I'm not asking precisely for the exact number of methods, but rather for the various mathods available to show a set is infinite. I'm interested in a sort of methodological review.
In order to show a set S is countably infinite I think one can :
show that S has ( at least) one proper subset T such that there is ( at least) one bijection from T to S.
show that there is a bijection from S to the set of natural numbers
In order to show a set S is uncountably infinite, I think one can :
show that there is (at least) one bijection from S to the set of real numbers
show that there is a function from N ( set of natural numbers) to S such that this function is injective but not surjective ( which would leave some elements of S " alone" and prove that S has more elements than N has) [ WRONG SUGGESTION ACCORDING TO COMMENTS , SEE BELOW]
I cannot go further.
Are there other available methods?