In article <firstname.lastname@example.org>, WM <email@example.com> wrote:
> On 8 Jan., 14:44, mstem...@walkabout.empros.com (Michael Stemper) > > > However, ANY real can be distinguished from ANY other by SOME finite > > initial segment of its decimal, or other base, representation. > > The problem is not distinguishing given reals, but how reals can > uniquely by *given*. That requires a finite definition. But I am > afraid that you don't even understand what that means. > > Regards, WM
A set is determined by any rule for distinguishing whether a test object is or is not a member. If more than any finite number of things satisfy such a rule, then the set will have more than any finite number of members.
If one can show that there is a surjection from N to such a set then that set is defined to be a countable set.
If one can show that there cannot be any surjection from N to such a set then that set is defined to be an uncountable set.