4.5 Examples Across Algebra and Geometry

Model theory treats many familiar mathematical objects as structures for suitable first order languages, and this viewpoint allows groups, rings, fields, ordered sets, graphs, and geometric systems to be studied using a common formal framework.

Algebraic Structures

Algebraic structures are among the most natural examples of first order structures, because they are usually described by a domain together with operations and distinguished constants.

A group may be represented in the language:

(\cdot, e, {}^{-1})

where $\cdot$ is a binary function symbol, $e$ is a constant symbol, and ${}^{-1}$ is a unary function symbol.

A group structure is then:

\mathcal G = (G; \cdot^{\mathcal G}, e^{\mathcal G}, ({}^{-1})^{\mathcal G})

where $G$ is the underlying set, $\cdot^{\mathcal G}$ is the group operation, $e^{\mathcal G}$ is the identity element, and $({}^{-1})^{\mathcal G}$ sends each element to its inverse.

Example 4.40

The integers under addition form a group:

(\mathbb Z; +, 0, -)

Here $+$ is interpreted as ordinary addition, $0$ is interpreted as the integer zero, and $-$ is interpreted as the unary operation sending $n$ to $-n$ .

The group law:

\forall x \forall y \forall z \, ((x+y)+z=x+(y+z))

is true in this structure.

The identity law:

\forall x \, (x+0=x)

is also true in this structure.

Rings

A ring is naturally expressed in the language:

(0, 1, +, -, \cdot)

where $0$ and $1$ are constant symbols, $+$ and $\cdot$ are binary function symbols, and $-$ is a unary function symbol.

A ring structure has the form:

\mathcal R = (R; 0^{\mathcal R}, 1^{\mathcal R}, +^{\mathcal R}, -^{\mathcal R}, \cdot^{\mathcal R})

The ring axioms can be written as first order sentences in this language, because associativity, distributivity, identities, and additive inverses all quantify only over elements of the ring.

Example 4.41

The ring of integers:

(\mathbb Z; 0, 1, +, -, \cdot)

is a structure in the language of rings.

The distributive law is expressed by the sentence:

\forall x \forall y \forall z \, (x \cdot (y+z) = x \cdot y + x \cdot z)

This sentence is true in the integers, and it is also true in every ring.

Fields

A field may be described in the same ring language:

(0, 1, +, -, \cdot)

The field axioms are first order, but the multiplicative inverse operation is usually expressed without adding a separate inverse function symbol, because inversion is undefined at $0$ .

The axiom saying that every nonzero element has a multiplicative inverse is:

\forall x \, (x \ne 0 \to \exists y \, (x \cdot y = 1))

This is a first order sentence because it quantifies over elements of the field.

Example 4.42

The rational numbers form a field:

(\mathbb Q; 0, 1, +, -, \cdot)

The real numbers also form a field:

(\mathbb R; 0, 1, +, -, \cdot)

These two structures satisfy the field axioms, but they differ on additional first order sentences in the language of ordered fields once an order relation is added.

Ordered Structures

Ordered structures are described using relation symbols.

The language of linear orders has one binary relation symbol:

<

A structure for this language has the form:

\mathcal A = (A; <^{\mathcal A})

The sentence expressing transitivity is:

\forall x \forall y \forall z \, ((x<y \land y<z) \to x<z)

The sentence expressing totality is:

\forall x \forall y \, (x<y \lor x=y \lor y<x)

The sentence expressing irreflexivity is:

\forall x \, \neg(x<x)

Example 4.43

The structure:

(\mathbb N; <)

is a linear order with a least element.

The structure:

(\mathbb Z; <)

is a linear order without a least element and without a greatest element.

The sentence:

\exists x \forall y \, (x=y \lor x<y)

is true in $(\mathbb N; <)$ , because $0$ is the least element, but it is false in $(\mathbb Z; <)$ .

Dense Linear Orders

A dense linear order is a linear order in which between any two distinct elements there is another element.

Density is expressed by:

\forall x \forall y \, (x<y \to \exists z \, (x<z \land z<y))

The rational numbers with their usual order:

(\mathbb Q; <)

form a dense linear order without endpoints.

The real numbers with their usual order:

(\mathbb R; <)

also form a dense linear order without endpoints.

Although these structures have different cardinalities, first order logic in the language of order alone cannot distinguish them by any sentence.

Graphs

A graph can be treated as a first order structure using one binary relation symbol:

E

A graph structure is:

\mathcal G = (V; E^{\mathcal G})

where $V$ is the set of vertices and $E^{\mathcal G}$ is the edge relation.

For an undirected graph without loops, the usual axioms are:

\forall x \, \neg E(x,x)

and:

\forall x \forall y \, (E(x,y) \to E(y,x))

The first sentence says that no vertex is adjacent to itself, and the second says that adjacency is symmetric.

Example 4.44

Let $\mathcal G$ be a graph with vertices:

\{a,b,c\}

and edges:

E(a,b), \quad E(b,a), \quad E(b,c), \quad E(c,b)

Then $\mathcal G$ is an undirected path of length two.

The sentence:

\exists x \exists y \exists z \, (E(x,y) \land E(y,z) \land x \ne z)

is true in this graph, because there are vertices connected through a middle vertex.

Geometric Structures

Geometric systems can also be described as first order structures, provided that the basic objects and relations are chosen explicitly.

For example, one may use a language with a binary relation:

\mathrm{Inc}(p,\ell)

where $\mathrm{Inc}(p,\ell)$ means that the point $p$ lies on the line $\ell$ .

In a many sorted version, points and lines are treated as different sorts, while in a one sorted version, predicates are added to distinguish points from lines.

Example 4.45

A simple incidence geometry may use predicates:

\mathrm{Point}(x)

and:

\mathrm{Line}(x)

together with an incidence relation:

\mathrm{Inc}(x,y)

The statement that every two distinct points lie on a common line can be written as:

\forall p \forall q \, ((\mathrm{Point}(p) \land \mathrm{Point}(q) \land p \ne q) \to \exists \ell \, (\mathrm{Line}(\ell) \land \mathrm{Inc}(p,\ell) \land \mathrm{Inc}(q,\ell)))

This is a first order sentence because it quantifies only over objects in the domain and uses the relation symbols of the language.

Metric and Ordered Geometry

Some geometric notions require care because first order logic can express relations between elements but cannot directly quantify over arbitrary sets or functions unless those sets or functions are encoded as elements of the structure.

For ordered geometry, one may use a ternary betweenness relation:

B(x,y,z)

where $B(x,y,z)$ means that $y$ lies between $x$ and $z$ .

A basic axiom might say:

\forall x \forall y \forall z \, (B(x,y,z) \to B(z,y,x))

which expresses the symmetry of betweenness with respect to the endpoints.

Vector Spaces

Vector spaces can be treated as structures, but the language must be chosen carefully.

If the field is fixed, such as $\mathbb Q$ , then scalar multiplication by each scalar can be represented by a unary function symbol:

\lambda_q

for each:

q \in \mathbb Q

Then a vector space over $\mathbb Q$ can be described in a first order language containing:

0, +, -, \lambda_q

for every rational number $q$ .

Example 4.46

A vector space over $\mathbb Q$ is represented as:

(V; 0, +, -, (\lambda_q)_{q \in \mathbb Q})

where:

\lambda_q(v)=qv

The axiom:

\forall x \, (\lambda_1(x)=x)

says that scalar multiplication by $1$ acts as the identity.

The axiom:

\forall x \, (\lambda_0(x)=0)

says that scalar multiplication by $0$ sends every vector to the zero vector.

Lemma 4.47 (Algebraic Axioms are First Order)

The usual axioms for groups, rings, fields, linear orders, graphs, and many incidence geometries can be written as first order sentences in suitable signatures.

Proof

Each axiom in these classes refers only to elements of the underlying domain and uses a fixed finite pattern of variables, operations, relations, equality, connectives, and quantifiers.

For groups, associativity, identity, and inverse laws quantify over group elements and use only the group operation, identity, and inverse.

For rings and fields, the algebraic laws quantify over ring or field elements and use only addition, multiplication, constants, and negation.

For orders, the axioms quantify over elements and use only the order relation.

For graphs, the basic properties of adjacency are expressed using the edge relation.

In each case, the statement has the form of a first order sentence in the chosen signature.

Expressive Limits

Although many mathematical structures fit naturally into first order logic, not every familiar mathematical property is first order expressible in a given language.

For example, in the language of orders, the property of being complete like the real line requires quantifying over sets of elements, not merely over individual elements.

Similarly, the property of being finite cannot be captured by a single first order sentence.

These limitations are central features of model theory, because they explain why different structures can satisfy the same first order sentences while still being different from a broader mathematical point of view.