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The phrase "a finite set" is correct and usable in written English.
It is typically used in mathematical or logical contexts to refer to a set that contains a limited number of elements.
Example: "In set theory, a finite set is one that has a countable number of elements, such as {1, 2, 3}."
Alternatives: "a limited set" or "a bounded set".
Exact(59)
random variable which takes values in a finite set.
Arguing indirectly, assume F is a finite set.
Let X denote a finite set.
This constrains U to belong to a finite set.
Let be a collection of subsets of a finite set.
Let be a universe of species and S = s 1, …, s n ⊆ S a finite set.
A directed graph or digraph (V, A) consists of a finite set V of vertices and a finite set A of arcs, each consisting of an ordered pair (u, v) where u ∈ V, v ∈ V, u ≠ v.
Of a finite set of factors.
Well, he's merely showing you a finite set of swans.
containing values from a finite set of constants.
Similar(1)
The union of a countable set and a finite set is also countable.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com