Ideas from 'A Tour through Mathematical Logic' by Robert S. Wolf [2005], by Theme Structure

[found in 'A Tour Through Mathematical Logic' by Wolf,Robert S. [Carus Maths Monographs 2005,0-88385-036-2]].

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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

13519

Model theory uses sets to show that mathematical deduction fits mathematical truth

13531

Model theory reveals the structures of mathematics

13532

Model theory 'structures' have a 'universe', some 'relations', some 'functions', and some 'constants'

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