Sets with same Cardinality, but no Explicit Bijection? Are there any good examples of sets where we know that they have the same cardinality, but have not found any explicit bijection between them?
The real numbers have the same cardinality as the elements of a basis of the real numbers regarded as a vector space over the rationals. The existence of such a basis needs some appropriate form of the axiom of choice, so the basis cannot be given explicitly, let alone any bijection.
The set of transcendental numbers, set of irrational numbers and the set of real numbers are all uncountable and have the same cardinality. It is not clear (at least to me) how to construct a bijections among them.

The set of rational numbers  and the set of algebraic numbers are both infinite countable sets. Here also it is not clear how to construct a bijection.