If your question could mean:,if there is a proper subset s of real number, and the cardinal of s is equal to the cardinal of real number, and is countable. That is a contradiction since the cardinal of real is alepth_1 assuming continuum hypothesis hold, which is not equal to alepth_0, the cardinal of integer.

If your question is: If there is a proper subset s of real, and the cardinal of s is aleph_0, than, the set of rational number, the set of integer, the set of algebraic number are all the same which is equal to integer, and your question is a tautology.