in mathematics we seem to have many natural instances of countably infinite sets: The set of natural numbers ℕ; the set of integers ℤ; the set of rational numbers ℚ; the set of finite binary sequences 2<ℕ; the set of integer polynomials ℤ[x]. We seem also to have many natural instances of sets of size continuum: The set of real numbers ℝ; the set of complex numbers ℂ; Cantor space (infinite binary sequences) 2ℕ; the power set of the natural numbers P(ℕ); the space of continuous functions f: ℝ → ℝ. But we seem to have no sets provably of intermediate cardinality between ℕ and ℝ.
