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Completeness and Model-Completeness
Paul Runnion
Abstract. Completeness and model-completeness are important in model
theory and heavily tied to field theory. Following an introduction with
definitions of crucial background concepts, we examine the basics of
completeness, including Vaught’s test and an important tie to field theory.
We also explore the basics of model-completeness, emphasizing its
relationship to completeness and to field theory.
0. Introduction
The related concepts of completeness and model-completeness are fundamental concepts
of model theory. These concepts are also heavily tied to algebra—specifically, field theory.
Before tackling these concepts, we must first explore some necessary background information.
As we begin our investigation of completeness, an understanding of deducibility will prove
important.
0.1.
Definition.
A deduction (or proof) of φ from Γ is a finite sequence  0 , 1 ,..., n of
formulas such that αn is φ and for each k  n , either
(a) αk is in    (where Λ is an infinite set of logical axioms), or
(b) αk is obtained by modus ponens from two earlier formulas in the sequence
Whenever φ is deducible from Γ, we write   , and we say that φ is a theorem of Γ.1
Additionally, we can connect the idea of deducibility with the idea of closure:
1
See section 2.4 of Enderton [2] and Section 2.1 of Bell and Slomson [1]
1
0.2.
Definition.
The deductive closure of a set of sentences Γ is the set of all sentences
deducible from Γ. If Γ is its own deductive closure, it is called deductively closed.2
Prior to continuing, we also must examine some important ideas in field theory. First is
that of an extension field.
0.3.
Definition.
A field E is an extension field of a field F if F is a subfield of E and the
operations of F are those of E restricted to F.3
This concept finds itself in a result sometimes referred to as the Fundamental Theorem of Field
Theory, a hint at the importance of extension fields. In this exploration, we will encounter a
characterization based on a specific type of extension field known as an algebraic extension.
0.4.
Definition.
An extension field E of a field F is an algebraic extension of F if every
element of E is algebraic over F; that is, if every element a  E is the zero of some
nonzero polynomial in F[x] (the ring of polynomials with coefficients in F).
0.5.
Definition.
A field without a proper algebraic extension is algebraically closed.
As expected, we study algebraically closed fields because they possess certain convenient
properties, such as the following:
0.6.
Theorem.
Let E be an algebraically closed field. Then, every polynomial in E[x]
splits in E (that is, every polynomial in E[x] can be factored as a product of linear
factors).
The result in Theorem 0.6 follows from Kronecker’s Theorem, which tells us that no
algebraically closed field could exist if Theorem 0.6 did not hold. We shall revisit algebraically
closed fields in our discussion on completeness.4
Another important definition is that of a normal form for predicate logic:
2
See Section 5.1 of Hedman [4]
See chapter 20 of Gallian [3]
4
See chapter 21 of Gallian [3]
3
2
0.7.
Definition.
A formula φ is in prenex normal form if it is of the form
Q1 x1 Q2 x2 ...Qn xn  , where each Qi is a quantifier symbol (either existential or
universal) and ψ is a formula which contains no quantifiers. ψ is referred to as the matrix
of φ.
Since we have (somewhat prematurely) stated that the form in Definition 0.7 is a normal form,
the following theorem is not surprising.
0.8.
Theorem.
For every formula φ, there exists a formula ψ containing the same
predicate letters and free variables as φ which is in prenex normal form such that
   .
We shall further comment that any formula in prenex normal form for which every Qi is an
existential quantifier symbol is called an existential formula, and an existential formula whose
matrix is a conjunction of atomic formulas and negations of atomic formulas shall be called
primitive. Prenex normal form (and specifically primitive formulas) will aid us later in
establishing a test for model-completeness.5
Now that we have established the necessary background, we are ready to begin our
investigation of completeness and model-completeness. We shall start with completeness.
1. Completeness
1.1.
Definition.
Let K be any set of sentences and let X be any sentence found in the
vocabulary of K. If either X or ¬X is deducible from K, then K is called complete.6
Although this definition is of conceptual value, it is frequently impractical as a test for
completeness. Vaught’s test (Theorem 1.2) provides a direct test for completeness:
5
6
See sections 9.3 and 9.4 of Bell and Slomson [1]
See section 3.2 of Robinson [5]
3
1.2.
Theorem.
Let K be a consistent set of axioms which include a relation of equality
and let α be the cardinal of K. Suppose that all models of K are infinite. Further suppose
that all models of K are isomorphic for some infinite cardinal α’≥α (that is, K is
α’-categorical). Then, K is complete.
Some scholars prefer to use a characterization of incompleteness rather than the direct version of
Vaught’s test, as given in the following corollary:
1.3.
Corollary.
Let T be a deductively closed theory. Then, T is incomplete iff there exist
models M and N of T that are not elementarily equivalent (that is, if M and N are not
structures with the same vocabulary which model the same sentences in that vocabulary).
The connection between Vaught’s test and Corollary 1.3 is clear.7
Above, we visited the concept of extension fields (Def. 0.3). Not surprisingly, we
quickly encounter a very similar concept regarding completeness.
1.4.
Theorem.
Let K be a non-empty and consistent set of sentences. Then, there exists
an extension K’ of K which contains no additional relations or individuals, such that K’ is
consistent and complete.
In the case of extension fields, we are restricted by the field operations. In the case of Theorem
1.4, we see that an extension of a consistent non-empty sentence set is restricted by the relations
of the original set. This similarity is no coincidence:
1.5.
Result.
The concept of an algebraically closed field of specified characteristic is
complete.
It is shown in the Bell and Slomson text that algebraically closed fields of specified characteristic
are m-categorical for each uncountable cardinal m. Therefore, by Vaught’s test, we obtain Result
1.5 as desired.8
7
See section 4.1 of Robinson [5] and Section 5.1 of Hedman [4]
4
Unfortunately, there are still situations in which we cannot establish completeness using
Vaught’s test. In an effort to address these cases, we examine in the next section a modified
form of completeness known as model-completeness.
2. Model-Completeness
As we begin to examine this modified form of completeness, we must first define what is
meant by the diagram of a structure.
2.1.
Definition.
The diagram of a structure M consists of the set of all atomic sentences
(that is, atomic formulas in which no variable occurs free) of the language of M which
hold in M together with the negations of all atomic sentences which do not hold in M.9
Notice that a diagram provides atomic formulas for a structure M in a similar manner to that in
which an operation table provides definitions for operations in a group, ring, or field and
provides us with the ability to characterize the group, ring, or field (for example, the ability to
characterize a group as abelian). Now, with this definition in hand, we are ready to define model
completeness.
2.2.
Definition.
Let K be a non-empty and consistent set of sentences. If for every model
M of K with diagram D the set K  D is complete, we say K is model-complete.
As evidenced by the definition, we could (in theory) determine if a set K is model-complete
through multiple applications of Vaught’s test. However, as was the case with completeness, a
proof of model-completeness is often easier done without strictly using the definition. The
following test was developed by Abraham Robinson:
8
9
See section 4.1 of Robinson [5] and Section 9.1 of Bell and Slomson [1]
See section 2.1 of Robinson [5] and Section 9.2 of Bell and Slomson [1]
5
2.3.
Theorem.
Let K be a non-empty and consistent set of sentences. If, for any two
models M and M’ of K such that M  M ' , any primitive sentence which is defined in M
holds in M’ only if it holds in M, then K is model-complete.10
It should be noted that this is both a necessary and sufficient condition for model-completeness.
Before we further tie model-completeness with completeness and explore a connection
between model-completeness and field theory, we must first examine the definition of a prime
model and one related theorem.
2.4.
Definition.
Let K be a non-empty and consistent set of sentences, and let Mo be a
model of K. If every model M of K possesses a substructure isomorphic to Mo, then Mo is
called a prime model.
2.5.
Theorem.
Let K be a set of sentences with prime model Mo and let Do be the diagram
of Mo. Then, any sentence X defined in K and deducible from K  Do is deducible
from K.
The following theorem uses Definition 2.4 and Theorem 2.5 to provide a method for proving the
completeness of a model-complete set often referred to as the Prime Model Test.
2.6.
Theorem.
Let K be a model-complete set of sentences which possesses a prime
model. Then, K is complete.
Proof:
Let K be a model-complete set of sentences which possesses a prime model Mo, and let X
be any sentence which is defined in K. Let Do be the diagram of Mo. We shall proceed
using Definition 1.1 rather than Vaught’s test. Since X is defined in K, we know X is
defined in Mo and therefore we know that either X or ¬X holds in Mo. Without loss of
10
See section 4.2 of Robinson [5]
6
generality, consider the case where X holds in Mo. Since K is model-complete, we know
K  Do is complete, and it follows that X is deducible from K  Do . Therefore, by
Theorem 2.5, X is deducible from K. This proves the theorem. 11
As was the case with completeness, we soon encounter an intriguing result tying model
completeness to field theory:
2.7.
Result.
The concept of an algebraically closed field is complete.
The similarity of this result to Result 1.5 is no coincidence. Note, however, that Result 1.4
applies to the concept of an algebraically closed field of specified characteristic, not to the
concept of an algebraically closed field in general. If we add to the concept of an algebraically
closed field an axiom or set of axioms to fix the characteristic, the resulting set has a prime
model—the algebraic closure of the prime field of given characteristic. Since this resulting set is
still model-complete, we can apply Theorem 2.6 to prove Result 1.4 in a different manner than
the sketch provided earlier.
References
[1]
Bell, J. L. and A. B. Slomson. Models and Ultraproducts: An Introduction. Second
revised printing. North Holland, 1971.
[2]
Enderton, Herbert B. A Mathematical Introduction to Logic. Second ed. Harcourt
Academic Press, 2002.
[3]
Gallian, Joseph A. Contemporary Abstract Algebra. Fifth ed. Houghton Mifflin, 2002.
[4]
Hedman, Shawn. A First Course in Logic. Oxford Texts in Logic 1, 2004.
[5]
Robinson, Abraham. Introduction to Model Theory and the Metamathematics of
Algebra. North Holland, 1965
11
See section 4.2 of Robinson [5] and Section 9.2 of Bell and Slomson [1]
7