The Upward Lowenheim-Skolem theory decrees that there must be a (non-standard) model of the naturals of cardinality the same as that of the standard model of the Reals.
For any combined theory of the Reals and the Naturals, a proof similar to that of Upward Lowenheim-Skolem means that there must be a model with any infinite cardinality of Naturals. But, is there a model where both the Reals and the Naturals have the same cardinality?
Equivalently, is the statement "There is a bijection between the reals and the naturals" satisfiable in some combined theory of naturals and reals where the standard Peano axoims and the axioms of the reals hold / can be encoded.
Or, is $\forall r, \exists i, \textrm{isInteger}(i) \land f(i) = r$ first-order provable in some minimal axiom scheme that can prove every thing that Peano axioms and the real axioms can prove?
If you take the real-closed field axioms RCF as your theory of reals, then it is easy to get what you want, since the computable reals satisfy RCF, and the computable reals are definitely countable. If you add an axiom Sup that every upper-bounded definable sequence of reals (over your theory) has a supremum, then the new theory RCF+Sup is no longer satisfied by the computable reals, but is satisfied by the collection of reals that can be computed by some finite Turing jump, which is still countable.