Combining Texts
Ideas for
'On the Question of Absolute Undecidability', 'Letters to Antoine Arnauld' and 'The Evolution of Logic'
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12 ideas
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
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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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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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
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 / 4. Using Numbers / a. Units
12920
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There is no multiplicity without true units [Leibniz]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
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 / i. Cardinal infinity
17890
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There are at least eleven types of large cardinal, of increasing logical strength [Koellner]
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6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
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 / 4. Axioms for Number / d. Peano arithmetic
17887
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PA is consistent as far as we can accept, and we expand axioms to overcome limitations [Koellner]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
17891
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Arithmetical undecidability is always settled at the next stage up [Koellner]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
13471
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Mathematics makes existence claims, but philosophers usually say those are never analytic [Hart,WD]
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