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