Combining Philosophers

Ideas for Michael Burke, Jaegwon Kim and William D. Hart

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8 ideas

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
There are at least as many infinite cardinals as transfinite ordinals (because they will map) [Hart,WD]
The axiom of infinity with separation gives a least limit ordinal ω [Hart,WD]
Von Neumann's ordinals generalise into the transfinite better, because Zermelo's ω is a singleton [Hart,WD]
The less-than relation < well-orders, and partially orders, and totally orders the ordinal numbers [Hart,WD]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
19th century arithmetization of analysis isolated the real numbers from geometry [Hart,WD]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
We can establish truths about infinite numbers by means of induction [Hart,WD]
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
Euclid has a unique parallel, spherical geometry has none, and saddle geometry has several [Hart,WD]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
Mathematics makes existence claims, but philosophers usually say those are never analytic [Hart,WD]