The purpose of my question is trying to understand whether, in some cases, we can achieve greater certainty of reasoning (say when dealing with statements about natural numbers, integers or rational numbers) by using countable subsets of reals instead of using reals as a whole. I will try to explain my question in some detail, because otherwise it might seem completely unmotivated (and perhaps not making much sense).


For concreteness, consider a certain countable subset of real numbers $R$, which we might call the set of $r$-numbers (for the lack of a better word). As one specific case, call $A$ the collection of all "arithmetical functions" (functions computable using some $0^{(n)}$ oracle for some finite ordinal $n$). Some functions within $A$ are accepted as members of the set $R$ (the collection of $r$-numbers). Let's just briefly decide the format when a given function $f:\mathbb{N} \rightarrow \mathbb{N}$ is accepted as an $r$-number:


$\\f(0) \le 1$ (interpreted as sign)


$f(a) \le 9$ when $a \ge 2$ (interpreted as decimal expansion)


Now for $r$-numbers, we define the operations of equality, comparison, addition, subtraction, multiplication and division using appropriate (infinitary) modifications of highschool algorithm. We can also state the "psuedo-completeness" property:
"*Every non-empty "arithmetical collection" of $r$-numbers, that is bounded above, has a least upper bound which is also an $r$-number.*" Here a non-empty "arithmetical collection" being defined by a suitable "arithmetical function" of the form $f:\mathbb{N^2} \rightarrow \mathbb{N}$ (each row of the function being interpreted as an $r$-number). 




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Now it possibly happens often enough that when reasoning about a statement involving naturals,integers or rationals that mathematicians "switch" to some form of reasoning involving continuous objects. After after some reasoning steps, they possibly switch back to the desired result involving naturals, integers or rationals.


Suppose that during some argument (involving $\mathbb{N}$) 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 $\mathbb{R}$ can be "replaced" by a switch to a suitable "countable subset of $\mathbb{R}$" without having to change the argument entirely/substantially. My question is not limited to specific set $R$ I described (it was just meant as an example) .... any bigger countable subset of reals can be used. 


P.S. In case the question is received well-enough, I am afraid I am not qualified enough to judge the answers. I can't think of any such examples (possibly due to lack of knowledge or perhaps that finding such examples is little harder). 



Reasoning Using Countable Subsets of Real Numbers

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> 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.
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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.


