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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 Model theory uses sets to show that mathematical deduction fits mathematical truth
Pref p.-8 Modern mathematics has unified all of its objects within set theory
1.2 p.11 A 'tautology' must include connectives
1.3 p.20 Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x)
1.3 p.20 Universal Specification: ∀xP(x) implies P(t). True for all? Then true for an instance
1.3 p.20 Existential Generalization (or 'proof by example'): if we can say P(t), then we can say something is P
1.3 p.31 Deduction Theorem: T∪{P}|-Q, then T|-(P→Q), which justifies Conditional Proof
1.7 p.54 Most deductive logic (unlike ordinary reasoning) is 'monotonic' - we don't retract after new givens
2.2 p.62 Comprehension Axiom: if a collection is clearly specified, it is a set
2.2 p.64 Frege's cardinals (equivalences of one-one correspondences) is not permissible in ZFC
2.2 p.67 Continuum Hypothesis: there are no sets between N and P(N)
2.3 p.70 Empty Set: ∃x∀y ¬(y∈x). The unique empty set exists
2.4 p.77 An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive
5.1 p.165 Model theory reveals the structures of mathematics
5.2 p.167 Model theory 'structures' have a 'universe', some 'relations', some 'functions', and some 'constants'
5.3 p.172 First-order model theory rests on completeness, compactness, and the Löwenheim-Skolem-Tarski theorem
5.3 p.172 In first-order logic syntactic and semantic consequence (|- and |=) nicely coincide
5.3 p.174 First-order logic is weakly complete (valid sentences are provable); we can't prove every sentence or its negation
5.4 p.181 An 'isomorphism' is a bijection that preserves all structural components
5.5 p.191 If a theory is complete, only a more powerful language can strengthen it
5.7 p.224 The LST Theorem is a serious limitation of first-order logic