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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 wellorderings, 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

Firstorder model theory rests on completeness, compactness, and the LöwenheimSkolemTarski theorem

5.3

p.172

13534

In firstorder logic syntactic and semantic consequence ( and =) nicely coincide

5.3

p.174

13535

Firstorder 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 firstorder logic
