In article <firstname.lastname@example.org>, William Hughes <email@example.com> wrote:
> 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.
But the constraints of your "constructive mathematics' are not required in classical mathematics when not doing your constructive mathematics, so are not relevant in classical mathematics. --