Model-Theoretic Connections

Satisfaction, Truth and Elementary Equivalence

Tarski's definition of satisfaction is a recursion on formula structure. Everything model-theoretic rests on getting it exactly right, including the quantifier clauses that look circular and are not.

Engineering · Mathematics5 min readKV-MATH-0248

Learning objectives

State Tarski's recursive definition of satisfaction.
Distinguish satisfaction of a formula from truth of a sentence.
Define a theory, its models, and Th(K).
Define elementary equivalence and give non-isomorphic equivalent structures.
Explain why first-order logic cannot pin down infinite structures up to isomorphism.
Relate Th(K) to Id(K) from the equational setting.

01Assignments and satisfaction

Satisfaction is a relation between a structure, a formula and an assignment of elements to the free variables, defined by recursion on the formula.

ProcedureTarski's recursion

in: structure A, formula φ, assignment a⃗ → out: satisfaction or not

let A be a structure and a⃗ an assignment to the variables
atomic: A ⊨ p ≈ q [a⃗] iff p^A(a⃗) = q^A(a⃗)
A ⊨ r(p⃗) [a⃗] iff ⟨p₁^A(a⃗),…⟩ ∈ r^A
negation: A ⊨ ¬φ [a⃗] iff not A ⊨ φ [a⃗]
conjunction: A ⊨ (φ & ψ)[a⃗] iff A ⊨ φ[a⃗] and A ⊨ ψ[a⃗]
universal: A ⊨ ∀x φ [a⃗] iff A ⊨ φ [a⃗ with x ↦ b] for EVERY b ∈ A
existential: A ⊨ ∃x φ [a⃗] iff A ⊨ φ [a⃗ with x ↦ b] for SOME b ∈ A
a sentence has no free variables, so the assignment is irrelevant:
A ⊨ σ is well defined without reference to a⃗

The recursion is on the structure of the formula, not on the structure A, which is why it terminates: every formula is finite. Caveat: the quantifier clauses quantify over A in the metatheory — this is a definition, not a circularity.

CautionThe quantifier clauses are not circular

Defining the meaning of ∀ by saying 'for every b ∈ A' looks like using universal quantification to define universal quantification. It is not: the metatheoretic quantifier ranges over a set in the ambient set theory, while the object-language ∀ is a symbol being interpreted. Tarski's achievement was making this separation precise.

02Theories and models

A theory is a set of sentences. A model of a theory is a structure satisfying every sentence in it. Two derived operators mirror M and Id from the equational setting.

The first-order Galois connection
Operator	Definition	Equational analogue
Mod(Σ)	all structures satisfying every sentence in Σ	M(Σ)
Th(K)	all sentences true in every member of K	Id(K)
Closed classes	elementary classes	varieties
Closed theories	deductively closed theories	equational theories

The Galois connection is formally identical. What differs is the characterisation of the closed sets: varieties are HSP-closed classes, whereas elementary classes have no comparably clean algebraic description. They are characterised by closure under ultraproducts and elementary equivalence, which is a genuine theorem but a much less usable one.

03Elementary equivalence

Two structures are elementarily equivalent when they satisfy exactly the same sentences.

A ≡ B ⟺ Th({A}) = Th({B})

Isomorphic structures are elementarily equivalent, by induction on formulas. The converse fails badly for infinite structures.

Isomorphism

Structural identity

A bijection commuting with all operations and relations. Implies elementary equivalence.

Elementary equivalence

Indistinguishable by sentences

Weaker. Two structures can satisfy the same first-order sentences while being of different cardinality.

Finite case

The two coincide

For finite structures in a finite language, elementary equivalence implies isomorphism — the structure can be described up to isomorphism by a single sentence.

04Why isomorphism cannot be captured

The failure is not an accident but a consequence of compactness and Löwenheim–Skolem.

Compactness produces large models
Any theory with arbitrarily large finite models has an infinite model, and any theory with an infinite model has models of every infinite cardinality above the language size.
So cardinality is invisible
No first-order theory with an infinite model can pin down the cardinality of its models. Elementary equivalence therefore cannot imply isomorphism.
Standard example
The reals as an ordered field and the real algebraic numbers as an ordered field are elementarily equivalent — both are real closed fields — and are not isomorphic, having different cardinalities.
Another
The standard model of arithmetic is elementarily equivalent to non-standard models containing infinite elements. Compactness constructs them directly.

Key resultFirst-order logic sees theories, not structures

A first-order theory determines its models only up to elementary equivalence, which for infinite structures is much weaker than isomorphism. Any property distinguishing two elementarily equivalent structures — well-ordering, cardinality, archimedeanness, torsion-freeness in the infinite case — is not first-order expressible.

05Complete theories

A theory is complete when for every sentence, either it or its negation is a consequence. Equivalently, all models are elementarily equivalent.

Completeness in practice
Theory	Complete?	Note
Th({A}) for any structure A	Yes	By construction — every sentence is decided.
Dense linear orders without endpoints	Yes	ℵ₀-categorical, hence complete by Vaught's test.
Atomless Boolean algebras	Yes	ℵ₀-categorical.
Algebraically closed fields of fixed characteristic	Yes	Categorical in uncountable cardinalities.
Group theory	No	Abelian and non-abelian groups both model it.
Peano arithmetic	No	Gödel's first incompleteness theorem.

Vaught's test is the standard tool: a theory with no finite models that is categorical in some infinite cardinality is complete. The atomless Boolean algebra case from the Boolean stream is the instance most relevant here.

06Relation to the equational setting

Id(K)

Equations only

A fully invariant congruence on the term algebra. Determines V(K) exactly, by Birkhoff. Finitely many closure conditions characterise it.

Th(K)

All sentences

A deductively closed theory. Determines the elementary class of K, which is generally much smaller than V(K) and has no HSP-style characterisation.

The two operators sit at different strengths: Id(K) ⊆ Th(K) always, since identities are sentences. Knowing Th(K) determines Id(K) but not conversely. This is why model-theoretic hypotheses are stronger than equational ones, and why results such as the finite basis theorems — which use model theory to prove equational conclusions — count as a genuine transfer.

Frequently asked

Does elementary equivalence preserve finiteness?

Yes for a fixed finite size — a structure of size n can be described by a sentence asserting exactly n elements exist. But 'finite' as a property is not first-order: no theory has exactly the finite structures as models, by compactness.

Is Th({A}) always complete?

Yes, trivially — for each sentence, either A satisfies it or A satisfies its negation, and the theory contains whichever holds. Completeness of a theory presented by axioms is the substantive question; completeness of the full theory of a structure is automatic.

Why does the finite case behave so differently?

Because a finite structure in a finite language can be described up to isomorphism by a single sentence specifying the number of elements and the full operation and relation tables. That sentence is satisfiable only by isomorphic copies, so elementary equivalence collapses to isomorphism. No such sentence exists for infinite structures.

Sources and further reading

S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, Millennium Edition (a corrected re-typesetting of Springer GTM 78, 1981).
G. Grätzer, Universal Algebra, 2nd edition, Springer.
R. McKenzie, G. McNulty and W. Taylor, Algebras, Lattices, Varieties, Volume I.

Original KEVOS® explanatory article. Written from the topic map of the cited works; no text is reproduced from them.