> The article doesn’t really tell us what is gained by rejecting infinity.

Decidability. The issues around undecidability all involve the lack of an upper bound. In a finite deterministic space, everything is decidable, although some things may be too costly computationally to decide.

There are several ways to go for decidability. The brute force way is computer arithmetic - there is no number larger than 2^64-1. That's how we get things done on computers, but proofs about numbers with finite upper bounds need lots of special cases. Mathematicians hate that.

I used to work on this sort of thing, using Boyer-Moore theory. That's a lot like the Peano axioms. There is (ZERO), and (ADD1 (ZERO)), and (ADD1 (ADD1 (ZERO))), etc. Everything is constructive and has an unambiguous representation in a LISP-like form. You can have recursive functions. But they must be proven to terminate, by having a nonnegative value which decreases on each recursive call. There is a distinction between "infinite" and "arbitrarily large". You can talk about arbitrarily large numbers, but you cannot get to 1/2 + 1/4 + 1/8 ... = 1. You can have integers and rational numbers of arbitrary size, but not reals.

Set theory was interesting. Rather than axiomatic set theory, I was using lists as sets, with the constraints that no value could be duplicated and the list must be ordered. Equality is strict - two things are equal only if the elements are all equal, compared element by element. It's possible to prove the usual axioms of set theory via this route. The ordered criterion requires proving things about ordered list insertion to get there. It's ugly and needs machine proofs.

I was doing this back in the early 1980s, when machine proofs were frowned upon. Mathematicians were still upset about the four-color theorem proof. It's all case analysis, with thousands of cases. That's more acceptable today.

Looked at in this light, infinity is a labor-saving device to eliminate special cases, at a potential cost in soundness.