### All the ideas for Michael Burke, Mark Steiner and A.George / D.J.Velleman

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52 ideas

###### 2. Reason / D. Definition / 7. Contextual Definition
 9955 Contextual definitions replace a complete sentence containing the expression [George/Velleman]
###### 2. Reason / D. Definition / 8. Impredicative Definition
 10031 Impredicative definitions quantify over the thing being defined [George/Velleman]
###### 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]
 10099 The 'ordered pair' , 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
 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 / 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]
###### 5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
 10111 Asserting Excluded Middle is a hallmark of realism about the natural world [George/Velleman]
###### 5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
 10129 A 'model' is a meaning-assignment which makes all the axioms true [George/Velleman]
###### 5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
 10105 Differences between isomorphic structures seem unimportant [George/Velleman]
###### 5. Theory of Logic / K. Features of Logics / 2. Consistency
 10119 Consistency is a purely syntactic property, unlike the semantic property of soundness [George/Velleman]
 10126 A 'consistent' theory cannot contain both a sentence and its negation [George/Velleman]
###### 5. Theory of Logic / K. Features of Logics / 3. Soundness
 10120 Soundness is a semantic property, unlike the purely syntactic property of consistency [George/Velleman]
###### 5. Theory of Logic / K. Features of Logics / 4. Completeness
 10127 A 'complete' theory contains either any sentence or its negation [George/Velleman]
###### 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
 10106 Rational numbers give answers to division problems with integers [George/Velleman]
 10102 The integers are answers to subtraction problems involving natural numbers [George/Velleman]
###### 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
 10107 Real numbers provide answers to square root problems [George/Velleman]
###### 6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / g. Applying mathematics
 9946 Logicists say mathematics is applicable because it is totally general [George/Velleman]
###### 6. Mathematics / A. Nature of Mathematics / 5. The Infinite / d. Actual infinite
 10125 The classical mathematician believes the real numbers form an actual set [George/Velleman]
###### 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order
 17899 Second-order induction is stronger as it covers all concepts, not just first-order definable ones [George/Velleman]
###### 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
 10128 The Incompleteness proofs use arithmetic to talk about formal arithmetic [George/Velleman]
###### 6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers
 17902 A successor is the union of a set with its singleton [George/Velleman]
###### 6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle
 10133 Frege's Theorem shows the Peano Postulates can be derived from Hume's Principle [George/Velleman]
###### 6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
 10130 Set theory can prove the Peano Postulates [George/Velleman]
###### 6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
 10089 Talk of 'abstract entities' is more a label for the problem than a solution to it [George/Velleman]
###### 6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
 10131 If mathematics is not about particulars, observing particulars must be irrelevant [George/Velleman]
###### 6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
 10092 In the unramified theory of types, the types are objects, then sets of objects, sets of sets etc. [George/Velleman]
 10094 The theory of types seems to rule out harmless sets as well as paradoxical ones. [George/Velleman]
 10095 Type theory has only finitely many items at each level, which is a problem for mathematics [George/Velleman]
 17901 Type theory prohibits (oddly) a set containing an individual and a set of individuals [George/Velleman]
###### 6. Mathematics / C. Sources of Mathematics / 8. Finitism
 10114 Bounded quantification is originally finitary, as conjunctions and disjunctions [George/Velleman]
 10134 Much infinite mathematics can still be justified finitely [George/Velleman]
###### 6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
 10123 The intuitionists are the idealists of mathematics [George/Velleman]
 10124 Gödel's First Theorem suggests there are truths which are independent of proof [George/Velleman]
###### 9. Objects / A. Existence of Objects / 5. Individuation / e. Individuation by kind
 16235 Persistence conditions cannot contradict, so there must be a 'dominant sortal' [Burke,M, by Hawley]
 14753 The 'dominant' of two coinciding sortals is the one that entails the widest range of properties [Burke,M, by Sider]
###### 9. Objects / B. Unity of Objects / 1. Unifying an Object / b. Unifying aggregates
 16072 'The rock' either refers to an object, or to a collection of parts, or to some stuff [Burke,M, by Wasserman]
###### 9. Objects / B. Unity of Objects / 3. Unity Problems / b. Cat and its tail
 14751 Tib goes out of existence when the tail is lost, because Tib was never the 'cat' [Burke,M, by Sider]
###### 9. Objects / B. Unity of Objects / 3. Unity Problems / c. Statue and clay
 13278 Maybe the clay becomes a different lump when it becomes a statue [Burke,M, by Koslicki]
 16234 Burke says when two object coincide, one of them is destroyed in the process [Burke,M, by Hawley]
 16071 Sculpting a lump of clay destroys one object, and replaces it with another one [Burke,M, by Wasserman]
###### 9. Objects / B. Unity of Objects / 3. Unity Problems / d. Coincident objects
 14750 Two entities can coincide as one, but only one of them (the dominant sortal) fixes persistence conditions [Burke,M, by Sider]
###### 9. Objects / D. Essence of Objects / 3. Individual Essences
 13230 Particular essence is often captured by generality [Steiner,M]
###### 14. Science / D. Explanation / 2. Types of Explanation / e. Lawlike explanations
 13229 Maybe an instance of a generalisation is more explanatory than the particular case [Steiner,M]
###### 14. Science / D. Explanation / 2. Types of Explanation / m. Explanation by proof
 13231 Explanatory proofs rest on 'characterizing properties' of entities or structure [Steiner,M]
###### 18. Thought / D. Concepts / 1. Concepts / a. Nature of concepts
 10110 Corresponding to every concept there is a class (some of them sets) [George/Velleman]