Combining Texts
Ideas for
'Actualism and Possible Worlds', 'Infinity: Quest to Think the Unthinkable' and 'Grundlagen der Arithmetik (Foundations)'
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17 ideas
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
10859
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A set is 'well-ordered' if every subset has a first element [Clegg]
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4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
9157
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The null set is only defensible if it is the extension of an empty concept [Frege, by Burge]
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9835
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It is because a concept can be empty that there is such a thing as the empty class [Frege, by Dummett]
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4. Formal Logic / F. Set Theory ST / 3. Types of Set / d. Infinite Sets
10857
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Set theory made a closer study of infinity possible [Clegg]
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10864
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Any set can always generate a larger set - its powerset, of subsets [Clegg]
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4. Formal Logic / F. Set Theory ST / 3. Types of Set / e. Equivalence classes
9854
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We can introduce new objects, as equivalence classes of objects already known [Frege, by Dummett]
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9883
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Frege introduced the standard device, of defining logical objects with equivalence classes [Frege, by Dummett]
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / b. Axiom of Extensionality I
10872
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Extensionality: Two sets are equal if and only if they have the same elements [Clegg]
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / c. Axiom of Pairing II
10875
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Pairing: For any two sets there exists a set to which they both belong [Clegg]
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / d. Axiom of Unions III
10876
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Unions: There is a set of all the elements which belong to at least one set in a collection [Clegg]
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V
18104
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Frege, unlike Russell, has infinite individuals because numbers are individuals [Frege, by Bostock]
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10878
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Infinity: There exists a set of the empty set and the successor of each element [Clegg]
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / g. Axiom of Powers VI
10877
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Powers: All the subsets of a given set form their own new powerset [Clegg]
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
10879
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Choice: For every set a mechanism will choose one member of any non-empty subset [Clegg]
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / k. Axiom of Existence
10871
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Axiom of Existence: there exists at least one set [Clegg]
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / l. Axiom of Specification
10874
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Specification: a condition applied to a set will always produce a new set [Clegg]
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4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / c. Logical sets
9834
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A class is, for Frege, the extension of a concept [Frege, by Dummett]
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