Combining Philosophers

Ideas for Baron,S/Miller,K, A.George / D.J.Velleman and Rayo,A/Uzquiasno,G

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13 ideas

4. Formal Logic / F. Set Theory ST / 1. Set Theory
13451 The two best understood conceptions of set are the Iterative and the Limitation of Size [Rayo/Uzquiano]
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
10098 The 'power set' of A is all the subsets of A [George/Velleman]
10101 Cartesian Product A x B: the set of all ordered pairs in which a∈A and b∈B [George/Velleman]
10099 The 'ordered pair' <a, b>, for two sets a and b, is the set {{a, b},{a}} [George/Velleman]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / e. Equivalence classes
10103 Grouping by property is common in mathematics, usually using equivalence [George/Velleman]
10104 'Equivalence' is a reflexive, symmetric and transitive relation; 'same first letter' partitions English words [George/Velleman]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
10096 Even the elements of sets in ZFC are sets, resting on the pure empty set [George/Velleman]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / b. Axiom of Extensionality I
10097 Axiom of Extensionality: for all sets x and y, if x and y have the same elements then x = y [George/Velleman]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / c. Axiom of Pairing II
10100 Axiom of Pairing: for all sets x and y, there is a set z containing just x and y [George/Velleman]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / m. Axiom of Separation
13452 Some set theories give up Separation in exchange for a universal set [Rayo/Uzquiano]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / p. Axiom of Reducibility
17900 The Axiom of Reducibility made impredicative definitions possible [George/Velleman]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / a. Sets as existing
10109 ZFC can prove that there is no set corresponding to the concept 'set' [George/Velleman]
4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
10108 As a reduction of arithmetic, set theory is not fully general, and so not logical [George/Velleman]