any subset of a countable set is also countable. at least outside of WMythology. What the rules are for goes on inside or WMythology is only accessible to WM himself. > > The set of all finite definitions is countable. > The set of all finitely defined reals cannot be put in bijection with | > N, but it is a subset of all finite definitions, isn't it?
There is nothing in any definition of a complete ordered field of reals that requires each of its members to be finitely definable.