Ideas of Robert S. Wolf, by Theme

[American, fl. 2005, Teaches mathematics at California Polytechnic State University.]

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4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / b. Terminology of PL
A 'tautology' must include connectives
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / c. Derivation rules of PL
Deduction Theorem: T∪{P}|-Q, then T|-(P→Q), which justifies Conditional Proof
4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / d. Universal quantifier ∀
Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x)
Universal Specification: ∀xP(x) implies P(t). True for all? Then true for an instance
4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / e. Existential quantifier ∃
Existential Generalization (or 'proof by example'): if we can say P(t), then we can say something is P
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / e. Axiom of the Empty Set IV
Empty Set: ∃x∀y ¬(y∈x). The unique empty set exists
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / n. Axiom of Comprehension
Comprehension Axiom: if a collection is clearly specified, it is a set
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
In first-order logic syntactic and semantic consequence (|- and |=) nicely coincide
First-order logic is weakly complete (valid sentences are provable); we can't prove every sentence or its negation
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
Model theory uses sets to show that mathematical deduction fits mathematical truth
Model theory reveals the structures of mathematics
Model theory 'structures' have a 'universe', some 'relations', some 'functions', and some 'constants'
First-order model theory rests on completeness, compactness, and the Löwenheim-Skolem-Tarski theorem
5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
An 'isomorphism' is a bijection that preserves all structural components
5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
The LST Theorem is a serious limitation of first-order logic
5. Theory of Logic / K. Features of Logics / 4. Completeness
If a theory is complete, only a more powerful language can strengthen it
5. Theory of Logic / K. Features of Logics / 10. Monotonicity
Most deductive logic (unlike ordinary reasoning) is 'monotonic' - we don't retract after new givens
6. Mathematics / A. Nature of Mathematics / 3. Numbers / e. Ordinal numbers
An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / g. Continuum Hypothesis
Continuum Hypothesis: there are no sets between N and P(N)
6. Mathematics / B. Foundations for Mathematics / 4. Definitions of Number / d. Hume's Principle
Frege's cardinals (equivalences of one-one correspondences) is not permissible in ZFC
6. Mathematics / B. Foundations for Mathematics / 5. Mathematics as Set Theory / a. Mathematics is set theory
Modern mathematics has unified all of its objects within set theory