# 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. 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

4.7. The
Incompleteness Theorem

Self-quotation theorem

The Truth Undefinability Theorem

The Incompleteness Theorem

Löb's theorem