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### Ideas of Robert S. Wolf, by Text

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

 2005 A Tour through Mathematical Logic
 Pref p.-8 13519 Model theory uses sets to show that mathematical deduction fits mathematical truth
 Pref p.-8 13518 Modern mathematics has unified all of its objects within set theory
 1.2 p.11 13520 A 'tautology' must include connectives
 1.3 p.20 13522 Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x)
 1.3 p.20 13521 Universal Specification: ∀xP(x) implies P(t). True for all? Then true for an instance
 1.3 p.20 13523 Existential Generalization (or 'proof by example'): if we can say P(t), then we can say something is P
 1.3 p.31 13524 Deduction Theorem: T∪{P}|-Q, then T|-(P→Q), which justifies Conditional Proof
 1.7 p.54 13525 Most deductive logic (unlike ordinary reasoning) is 'monotonic' - we don't retract after new givens
 2.2 p.62 13526 Comprehension Axiom: if a collection is clearly specified, it is a set
 2.3 p.70 13529 Empty Set: ∃x∀y ¬(y∈x). The unique empty set exists
 2.4 p.77 13530 An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive
 5.1 p.165 13531 Model theory reveals the structures of mathematics
 5.2 p.167 13532 Model theory 'structures' have a 'universe', some 'relations', some 'functions', and some 'constants'
 5.3 p.172 13533 First-order model theory rests on completeness, compactness, and the Löwenheim-Skolem-Tarski theorem
 5.3 p.172 13534 In first-order logic syntactic and semantic consequence (|- and |=) nicely coincide
 5.3 p.174 13535 First-order logic is weakly complete (valid sentences are provable); we can't prove every sentence or its negation
 5.4 p.181 13537 An 'isomorphism' is a bijection that preserves all structural components
 5.5 p.191 13538 If a theory is complete, only a more powerful language can strengthen it
 5.7 p.224 13539 The LST Theorem is a serious limitation of first-order logic