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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 Ref
-: '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]
13011 | New axioms are being sought, to determine the size of the continuum [Maddy] |
13013 | The Axiom of Extensionality seems to be analytic [Maddy] |
13014 | Extensional sets are clearer, simpler, unique and expressive [Maddy] |
13019 | The Iterative Conception says everything appears at a stage, derived from the preceding appearances [Maddy] |
13018 | Limitation of Size is a vague intuition that over-large sets may generate paradoxes [Maddy] |
13021 | The Axiom of Infinity states Cantor's breakthrough that launched modern mathematics [Maddy] |
13022 | Infinite sets are essential for giving an account of the real numbers [Maddy] |
13023 | The Power Set Axiom is needed for, and supported by, accounts of the continuum [Maddy] |
13024 | Efforts to prove the Axiom of Choice have failed [Maddy] |
13025 | Modern views say the Choice set exists, even if it can't be constructed [Maddy] |
13026 | A large array of theorems depend on the Axiom of Choice [Maddy] |