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 2005,0883850362]].
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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. FirstOrder Logic
13534

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

13535

Firstorder 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
13531

Model theory reveals the structures of mathematics

13532

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

13519

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

13533

Firstorder model theory rests on completeness, compactness, and the LöwenheimSkolemTarski 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. LöwenheimSkolem Theorems
13539

The LST Theorem is a serious limitation of firstorder 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. Numbers / e. Ordinal numbers
13530

An ordinal is an equivalence class of wellorderings, or a transitive set whose members are transitive

6. Mathematics / A. Nature of Mathematics / 4. The Infinite / g. Continuum Hypothesis
13528

Continuum Hypothesis: there are no sets between N and P(N)

6. Mathematics / B. Foundations for Mathematics / 4. Definitions of Number / d. Hume's Principle
13527

Frege's cardinals (equivalences of oneone correspondences) is not permissible in ZFC

6. Mathematics / B. Foundations for Mathematics / 5. Mathematics as Set Theory / a. Mathematics is set theory
13518

Modern mathematics has unified all of its objects within set theory
