It means you construct the set of all reals in one go. The set of reals is definable and constructable. Individual real numbers are not.Numbers like $\pi$ are computable, so I believe we could assert the sets $A$ and $B$ given there is a formula that the axiom schema of separation could utilize to do so. My question centers around using Dedekind cuts to â€˜constructâ€™ real numbers when in reality weâ€™re just assuming the existence of the reals and then embedding them in $\mathcal{P}(\mathbb{Q})$. Itâ€™s the uncountably many reals that we have no way of computing, or even defining if restricting our definitions to that which could be created using only finite sentences from a finite formal language, that leads me to question the purpose of Dedekind cuts and what we mean when we generally assert that we use them to construct the reals.

It is easier to construct the set of reals, than to construct specific real numbers.

As a (very close) analogy, assume that natural numbers $\mathbb{N}$. The power set $\mathcal{P}(\mathbb{N})$ kind of exists by axiom. In either case, it is a definable set. That doesn't mean we can actually exhibit all elements of $\mathcal{P}(\mathbb{N})$ easily. We can't, since many of these elements are undefinable.

Much of the same considerations hold for the real numbers.