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Full Idea
Many theorems depend on the Axiom of Choice, including that a countable union of sets is countable, and results in analysis, topology, abstract algebra and mathematical logic.
Gist of Idea
A large array of theorems depend on the Axiom of Choice
Source
Penelope Maddy (Believing the Axioms I [1988], §1.7)
Book Ref
-: 'Journal of Symbolic Logic' [-], p.488
A Reaction
The modern attitude seems to be to admit anything if it leads to interesting results. It makes you wonder about the modern approach of using mathematics and logic as the cutting edges of ontological thinking.
| 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] |