On Dec 22, 7:04 pm, Virgil <vir...@ligriv.com> wrote: > In article > <2263ace1-dfa7-466d-8341-c50692402...@v7g2000yqv.googlegroups.com>, > William Hughes <wpihug...@gmail.com> wrote: > > > Note, that > > subcountable > > does not mean countable. > > I am not at all sure of what you mean by subcountable.
A set X is subcountable if we can associate a different natural number with every element x of X, call it f(x) In classical mathematics subcountable implies countable because f(X) must be a subset of the natural numbers. However, if we take a contructivist viewpoint, then we do not know that f(X) is a subset (it may not be contructable). So in constructive mathematics the fact that X is subcountable, does not mean we can find a bijection between X and some subset of the naturals, so X might not be countable. E.g. in constructive mathematics the (constructive) reals are subcountable but not countable.
So the fact that a set is uncountable need not mean it is "bigger" than the natural numbers.