Combining Philosophers

Ideas for José Ortega y Gassett, Penelope Maddy and A.George / D.J.Velleman

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30 ideas

4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
18194 'Forcing' can produce new models of ZFC from old models [Maddy]
10098 The 'power set' of A is all the subsets of A [George/Velleman]
10099 The 'ordered pair' <a, b>, for two sets a and b, is the set {{a, b},{a}} [George/Velleman]
10101 Cartesian Product A x B: the set of all ordered pairs in which a∈A and b∈B [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
18195 A Large Cardinal Axiom would assert ever-increasing stages in the hierarchy [Maddy]
13011 New axioms are being sought, to determine the size of the continuum [Maddy]
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
13013 The Axiom of Extensionality seems to be analytic [Maddy]
13014 Extensional sets are clearer, simpler, unique and expressive [Maddy]
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 / f. Axiom of Infinity V
13021 The Axiom of Infinity states Cantor's breakthrough that launched modern mathematics [Maddy]
13022 Infinite sets are essential for giving an account of the real numbers [Maddy]
18191 Axiom of Infinity: completed infinite collections can be treated mathematically [Maddy]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / g. Axiom of Powers VI
13023 The Power Set Axiom is needed for, and supported by, accounts of the continuum [Maddy]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / i. Axiom of Foundation VIII
18193 The Axiom of Foundation says every set exists at a level in the set hierarchy [Maddy]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
13024 Efforts to prove the Axiom of Choice have failed [Maddy]
13025 Modern views say the Choice set exists, even if it can't be constructed [Maddy]
13026 A large array of theorems depend on the Axiom of Choice [Maddy]
17610 The Axiom of Choice paradoxically allows decomposing a sphere into two identical spheres [Maddy]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / p. Axiom of Reducibility
18169 Axiom of Reducibility: propositional functions are extensionally predicative [Maddy]
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 / 5. Conceptions of Set / e. Iterative sets
13019 The Iterative Conception says everything appears at a stage, derived from the preceding appearances [Maddy]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
13018 Limitation of Size is a vague intuition that over-large sets may generate paradoxes [Maddy]
4. Formal Logic / F. Set Theory ST / 7. Natural Sets
17824 The master science is physical objects divided into sets [Maddy]
8755 Maddy replaces pure sets with just objects and perceived sets of objects [Maddy, by Shapiro]
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]