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8920 | Equivalence relations are reflexive, symmetric and transitive, and classify similar objects [Lipschutz] |
Full Idea: A relation R on a non-empty set S is an equivalence relation if it is reflexive (for each member a, aRa), symmetric (if aRb, then bRa), and transitive (aRb and bRc, so aRc). It tries to classify objects that are in some way 'alike'. | |
From: Seymour Lipschutz (Set Theory and related topics (2nd ed) [1998], 3.9) | |
A reaction: So this is an attempt to formalise the common sense notion of seeing that two things have something in common. Presumably a 'way' of being alike is going to be a property or a part |
17610 | The Axiom of Choice paradoxically allows decomposing a sphere into two identical spheres [Maddy] |
Full Idea: One feature of the Axiom of Choice that troubled many mathematicians was the so-called Banach-Tarski paradox: using the Axiom, a sphere can be decomposed into finitely many parts and those parts reassembled into two spheres the same size as the original. | |
From: Penelope Maddy (Defending the Axioms [2011], 1.3) | |
A reaction: (The key is that the parts are non-measurable). To an outsider it is puzzling that the Axiom has been universally accepted, even though it produces such a result. Someone can explain that, I'm sure. |