Axiom of infinity4.2. The Completeness Theorem

Finite cardinalities Countability of ℕ×ℕ

Countability of finite sequences of integers

Rebuilding recursion

Existence of countable term algebras4.3. Infinity and the axiom of choice

The Completeness Theorem

Skolem's Paradox

The undecidability of the axiom of choice4.4. Non-standard models of Arithmetic

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

Non-standard models of elementary arithmetic4.5. How mathematical theories develop

Non-standard models of Presburger Arithmetic

Non-standard models of full first-order arithmetic

Proofs4.6. The Incompleteness Theorem

Definitions

Constructions

Self-quotation theoremMoved to 5 :

The Truth Undefinability Theorem

The Incompleteness Theorem

Löb's theorem

Second-order logic

Weak second-order theories, separation axiomsSecond-order arithmetic

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 as a possible foundation for mathematics

The need of the powerset

The undecidability of the axiom of choice