Ideas of A.George / D.J.Velleman, by Theme

[American, fl. 2002, Two professors at Amherst College.]

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2. Reason / D. Definition / 7. Contextual Definition
Contextual definitions replace a complete sentence containing the expression
2. Reason / D. Definition / 8. Impredicative Definition
Impredicative definitions quantify over the thing being defined
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
The 'power set' of A is all the subsets of A
The 'ordered pair' <a, b>, for two sets a and b, is the set {{a, b},{a}}
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
Grouping by property is common in mathematics, usually using equivalence
'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
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
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
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
The Axiom of Reducibility made impredicative definitions possible
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / a. Sets as existing
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
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
Asserting Excluded Middle is a hallmark of realism about the natural world
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
A 'model' is a meaning-assignment which makes all the axioms true
5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
Differences between isomorphic structures seem unimportant
5. Theory of Logic / K. Features of Logics / 2. Consistency
Consistency is a purely syntactic property, unlike the semantic property of soundness
A 'consistent' theory cannot contain both a sentence and its negation
5. Theory of Logic / K. Features of Logics / 3. Soundness
Soundness is a semantic property, unlike the purely syntactic property of consistency
5. Theory of Logic / K. Features of Logics / 4. Completeness
A 'complete' theory contains either any sentence or its negation
6. Mathematics / A. Nature of Mathematics / 3. Numbers / b. Types of number
Rational numbers give answers to division problems with integers
The integers are answers to subtraction problems involving natural numbers
6. Mathematics / A. Nature of Mathematics / 3. Numbers / g. Real numbers
Real numbers provide answers to square root problems
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / d. Actual infinite
The classical mathematician believes the real numbers form an actual set
6. Mathematics / A. Nature of Mathematics / 7. Application of Mathematics
Logicists say mathematics is applicable because it is totally general
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / e. Peano arithmetic 2nd-order
Second-order induction is stronger as it covers all concepts, not just first-order definable ones
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / g. Incompleteness of Arithmetic
The Incompleteness proofs use arithmetic to talk about formal arithmetic
6. Mathematics / B. Foundations for Mathematics / 4. Definitions of Number / c. Fregean numbers
A successor is the union of a set with its singleton
6. Mathematics / B. Foundations for Mathematics / 4. Definitions of Number / d. Hume's Principle
Frege's Theorem shows the Peano Postulates can be derived from Hume's Principle
6. Mathematics / B. Foundations for Mathematics / 5. Mathematics as Set Theory / a. Mathematics is set theory
Set theory can prove the Peano Postulates
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
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
If mathematics is not about particulars, observing particulars must be irrelevant
6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
In the unramified theory of types, the types are objects, then sets of objects, sets of sets etc.
Type theory has only finitely many items at each level, which is a problem for mathematics
The theory of types seems to rule out harmless sets as well as paradoxical ones.
Type theory prohibits (oddly) a set containing an individual and a set of individuals
6. Mathematics / C. Sources of Mathematics / 8. Finitism
Much infinite mathematics can still be justified finitely
Bounded quantification is originally finitary, as conjunctions and disjunctions
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
The intuitionists are the idealists of mathematics
Gödel's First Theorem suggests there are truths which are independent of proof
18. Thought / D. Concepts / 1. Concepts / a. Concepts
Corresponding to every concept there is a class (some of them sets)