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Full Idea
The Axiom of Infinity is a simple statement of Cantor's great breakthrough. His bold hypothesis that a collection of elements that had lurked in the background of mathematics could be infinite launched modern mathematics.
Gist of Idea
The Axiom of Infinity states Cantor's breakthrough that launched modern mathematics
Source
Penelope Maddy (Believing the Axioms I [1988], §1.5)
Book Ref
-: 'Journal of Symbolic Logic' [-], p.486
A Reaction
It also embodies one of those many points where mathematics seems to depart from common sense - but then most subjects depart from common sense when they get more sophisticated. Look what happened to art.
Related Idea
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] |