Misplaced Pages

The Feferman–Vaught theorem in model theory is a theorem by Solomon Feferman and Robert Lawson Vaught that shows how to reduce, in an algorithmic way, the first-order theory of a product of structures to the first-order theory of elements of the structure.

The theorem is considered to be one of the standard results in model theory. The theorem extends the previous result of Andrzej Mostowski on direct products of theories. It generalizes (to formulas with arbitrary quantifiers) the property in universal algebra that equalities (identities) carry over to direct products of algebraic structures (which is a consequence of one direction of Birkhoff's theorem).

Direct product of structures

Consider a first-order logic signature L. The definition of product structures takes a family of L-structures ${\displaystyle \mathbf {A} _{i}}$ for ${\displaystyle i\in I}$ for some index set I and defines the product structure ${\displaystyle \mathbf {A} =\prod _{i\in I}\mathbf {A} _{i}}$, which is also an L-structure, with all functions and relations defined pointwise.

The definition generalizes the direct product in universal algebra to structures for languages that contain not only function symbols but also relation symbols.

If ${\displaystyle r(a_{1},\ldots ,a_{p})}$ is a relation symbol with ${\displaystyle p}$ arguments in L and ${\displaystyle a_{1},\ldots ,a_{n}\in \Pi _{i\in I}\mathbf {A} _{i}}$ are elements of the cartesian product, we define the interpretation of ${\displaystyle r}$ in ${\displaystyle \mathbf {A} }$ by

${\displaystyle \mathbf {A} \models r(a_{1},\ldots ,a_{p})\iff \forall i\in I,\ \mathbf {A} _{i}\models r(a_{1}(i),\ldots ,a_{p}(i))}$

When ${\displaystyle r}$ is a functional relation, this definition reduces to the definition of direct product in universal algebra.

Statement of the theorem for direct products

For a first-order formula ${\displaystyle \phi ({\bar {x}})}$ in signature L with free variables a subset of ${\displaystyle {\bar {x}}}$, and for an interpretation ${\displaystyle {\bar {a}}}$ of the variables ${\displaystyle {\bar {x}}}$, we define the set of indices ${\displaystyle i}$ for which ${\displaystyle \phi ({\bar {a}})}$ holds in ${\displaystyle \mathbf {A} _{i}}$

${\displaystyle ||\phi ({\bar {a}})||=\{i\mid \mathbf {A} _{i}\models \phi ({\bar {a}}(i))\}}$

Given a first-order formula with free variables ${\displaystyle \phi ({\bar {x}})}$, there is an algorithm to compute its equivalent game normal form, which is a finite disjunction ${\displaystyle \bigvee _{i=1}^{k}\theta _{i}({\bar {x}})}$ of mutually contradictory formulas.

The Feferman–Vaught theorem gives an algorithm that takes a first-order formula ${\displaystyle \phi ({\bar {x}})}$ and constructs a formula ${\displaystyle \phi ^{*}}$ that reduces the condition that ${\displaystyle \phi ({\bar {a}})}$ holds in the product to the condition that ${\displaystyle \phi ^{*}}$ holds in the interpretation of ${\displaystyle k+1}$ sets of indices:

${\displaystyle I,||\theta _{i}({\bar {a}})||,\ldots ,||\theta _{k}({\bar {a}})||}$

The formula ${\displaystyle \phi ^{*}}$ is thus a formula with ${\displaystyle k+1}$ free set variables, for example, in the first-order theory of fields of sets.

Proof idea

The formula ${\displaystyle \phi ^{*}}$ can be constructed following the structure of the starting formula ${\displaystyle \phi }$. When ${\displaystyle \phi }$ is quantifier free then, by definition of direct product above it follows

${\displaystyle {\begin{array}{rl}\mathbf {A} \models \phi ({\bar {a}})&\iff \forall i\in I.\ \mathbf {A} _{i}\models \phi ({\bar {a}}(i))\\&\iff \{i\mid \mathbf {A} _{i}\models \phi ({\bar {a}}(i))\}=I\\&\iff ||\phi ({\bar {a}})||=I\end{array}}}$

Consequently, we can take ${\displaystyle \phi ^{*}(U,X_{1})}$ to be the equality ${\displaystyle U=X_{1}}$ in the language of fields of sets.

Extending the condition to quantified formulas can be viewed as a form of quantifier elimination, where quantification over product elements ${\displaystyle {\bar {a}}}$ in ${\displaystyle \phi }$ is reduced to quantification over subsets of ${\displaystyle I}$.

Generalized products

It is often of interest to consider substructure of the direct product structure. If the restriction that defines product elements that belong to the substructure can be expressed as a condition on the sets of index elements, then the results can be generalized.

An example is the substructure of product elements that are constant at all but finitely many indices. Assume that the language L contains a constant symbol ${\displaystyle c}$ and consider the substructure containing only those product elements ${\displaystyle a}$ for which the set

${\displaystyle \{i\mid {\textbf {A}}_{i}\models a(i)\neq c\}}$

is finite. The theorem then reduces the truth value in such substructure to a formula ${\displaystyle \phi ^{*}}$ in the field of sets, where certain sets are restricted to be finite.

One way to define generalized products is to consider those substructures where the sets ${\displaystyle ||\phi (a)||}$ belong to some field ${\displaystyle B}$ of sets ${\displaystyle X\subseteq I}$ of indices (a subset of the powerset set algebra ${\displaystyle 2^{I}}$), and where the product substructure admits gluing. Here admitting gluing refers to the following closure condition: if ${\displaystyle a,b}$ are two product elements and ${\displaystyle X\in B}$ is the element of the field of sets, then so is the element ${\displaystyle c}$ defined by "gluing" ${\displaystyle a}$ and ${\displaystyle b}$ according to ${\displaystyle X}$:

${\displaystyle c(i)=\left\{{\begin{array}{rl}a(i),&{\mbox{ if }}i\in X\\b(i),&{\mbox{ if }}i\in (I\setminus X)\end{array}}\right.}$

Consequences

The Feferman–Vaught theorem implies the decidability of Skolem arithmetic by viewing, via the fundamental theorem of arithmetic, the structure of natural numbers with multiplication as a generalized product (power) of Presburger arithmetic structures.

Given an ultrafilter on the set of indices ${\displaystyle I}$, we can define a quotient structure on product elements, leading to the theorem of Jerzy Łoś that can be used to construct hyperreal numbers.

v t e Mathematical logic
General	Axiom list Cardinality First-order logic Formal proof Formal semantics Foundations of mathematics Information theory Lemma Logical consequence Model Theorem Theory Type theory
Theorems (list)  and paradoxes	Gödel's completeness and incompleteness theorems Tarski's undefinability Banach–Tarski paradox Cantor's theorem, paradox and diagonal argument Compactness Halting problem Lindström's Löwenheim–Skolem Russell's paradox
Logics	Traditional Classical logic Logical truth Tautology Proposition Inference Logical equivalence Consistency Equiconsistency Argument Soundness Validity Syllogism Square of opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued logic 3 finite ∞ Predicate First-order list Second-order Monadic Higher-order Fixed-point Free Quantifiers Predicate Monadic predicate calculus
Set theory	Set hereditary Class (Ur-)Element Ordinal number Extensionality Forcing Relation equivalence partition Set operations: intersection union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter Recursive Fuzzy Universal Universe constructible Grothendieck Von Neumann Maps and cardinality Function/Map domain codomain image In/Sur/Bi-jection Schröder–Bernstein theorem Isomorphism Gödel numbering Enumeration Large cardinal inaccessible Aleph number Operation binary Set theories Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck Von Neumann–Bernays–Gödel Ackermann Constructive
Formal systems (list), language and syntax	Alphabet Arity Automata Axiom schema Expression ground Extension by definition conservative Relation Formation rule Grammar Formula atomic closed ground open Free/bound variable Language Metalanguage Logical connective ¬ ∨ ∧ → ↔ = Predicate functional variable propositional variable Proof Quantifier ∃ ! ∀ rank Sentence atomic spectrum Signature String Substitution Symbol function logical/constant non-logical variable Term Theory list Example axiomatic systems (list) of arithmetic: Peano second-order elementary function primitive recursive Robinson Skolem of the real numbers Tarski's axiomatization of Boolean algebras canonical minimal axioms of geometry: Euclidean: Elements Hilbert's Tarski's non-Euclidean Principia Mathematica
Proof theory	Formal proof Natural deduction Logical consequence Rule of inference Sequent calculus Theorem Systems axiomatic deductive Hilbert list Complete theory Independence (from ZFC) Proof of impossibility Ordinal analysis Reverse mathematics Self-verifying theories
Model theory	Interpretation function of models Model equivalence finite saturated spectrum submodel Non-standard model of arithmetic Diagram elementary Categorical theory Model complete theory Satisfiability Semantics of logic Strength Theories of truth semantic Tarski's Kripke's T-schema Transfer principle Truth predicate Truth value Type Ultraproduct Validity
Computability theory	Church encoding Church–Turing thesis Computably enumerable Computable function Computable set Decision problem decidable undecidable P NP P versus NP problem Kolmogorov complexity Lambda calculus Primitive recursive function Recursion Recursive set Turing machine Type theory
Related	Abstract logic Algebraic logic Automated theorem proving Category theory Concrete/Abstract category Category of sets History of logic History of mathematical logic timeline Logicism Mathematical object Philosophy of mathematics Supertask
Mathematics portal

References

1. Feferman, S; Vaught, R (1959). "The first order properties of products of algebraic systems". Fundamenta Mathematicae. 47 (1): 57–103.
2. Hodges, Wilfrid (1993). "Section 9.6: Feferman-Vaught theorem". Model theory. Cambridge University Press. ISBN 0521304423.
3. Karp, Carol (August 1967). "S. Feferman and R. L. Vaught. The first order properties of products of algebraic systems. Fundamenta mathematicae, vol, 47 (1959), pp. 57–103. (Review)". Journal of Symbolic Logic. 32 (2): 276–276. doi:10.2307/2271704.
4. Monk, J. Donald (1976). "23: Generalized Products". Mathematical Logic. Graduate Texts in Mathematics. Berlin, New York: Springer-Verlag. ISBN 978-0-387-90170-1.
5. Mostowski, Andrzej (March 1952). "On direct products of theories". Journal of Symbolic Logic. 17 (1): 1–31. doi:10.2307/2267454.
6. Hodges, Wilfrid (1993). "Section 9.6: Feferman-Vaught theorem". Model theory. Cambridge University Press. p. 459. ISBN 0521304423.

• Model theory