Comment by nrds
The claims in this article, such as those suggesting recursion has something to do with the infinite, are all relative to the set-theoretic foundation. This is not essential.
In contrast, in the type theories behind proof assistants like Coq, Lean, and Agda, recursion is intimately tied to _finite_ structures. Instead of the vague "intersection of all sets such that" which we see in the article here, recursion is a well-defined computational process, and defined in a rather obvious way once you're familiar with the background.
The "intersection of all sets such that" is not vague at all. It's perfectly formally defined in ZF* set theories. But it's impredicative. One of the guiding ideas behind type theories is to minimise impredicative constructions as much as possible. After all, impredicative definitions are circular ... Of course there is no free lunch and the power of impredicative constructions needs to be supplied in other ways in type theories ...