5.3 Applications to Algebra

Model theoretic methods, and in particular the compactness theorem and the Lowenheim Skolem theorems, have strong consequences in algebra, because many algebraic structures can be described by first order sentences, and therefore logical principles can be used to construct models and prove existence theorems that are difficult to obtain by direct algebraic arguments.

First Order Theories in Algebra

Many algebraic structures can be expressed in a first order language.

For example, the language of groups consists of a binary function symbol $\cdot$ , a constant symbol $e$ , and a unary function symbol for inverses.

The axioms of groups can be written as first order sentences:

\forall x \forall y \forall z \; ((x \cdot y) \cdot z = x \cdot (y \cdot z))

\forall x \; (e \cdot x = x \land x \cdot e = x)

\forall x \; (x \cdot x^{-1} = e \land x^{-1} \cdot x = e)

Thus the class of all groups is the class of models of a first order theory.

Similar descriptions exist for rings, fields, ordered structures, and many other algebraic systems.

Example 5.17 (Existence of Infinite Models)

Suppose that a theory of algebraic structures has models of arbitrarily large finite size.

For example, consider the theory of fields of characteristic $0$ together with additional axioms that do not bound the size of the field.

If for each natural number $n$ there exists a model with at least $n$ elements, then by compactness the theory has an infinite model.

To see this formally, for each $n$ consider the sentence:

\exists x_1 \cdots \exists x_n \; \bigwedge_{i \neq j} x_i \neq x_j

Add all such sentences to the theory.

Every finite subset requires only finitely many distinct elements, so it is satisfiable in a sufficiently large finite model.

By compactness, the full theory has a model, and that model must be infinite.

Example 5.18 (Fields with Special Properties)

Consider the theory of fields together with the requirement that there exist elements satisfying certain polynomial equations.

Suppose that for every finite collection of such requirements, there exists a field satisfying them.

Then by compactness, there exists a field satisfying all requirements simultaneously.

This allows one to construct fields with infinitely many specified properties, provided that no finite contradiction arises.

This technique is often used to build algebraic structures with prescribed behavior.

Algebraic Closure via Compactness

One important application of compactness is the existence of algebraic closures.

Given a field $F$ , one can consider the set of all polynomial equations over $F$ that are required to have roots.

For each polynomial $p(x)$ , introduce a sentence asserting the existence of a root:

\exists x \; p(x) = 0

Any finite collection of such sentences can be satisfied in a suitable extension field, since finitely many polynomials can be handled one at a time.

By compactness, there exists a field extension in which all these polynomials have roots.

This construction leads to an algebraically closed field containing $F$ .

Example 5.19 (Nonstandard Finite Structures)

Consider the theory of finite groups together with sentences asserting the existence of arbitrarily many elements.

Although no single finite group satisfies all such requirements, every finite subset of the requirements can be satisfied by a sufficiently large finite group.

By compactness, there exists a model satisfying all requirements, but this model cannot be finite.

Thus one obtains an infinite structure that satisfies all first order properties shared by finite groups, illustrating that first order logic cannot distinguish finite from infinite in this context.

Example 5.20 (Ultraproduct Motivation)

Compactness is closely related to the construction of ultraproducts.

Given a family of structures and an ultrafilter, one can form a new structure that satisfies exactly those sentences that hold in almost all of the original structures.

This construction allows one to pass from a sequence of finite structures to a single infinite structure that reflects their common properties.

Although ultraproducts require additional machinery, compactness explains why such constructions preserve first order truth.

Corollary 5.21 (Transfer of Properties)

If a first order property holds in all finite structures of a certain kind, and if those structures exist in arbitrarily large finite sizes, then there exists an infinite structure satisfying the same property.

This follows directly from compactness, since the property can be expressed as a set of sentences that are satisfied by all finite models.

Limitations

While compactness is powerful, it also shows that first order logic cannot express certain algebraic properties.

For example, the property that a group is finite cannot be expressed by any set of first order sentences, since compactness would force the existence of an infinite model satisfying those sentences.

Similarly, properties that depend on global size or cardinality cannot be captured in first order logic.

Conceptual Perspective

The use of compactness in algebra shows that logical methods provide a unifying framework for constructing and analyzing algebraic structures.

Instead of building structures explicitly, one specifies properties locally, checks that no finite contradiction arises, and then invokes compactness to guarantee the existence of a global model.

This approach is particularly effective when dealing with infinite systems of conditions, and it illustrates the deep interaction between logic and algebra.