Please note the use of the word “phenomenal” in the following.
“Final cause” in the phenomenal sense does not invoke the idea of telos [purpose, goal, end, limit] on cosmic scale. It only invokes the idea of finality of a process in nature. Such as a biological function — you know, those little things like metabolism, respiration, cell repair, etc., etc.
Given that, and then reading your Aristotle quote carefully … To wit:
“Further, the final cause is an end, and that sort of end which is not for the sake of something else, but for whose sake everything else is; so that if there is to be a last term of this sort, the process will not be infinite; but if there is no such term, there will be no final cause, but those who maintain the infinite series eliminate the Good without knowing it (yet no one would try to do anything if he were not going to come to a limit); nor would there be reason in the world; the reasonable man, at least, always acts for a purpose, and this is a limit; for the end is a limit. — Aristotle Metaphysics Book II, Part 2
… I’d say that Aristotle was looking at practical reality in explaining the phenomenal aspect in “infinity,” while saying that looking at things “infinite” is not reasonable.
Given that, I went to Ayn Rand for her thoughts on causality and found this.
”Only a process of final causation – i.e., the process of choosing a goal, then taking the steps to achieve it – can you give logical continuity, coherence and meaning to a man’s actions.
Thank you for your patience … it is one of Aristotle’s “virtues.”
The bold italics were added by you. But with this term, Aristotle indicates a logical transition. He is moving from the more general, to the more specific human case. He is qualifying his opening statement, about what is necessary in logic and reason — which he considers to be universal — and the way things are perceived by a human observer not fully conversant with natural truth.
With the bolds — possibly you should have supplied them earlier, at the line, "so that if there is to be a last term of this sort" — he is questioning man, asking him to reason about the significance of this "last term," and its import for the intelligibility of the natural world, of the universe, to the human mind.
Aristotle was right — it seems unquestionably the case in my mind — to assert the absolute necessity of a first uncaused case, of a Prime Mover, in order for a universe to occur in a way that is intelligible to the human mind.
Make of that what you will. The problem of first cause is not preeminently, nor exclusively a scientific problem. Still, science itself utterly depends on it for its own foundation in reason and logic.
As far as infinity is concerned, I can't speak for Aristotle. But I do know that great minds now agree on the proposition that the mathematical term infinity is not constructible in physics. Nor is it anything that a human mind can reach, let alone comprehend.