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Ideas from 'Philosophies of Mathematics' by A.George / D.J.Velleman [2002], by Theme Structure

[found in 'Philosophies of Mathematics' by George,A/Velleman D.J. [Blackwell 2002,0-631-19544-0]].

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2. Reason / D. Definition / 7. Contextual Definition
9955 Contextual definitions replace a complete sentence containing the expression
2. Reason / D. Definition / 8. Impredicative Definition
10031 Impredicative definitions quantify over the thing being defined
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
10099 The 'ordered pair' <a, b>, for two sets a and b, is the set {{a, b},{a}}
10101 Cartesian Product A x B: the set of all ordered pairs in which a∈A and b∈B
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
10104 'Equivalence' is a reflexive, symmetric and transitive relation; 'same first letter' partitions English words
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
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
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
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / p. Axiom of Reducibility
17900 The Axiom of Reducibility made impredicative definitions possible
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'
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
5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
10111 Asserting Excluded Middle is a hallmark of realism about the natural world
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
5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
10105 Differences between isomorphic structures seem unimportant
5. Theory of Logic / K. Features of Logics / 2. Consistency
10119 Consistency is a purely syntactic property, unlike the semantic property of soundness
10126 A 'consistent' theory cannot contain both a sentence and its negation
5. Theory of Logic / K. Features of Logics / 3. Soundness
10120 Soundness is a semantic property, unlike the purely syntactic property of consistency
5. Theory of Logic / K. Features of Logics / 4. Completeness
10127 A 'complete' theory contains either any sentence or its negation
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
10102 The integers are answers to subtraction problems involving natural numbers
10106 Rational numbers give answers to division problems with integers
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
10107 Real numbers provide answers to square root problems
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / g. Applying mathematics
9946 Logicists say mathematics is applicable because it is totally general
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / d. Actual infinite
10125 The classical mathematician believes the real numbers form an actual set
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
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
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
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
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
10130 Set theory can prove the Peano Postulates
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
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
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.
17901 Type theory prohibits (oddly) a set containing an individual and a set of individuals
10094 The theory of types seems to rule out harmless sets as well as paradoxical ones.
10095 Type theory has only finitely many items at each level, which is a problem for mathematics
6. Mathematics / C. Sources of Mathematics / 8. Finitism
10134 Much infinite mathematics can still be justified finitely
10114 Bounded quantification is originally finitary, as conjunctions and disjunctions
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
10123 The intuitionists are the idealists of mathematics
10124 Gödel's First Theorem suggests there are truths which are independent of proof
18. Thought / D. Concepts / 1. Concepts / a. Nature of concepts
10110 Corresponding to every concept there is a class (some of them sets)