Finitism denies the existence of infinite mathematical objects (e.g. quantification over infinite domains is not considered meaningful). Is the position that denies the existence of uncountable sets (but allows for countable, i.e. discrete, sets) defended in literature?
I searched for "discretism" without success.
There are a number of subtle variations that the denial of a natural powerset (the powerset of ℕ) can take. In Barton and Friedman[21] we find that 2ℵ0 is inflated to the size of a proper class (c.f. Matthews[21] as well as "pocket-sized set theory"), where proper classes might be taken to be in some sense "complete" infinities, but not as sets: they have elements but aren't themselves elements, e.g. for ORD (the proper class of ordinals) there is no {ORD}, etc. So there wouldn't be an {ℝ}, here.
On the other hand, one might deny the powerset of ℕ as a completed infinity altogether, too. This is more or less how it goes in predicativism, where a reference to "all" subsets of ℕ is taken to be impredicative. One can go even further and think through only the limit of predicativity "given the natural numbers":
We also note that the addition of inaccessible set axioms to a weak subsystem of CZF (with no set induction) produces a theory of strength Γ0, the ordinal singled out by Feferman and Schütte as the limit of predicativity given the natural numbers (Crosilla and Rathjen 2001; see also section 1.3).
Then even Γ0 becomes a sort of "barrier" within the domain of countable ordinals. That ordinal is, normally speaking, relatively small (or "short" might be a better description). What is this ordinal? Here's a rather perspicuous expression from the MathSE, in terms of an infinite extension of the addition-multiplication-exponentiation-tetration sequence:
So regarding "predicativity given the natural numbers," we might speak of a denial not only of uncountable sets, but even of countable ones as of the above.
Note that, without the axiom scheme of replacement, one has only countably many uncountable alephs in play and ℵω is equivalent to the universe of sets. Also, countable ordinals of length ω + ω and beyond are not provably sets in this context. One will still have uncountable sets, but with serious restrictions on the details. How much this is compatible with the intuitions behind countabilism proper depends on whose intuitions we are speaking to.
List of countabilist positions thus indicated:
This position is called "countabilism." The 2022 paper "In defense of Countabilism" by Builes and Wilson is freely available on the Web (e.g. from Wilson's website, here). It cites many recent works on the subject and is a good entry-point into the literature.
