### 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
 13520 A 'tautology' must include connectives
###### 4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / c. Derivation rules of PL
 13524 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 ∀
 13522 Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x)
 13521 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 ∃
 13523 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
 13529 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
 13526 Comprehension Axiom: if a collection is clearly specified, it is a set
###### 5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
 13534 In first-order logic syntactic and semantic consequence (|- and |=) nicely coincide
 13535 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
 13531 Model theory reveals the structures of mathematics
 13532 Model theory 'structures' have a 'universe', some 'relations', some 'functions', and some 'constants'
 13519 Model theory uses sets to show that mathematical deduction fits mathematical truth
 13533 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
 13537 An 'isomorphism' is a bijection that preserves all structural components
###### 5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
 13539 The LST Theorem is a serious limitation of first-order logic
###### 5. Theory of Logic / K. Features of Logics / 4. Completeness
 13538 If a theory is complete, only a more powerful language can strengthen it
###### 5. Theory of Logic / K. Features of Logics / 10. Monotonicity
 13525 Most deductive logic (unlike ordinary reasoning) is 'monotonic' - we don't retract after new givens
###### 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
 13530 An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive
###### 6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
 13518 Modern mathematics has unified all of its objects within set theory