4. Model Theory

4.1. Finiteness and countability
Axiom of infinity
Finite cardinalities Countability of ℕ×ℕ
Countability of finite sequences of integers
Rebuilding recursion
4.2. The Completeness Theorem
Existence of countable term algebras
The Completeness Theorem
Skolem's Paradox
4.3. Infinity and the axiom of choice
The undecidability of the axiom of choice
Proving the finite choice theorem
Schröder–Bernstein theorem
Axiom of dependent choice (DC)
Injections of ℕ into infinite sets
Counter-examples to the axiom of choice
AC vs measurability
4.4. Non-standard models of Arithmetic
Non-standard models of elementary arithmetic
Non-standard models of Presburger Arithmetic
Non-standard models of full first-order arithmetic
4.5. How mathematical theories develop
Proofs
Definitions
Constructions
4.6. The Incompleteness Theorem
Self-quotation theorem
The Truth Undefinability Theorem
The Incompleteness Theorem
Löb's theorem
Moved to 5 :
Second-order logic

Weak second-order theories, separation axioms
Translating second-order theories into first-order ones
The first-order expression by a schema of axioms (weakest method)
The theory with stable power type (Henkin semantics)
Set theoretical interpretation (strongest method)
Semantic completeness, logical incompleteness
Higher-order theories
Second-order arithmetic
Second-order arithmetic as a possible foundation for mathematics
The need of the powerset
The undecidability of the axiom of choice