Combining Texts
Ideas for
'Unconscious Cerebral Initiative', 'Introduction to Mathematical Logic' and 'Knowledge and the Philosophy of Number'
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10 ideas
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
17758
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Ordinals are transitive sets of transitive sets; or transitive sets totally ordered by inclusion [Walicki]
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17755
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Ordinals are the empty set, union with the singleton, and any arbitrary union of ordinals [Walicki]
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17756
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The union of finite ordinals is the first 'limit ordinal'; 2ω is the second... [Walicki]
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17760
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Two infinite ordinals can represent a single infinite cardinal [Walicki]
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17757
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Members of ordinals are ordinals, and also subsets of ordinals [Walicki]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / h. Ordinal infinity
23626
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Transfinite ordinals are needed in proof theory, and for recursive functions and computability [Hossack]
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6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
17762
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In non-Euclidean geometry, all Euclidean theorems are valid that avoid the fifth postulate [Walicki]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
17754
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Inductive proof depends on the choice of the ordering [Walicki]
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6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / b. Mathematics is not set theory
23621
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Numbers are properties, not sets (because numbers are magnitudes) [Hossack]
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6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
23622
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We can only mentally construct potential infinities, but maths needs actual infinities [Hossack]
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