Combining Texts
Ideas for
'works', 'Believing the Axioms I' and 'Phenomenology of Spirit'
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19 ideas
4. Formal Logic / F. Set Theory ST / 1. Set Theory
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15901
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Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Cantor, by Lavine]
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4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / c. Basic theorems of ST
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13444
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Cantor's Theorem: for any set x, its power set P(x) has more members than x [Cantor, by Hart,WD]
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18098
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Cantor proved that all sets have more subsets than they have members [Cantor, by Bostock]
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4. Formal Logic / F. Set Theory ST / 3. Types of Set / c. Unit (Singleton) Sets
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15505
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If a set is 'a many thought of as one', beginners should protest against singleton sets [Cantor, by Lewis]
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4. Formal Logic / F. Set Theory ST / 3. Types of Set / d. Infinite Sets
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10701
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Cantor showed that supposed contradictions in infinity were just a lack of clarity [Cantor, by Potter]
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10865
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The continuum is the powerset of the integers, which moves up a level [Cantor, by Clegg]
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
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13011
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New axioms are being sought, to determine the size of the continuum [Maddy]
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / b. Axiom of Extensionality I
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13013
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The Axiom of Extensionality seems to be analytic [Maddy]
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13014
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Extensional sets are clearer, simpler, unique and expressive [Maddy]
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / d. Axiom of Unions III
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13016
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The Axiom of Union dates from 1899, and seems fairly obvious [Cantor, by Maddy]
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V
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13021
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The Axiom of Infinity states Cantor's breakthrough that launched modern mathematics [Maddy]
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13022
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Infinite sets are essential for giving an account of the real numbers [Maddy]
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / g. Axiom of Powers VI
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13023
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The Power Set Axiom is needed for, and supported by, accounts of the continuum [Maddy]
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
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13024
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Efforts to prove the Axiom of Choice have failed [Maddy]
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13025
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Modern views say the Choice set exists, even if it can't be constructed [Maddy]
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13026
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A large array of theorems depend on the Axiom of Choice [Maddy]
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4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / b. Combinatorial sets
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14199
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Cantor's sets were just collections, but Dedekind's were containers [Cantor, by Oliver/Smiley]
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4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
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13019
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The Iterative Conception says everything appears at a stage, derived from the preceding appearances [Maddy]
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4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
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13018
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Limitation of Size is a vague intuition that over-large sets may generate paradoxes [Maddy]
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