Combining Texts
Ideas for
'Defending the Axioms', 'The Evolution of Logic' and 'Saundaranandakavya'
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11 ideas
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
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13463
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There are at least as many infinite cardinals as transfinite ordinals (because they will map) [Hart,WD]
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13492
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Von Neumann's ordinals generalise into the transfinite better, because Zermelo's ω is a singleton [Hart,WD]
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13459
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The less-than relation < well-orders, and partially orders, and totally orders the ordinal numbers [Hart,WD]
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13491
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The axiom of infinity with separation gives a least limit ordinal ω [Hart,WD]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
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13446
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19th century arithmetization of analysis isolated the real numbers from geometry [Hart,WD]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
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13509
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We can establish truths about infinite numbers by means of induction [Hart,WD]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
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17615
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Every infinite set of reals is either countable or of the same size as the full set of reals [Maddy]
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6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
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13474
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Euclid has a unique parallel, spherical geometry has none, and saddle geometry has several [Hart,WD]
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6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
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17618
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Set-theory tracks the contours of mathematical depth and fruitfulness [Maddy]
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6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
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17614
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The connection of arithmetic to perception has been idealised away in modern infinitary mathematics [Maddy]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
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13471
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Mathematics makes existence claims, but philosophers usually say those are never analytic [Hart,WD]
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