Reasoning Using Countable Subsets of Real Numbers

See my paper Analysis in $J_2$, where I explain how core mathematics, particularly abstract analysis, can be developed within a concrete countable set $J_2$ (the second set in Jensen's constructible hierarchy).

Suppose that during some argument (involving ℕ) one switches to real numbers and then back to discrete domain (before completing the argument). My question is, can one give examples where a switch to ℝ can be "replaced" by a switch to a suitable "countable subset of ℝ" without having to change the argument entirely/substantially.

A large portion of reverse mathematics consists of doing precisely this (see Simpson's "Subsystems of Second Order Arithmetic"): taking statements which appear to involve the reals (or other equivalent sets, like the power set of the natural numbers) and showing that the same results follow when using only certain axioms about the existence of reals.

Reverse math is usually presented axiomatically, but it's common to think in terms of $\omega$-models: to relate provability in the formal theory $ACA_0$ with those statements which hold when we only use reals which are definable by an arithmetic formula, and so on.