Full Idea
If one is interested in analysis then infinite sets are indispensable since even the notion of a real number cannot be developed by means of finite sets alone.
Gist of Idea
Infinite sets are essential for giving an account of the real numbers
Source
Penelope Maddy (Believing the Axioms I [1988], §1.5)
Book Reference
-: 'Journal of Symbolic Logic' [-], p.486
A Reaction
[Maddy is citing Fraenkel, Bar-Hillel and Levy] So Cantor's great breakthrough (Idea 13021) actually follows from the earlier acceptance of the real numbers, so that's where the departure from common sense started.
Related Ideas
Idea 13021 The Axiom of Infinity states Cantor's breakthrough that launched modern mathematics [Maddy]
Idea 15931 The iterative conception needs the Axiom of Infinity, to show how far we can iterate [Lavine]