Model Theory Basics — Mathematics Knowledge System

Overview

Model theory studies the relationship between formal first-order languages and the mathematical structures (models) that interpret them — where proof theory asks 'what can be derived syntactically,' model theory asks 'what is true in which structures.' A structure (or model) M for a language L consists of a nonempty domain (universe) together with an interpretation of each constant, function, and relation symbol of L; a sentence φ is satisfied by M, written M ⊨ φ, when φ comes out true under that interpretation. Two structures M and N are elementarily equivalent (M ≡ N) if they satisfy exactly the same first-order sentences, even if they are not isomorphic — a strictly weaker relationship than isomorphism, since (by Löwenheim–Skolem) elementarily equivalent structures can even have different cardinalities. A substructure N ⊆ M is an elementary substructure (N ≺ M) if it agrees with M not just on quantifier-free formulas but on ALL first-order formulas, including those with quantifiers ranging over the larger structure M — a much stronger condition than merely being a substructure, since a plain substructure can satisfy an existential sentence 'accidentally' using elements outside N, without N itself containing the required witness.

Intuition

Think of a structure as a physical instantiation of an abstract language: the language of group theory has symbols like · and e, and a structure supplies an actual set with an actual multiplication and identity element making those symbols meaningful. Elementary equivalence (M ≡ N) says two structures are indistinguishable by any first-order sentence — they might still be very different as sets (even different sizes, thanks to Löwenheim–Skolem) but no first-order statement can tell them apart, like two opaque boxes that give identical answers to every yes/no question you're allowed to ask. Being an elementary substructure (N ≺ M) is a much finer requirement than merely being a subset with compatible operations: it demands that whenever M can find a witness making an existential formula true, N could have found ITS OWN witness (inside N) doing the same job — so N is a faithful, self-sufficient miniature of M for every first-order question, not just a leftover piece that happens to sit inside M.

Formal Definition

Definition

Key model-theoretic relationships between structures M and N for a language L:

M \models \varphi \quad \text{(M satisfies sentence } \varphi \text{, i.e. } \varphi \text{ is true in } M\text{)}

Satisfaction

M \equiv N \;:\iff\; \forall\, \varphi\ \big(M \models \varphi \iff N \models \varphi\big)

Elementary equivalence

N \preceq M \;:\iff\; N \subseteq M \ \text{ and } \ \forall\, \varphi(\bar x),\ \forall\, \bar a \in N:\ N \models \varphi(\bar a) \iff M \models \varphi(\bar a)

Elementary substructure

M \cong N \implies M \equiv N \quad \text{(converse fails in general)}

Isomorphism implies elementary equivalence, not conversely

Worked Examples

Both ℚ‾ and ℂ are algebraically closed fields of characteristic 0. A foundational model-theory result: the theory of algebraically closed fields of a fixed characteristic is complete (any two models satisfy the same sentences) — it just isn't categorical in every cardinality.
$\text{ACF}_0 \text{ is a complete theory}$
Since ℚ‾ ⊨ ACF₀ and ℂ ⊨ ACF₀, and ACF₀ is complete, every sentence true in one is true in the other: ℚ‾ ≡ ℂ.
$\bar{\mathbb{Q}} \models \text{ACF}_0,\ \mathbb{C} \models \text{ACF}_0 \implies \bar{\mathbb{Q}} \equiv \mathbb{C}$
But |ℚ‾| = ℵ₀ (countable) while |ℂ| = 2^{ℵ₀} (uncountable), so no bijection exists between them at all — they cannot be isomorphic.
$|\bar{\mathbb{Q}}| = \aleph_0 \ne 2^{\aleph_0} = |\mathbb{C}| \implies \bar{\mathbb{Q}} \ncong \mathbb{C}$

Answer: ℚ‾ and ℂ are elementarily equivalent (same first-order truths) but not isomorphic (different cardinalities) — a clean illustration that ≡ is strictly weaker than ≅.

Practice Problems

Difficulty 5/10

If two structures M and N are isomorphic, must they be elementarily equivalent? Must the converse hold?

Difficulty 6/10

What extra condition distinguishes 'N is an elementary substructure of M' from merely 'N is a substructure of M'?

Difficulty 7/10

Explain what it means for a first-order theory T to be 'complete,' and why completeness of ACF₀ (algebraically closed fields of characteristic 0) implies any two algebraically closed fields of characteristic 0 and the same uncountable cardinality are isomorphic is a STRONGER fact (categoricity) than completeness alone provides.

Quiz

Summary

Model theory studies structures (models) that interpret a first-order language and the sentences they satisfy (M ⊨ φ).
Elementary equivalence (M ≡ N: same first-order truths) is strictly weaker than isomorphism — elementarily equivalent structures can even differ in cardinality, as ℚ‾ and ℂ show.
An elementary substructure (N ≺ M) demands agreement on all first-order formulas including quantifiers, which is much stronger than N merely being a substructure of M.

References

WebsiteWikipedia — Model theory
WebsiteWikipedia — Elementary equivalence