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 · Mathematics11 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.
Handbook application: from concept to controlled practice
Purpose. This expanded section turns the original page into a practical handbook. It preserves the supplied material and adds a repeatable way to apply, check and review Satisfaction, Truth and Elementary Equivalence. It does not replace a contract, legislation, a controlled standard, competent engineering judgement or specialist advice.
The operating aim is to turn a compact mathematical statement into a usable chain of definitions, claims, examples and checks. Read the original explanation first, then use the workflow and checks below to convert knowledge into evidence.
Treat Satisfaction, Truth and Elementary Equivalence as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—satisfaction, elementary, equivalence, theories, truth—should be made explicit before any proof or computation begins. Record the ambient set or structure, the permitted operations and the equality or equivalence relation in use. A compact theorem often changes meaning when the base field, finiteness condition, commutativity assumption or direction of an action changes.
For a proof, write the hypotheses as a checklist and mark the line at which each one is used. For a computation, state the representation of the input, the arithmetic model, the termination condition and the output invariant. For a classification problem, distinguish existence from uniqueness and distinguish an object from its representation. These separations prevent a correct local calculation from being mistaken for the general result.
A useful worked example should be small enough to inspect completely but rich enough to exercise the main mechanism. Compute the result in two ways where practical: symbolically and by substitution, structurally and numerically, or directly and through a normal form. Then include one near-miss example in which a hypothesis fails. The contrast explains why the theorem is shaped as it is and gives the reader a diagnostic pattern for later problems.
Verification is part of the mathematics. Check domains and codomains, substitute proposed solutions, test identity and zero cases, compare dimensions or cardinalities, and confirm that maps respect the required operations. In numerical work, report precision, conditioning and a residual rather than digits alone. In algorithmic work, separate mathematical correctness from implementation complexity and resource limits.
Step-by-step operating method
Fix the setting. State the objects, ambient structure, notation and assumptions before manipulating symbols.
Separate claims. Distinguish definitions, hypotheses, conclusions, equivalent conditions and consequences.
Choose a method. Select proof, construction, calculation or algorithm according to the question actually asked.
Work a small case. Use the smallest non-trivial example to expose the mechanism and test edge behaviour.
Verify independently. Substitute back, check invariants, test boundary cases or use an alternative derivation.
Worked-example protocol
Illustrative method—not a source theorem. Start with a small admissible input and list the definitions it must satisfy. Carry out each transformation on a separate line, citing the property that permits it. Preserve exact values until approximation is necessary. At the end, verify the output against the original definition and one invariant such as dimension, degree, determinant, order, norm or residual. Then alter one hypothesis and observe which step ceases to be valid. This protocol creates a reusable example without inventing a theorem-specific numerical answer.
Stage
Record
Quality check
Input
Objects, domain, notation, assumptions
Every symbol is defined
Method
Permitted operation or cited result at each step
All hypotheses hold
Output
Exact result and representation
Correct type, domain and form
Verification
Substitution, invariant or alternative derivation
Independent agreement
Boundary test
Zero, identity, degenerate or failed hypothesis
Scope is understood
Common failure modes and recovery actions
1. Watch for
Using a theorem without checking every hypothesis.
Recovery: Return to the governing definition or requirement and restate the decision in one sentence.
2. Watch for
Treating a suggestive example as a proof of the general case.
Recovery: Separate evidence from assumption, assign an owner and set a date for validation.
3. Watch for
Changing notation or conventions part-way through an argument.
Recovery: Run a small counterexample, boundary test, pilot or independent check before proceeding.
4. Watch for
Hiding a division-by-zero, convergence, finiteness or commutativity assumption.
Recovery: Record the consequence, decision and rationale, then update the controlled baseline.
5. Watch for
Reporting a computed result without a residual, substitution or structural check.
Recovery: Escalate when the issue affects safety, compliance, acceptance, material value or an agreed tolerance.
Review checklist
Can every symbol be traced to a definition or prior result?
Which hypothesis does each major step use?
Does the method cover zero, identity, degenerate and boundary cases?
Can the conclusion be checked by a second representation or calculation?
Are mandatory requirements distinguished from recommendations and illustrative values?
Are sources, assumptions, units, dates and versions recorded closely enough to reproduce the decision?
Have safety, legal, ethical, stakeholder and operational consequences been considered at the appropriate level?
Is there a named owner and a trigger for review, escalation, change or retirement?
Questions for deeper application
What is the most important distinction a practitioner must preserve when applying Satisfaction, Truth and Elementary Equivalence?
Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.
Which assumption about satisfaction would change the result most if it proved false?
Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.
What evidence would allow an independent reviewer to reproduce or challenge the conclusion?
Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.
Which boundary, exception or failure case has not yet been tested?
Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.
What must be handed over, monitored or reviewed after the immediate work is complete?
Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.
Authoritative references and use notes
The sources below were selected as institutional or primary guidance for the broader practice. They support the handbook method; they do not imply that every statement or clause in a source applies to every project. Confirm the current edition, jurisdiction, contract and application before treating any requirement as mandatory.
MIT OpenCourseWare — Algebra I — Massachusetts Institute of Technology. Used for groups, vector spaces, linear transformations and linear groups. Accessed 2026-08-13.
The Stacks Project — table of contents — The Stacks Project. Used for commutative algebra, homological algebra, modules and derived categories. Accessed 2026-08-13.
