more from this thinker | more from this text
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]
| 22288 | We have the idea of self, and an idea of that idea, and so on, so infinite ideas are available [Dedekind, by Potter] |
| 18104 | Frege, unlike Russell, has infinite individuals because numbers are individuals [Frege, by Bostock] |
| 14440 | We may assume that there are infinite collections, as there is no logical reason against them [Russell] |
| 14447 | Infinity says 'for any inductive cardinal, there is a class having that many terms' [Russell] |
| 13430 | Infinity: there is an infinity of distinguishable individuals [Ramsey] |
| 10051 | The axiom of infinity is not a truth of logic, and its adoption is an abandonment of logicism [Kneale,W and M] |
| 13037 | Infinity: ∃x (0 ∈ x ∧ ∀y ∈ x (S(y) ∈ x) [Kunen] |
| 17800 | The misnamed Axiom of Infinity says the natural numbers are finite in size [Mayberry] |
| 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] |
| 18191 | Axiom of Infinity: completed infinite collections can be treated mathematically [Maddy] |
| 10878 | Infinity: There exists a set of the empty set and the successor of each element [Clegg] |
| 13044 | Infinity: There is at least one limit level [Potter] |