**2.6 Model theory**

**2.6.1 Theories with quantifier elimination**

We now give five examples of theories that admit quantifier elimination and are complete. These examples are used in the rest of this work. The first example is the theory of nontrivial divisible ordered abelian groups.

38 Chapter 2. Preliminaries
**Definition 2.100.** Let *L*og= (0,+,⩽) be the language of ordered groups. The theory of
non-trivial divisible ordered abelian groups, denotedTh_{doag} consists of the*L*og-sentences expressing
the axioms of ordered abelian groups (associativity of addition, commutativity of addition, zero
as the neutral element, existence of an inverse, total order axioms, compatibility of order and
addition), the axiom of nontriviality*∃y, y̸*= 0, and an infinite series of axioms defining
divisi-bility: for every *n*⩾2,Th_{doag} contains a sentence *∀y*_{1}*,∃y*_{2}*, y*_{1} =*y*_{2}+*y*_{2}+*· · ·*+*y*_{2} (where the
addition in taken*n*times).

The theory of of nontrivial divisible ordered abelian groups admits quantifier elimination.

**Theorem 2.101** ([Mar02, Corollary 3.1.17]). *The theory*Th_{doag} *of nontrivial divisible ordered*
*abelian groups admits quantifier elimination and is complete.*

As a corollary, we see that if *Γ* is a nontrivial divisible ordered abelian group, then its
definable sets have a particularly simple structure. More precisely, we make the following
definition.

**Definition 2.102.** We say that a subsetS_{⊂}Γ* ^{n}* is a

*basic semilinear*set if it is of the form S=

*{g∈Γ*

*:*

^{n}*∀i*= 1, . . . , p, f

*i*(g)

*> h*

^{(i)}

*,∀i*=

*p*+ 1, . . . , q, f

*i*(g) =

*h*

^{(i)}

*},*

where *f*_{i}*∈* Z[X_{1}*, . . . , X** _{n}*] are homogeneous linear polynomials with integer coeﬃcients and

*h*

^{(i)}

*∈Γ*. We say thatS is

*semilinear*if it is a finite union of basic semilinear sets.

**Lemma 2.103.** *A set*S_{⊂}Γ^{n}*is definable in* *L*og *if and only if it is semilinear.*

*Sketch of the proof.* It is easy to see that every semilinear set is definable. To prove the converse,
note that a union of two semilinear sets is semilinear. Furthermore, an intersection of two basic
semilinear sets is basic semilinear. Hence, by the fact that intersection distributes over union, an
intersection of two semilinear sets is semilinear. Moreover, the complement of a basic semilinear
set is semilinear and hence the complement of any semilinear set is semilinear. Furthermore,
one can prove by induction that if *ϕ* is an *L*og-term with a nonempty set of free variables
*y*_{1}*, . . . , y** _{n}*, then there exist

*m*

_{1}

*, . . . , m*

_{n}*∈*N such that

*ϕ(y) =*

*m*

_{1}

*y*

_{1}+

*· · ·*+

*m*

_{n}*y*

*for any*

_{n}*y*

*∈*

*Γ*

*. Hence, any set defined by an atomic*

^{n}*L*og-formula is of the form

*{y*

*∈*

*Γ*

*:*

^{n}*f*(y) ⩽

*h}*or

*{y*

*∈*

*Γ*

*:*

^{n}*f*(y) =

*h}*, where

*f*

*∈*Z[X1

*, . . . , X*

*n*] is a homogeneous linear polynomial with integer coeﬃcients and

*h*

*∈Γ*. In particular, these sets are semilinear. Hence, sets defined by negations, conjunctions, and disjunctions of atomic formulas are semilinear. Moreover, one can prove by induction that every quantifier-free formula can be written in the form

∨*p*
*k=1*

*q*_{k}

∧

*ℓ=1*

*ϕ*_{k,ℓ}*,*

where every *ϕ**k,ℓ* is an atomic formula or its negation. Therefore, every set defined by a
quantifier-free formula is semilinear. The claim follows from Theorem 2.101.

*Example* 2.104. If*Γ* =R, then every basic semilinear set is a relative interior of a polyhedron.

Note that the converse is not true. For example, the set *{y* *∈*R^{2}:*y*1+*y*2 *> π}* is semilinear,
but*{y* *∈*R^{2}:*√*

2y_{1}+*y*_{2} *>*0*}* is not. If*Γ* =Q, then basic semilinear sets correspond precisely
to relatively open rational polyhedra.

2.6. Model theory 39
In this work, we use divisible ordered abelian groups which arise as value groups of
nonar-chimedean real closed fields. Since the valuation map may evaluate to *−∞*, we need to deal
with divisible ordered abelian groups with bottom element.

**Definition 2.105.** We denote by *L*ogb := (0,*−∞,*+,⩽) the language of ordered groups with
bottom element. The theory of nontrivial divisible ordered abelian groups with bottom
ele-ment, denoted Th_{doagb}, consists of the axioms of divisible ordered abelian groups (cf.
Defini-tion 2.100),^{4} the nontriviality axiom *∃y,*(y *̸= 0∧y* *̸=* *−∞), and the axioms that extend the*
addition and order to*−∞*, namely *∀y,* *−∞*+*y*=*−∞* and *∀y, y*⩾*−∞*.

Using Theorem 2.101, we can show that Th_{doagb} admits quantifier elimination and is
com-plete.

**Theorem 2.106.** *The theory*Thdoagb *admits quantifier elimination and is complete. Moreover,*
*anyL*ogb*-formulaϕ(y*_{1}*, . . . , y** _{m}*)

*withm*⩾1

*is equivalent to a quantifier-free formula of the form*

∨

*Σ⊂[m]*

((*∀σ∈Σ, y*_{σ}*̸*=*−∞*^{)} *∧* (*∀σ /∈Σ, y** _{σ}* =

*−∞*)

*∧*

*ψ*

_{Σ}^{)}

*,*(2.10)

*where the disjunction goes over all subsets of*[m], and every*ψ*_{Σ}*is a quantifier-freeL*og*-formula*
*such that* Fvar(ψ*Σ*)*⊂ {y**σ**}**σ**∈**Σ**.*

The proof of the above result is easy but technical. We have put it in Appendix B.1 for the
sake of completeness. As in the previous case, quantifier elimination allows us to describe the
class of definable sets. To do so, we need to observe that if *Γ* *∪ {−∞}* is a nontrivial divisible
ordered abelian group with bottom element, then it has a natural stratification.

**Definition 2.107.** If *x* *∈* (Γ *∪ {−∞}*)* ^{n}*, then we define its

*support*as the set of its finite coordinates,

*{k*

*∈*[n] :

*x*

_{k}*̸*=

*−∞}*. Given a nonempty subset

*K*

*⊂*[n], and a set S

*(Γ*

_{⊂}*∪*

*{−∞}*)

*, we define the*

^{n}*stratum of*S

*associated withK, denoted*S

_{K}

_{⊂}Γ*, as the subset of*

^{K}*Γ*

*formed by the projection of the points*

^{K}*x∈*Swith support

*K. More formally, a point*

*y∈Γ*

*belongs to S*

^{K}*if the point*

_{K}*x*

*∈*(Γ

*∪ {−∞}*)

*defined as*

^{n}*x*

*=*

_{k}*y*

*for all*

_{k}*k*

*∈K*and

*x*

*=*

_{k}*−∞*

otherwise belongs toS.

**Lemma 2.108.** *A subset* S* _{⊂}*(Γ

*∪ {−∞}*)

^{n}*is definable in*

*L*ogb

*if and only if all strata of*S

*are semilinear.*

Again, the technical proof of this claim can be found in Appendix B.1. The next theory that is used in this work is the theory of real closed fields.

**Definition 2.109.** The theory of real closed fields, denotedTh_{rcf}, is a theory in the language of
ordered rings *L*or := (0,1,+,*·,*⩽).^{5} It consists of the usual axioms of ordered fields, the axiom

*∀x*1*,*(x1⩾0*→ ∃x*2*, x*1 =*x*2*·x*2)that governs the existence of square roots, and an infinite set
of axioms that states the fact that every polynomial of an odd degree has a root. In other words,
for every*n*⩾1,Thrcf contains the axiom*∀x*0*, . . . ,∀x*2n*,∃y, y*^{2n+1}+*x*2n*y*^{2n}+*· · ·*+*x*1*y*+*x*0 = 0.

4With one change required by the presence of *−∞: the existence of an inverse should be formulated as*

*∀**y,*(y*̸*=*−∞ → ∃**z,*(z *̸*=*−∞ ∧**y*+*z*= 0)). The assumption that *z* *̸*=*−∞*is not needed (by the nontriviality
axiom) but it simplifies the presentation.

5We assume that+and*·*have type(1,1,1)and that⩽has type(1,1). It it clear that any ordered filed is an
*L*or-structure with the natural interpretation of the symbols.

40 Chapter 2. Preliminaries A classical result proved by Tarski states that the theory of real closed fields admits quantifier elimination and is complete.

**Theorem 2.110** ([Mar02, Theorem 3.3.15 and Corollary 3.3.16]). *The theory* Th_{rcf} *of real*
*closed fields admits quantifier elimination and is complete.*

As previously, ifK is a real closed field, then quantifier elimination allows us to characterize the class of definable sets.

**Lemma 2.111.** *A setS* *⊂*K^{n}*is definable in* *L*or *if and only if it is semialgebraic.*

*Sketch of the proof.* By induction, if *ϕ(x*_{1}*, . . . x** _{n}*) is an

*L*or-term with a nonempty set of free variables, then there exists a polynomial

*P*

*∈*N[X

_{1}

*, . . . , X*

*]with natural coeﬃcients such that*

_{n}*ϕ(x) =*

*P*(x) for all

*x*

*∈*K

*. Therefore, a set definable by an atomic formula is of the form*

^{n}*{x∈*K

*:*

^{n}*P(x) = 0}*or

*{x*

*∈*K

*:*

^{n}*P*(x) ⩾0

*}*, where

*P*

*∈*K[X1

*, . . . , X*

*n*]is a polynomial with coeﬃcients inK. The rest of the proof is the same as the proof of Lemma 2.103 (with the word

“semilinear” replaced by “semialgebraic”).

The final two theories used in this work concern valued fields. First, we present a
quan-tifier elimination result in henselian valued fields with angular component. This result was
obtained by Pas [Pas89, Pas90a] and based on the work of Denef [Den84, Den86]. Suppose
that (K*, Γ,*k*,*val,ac) is a valued field with a fixed residue field and an angular component.

Furthermore, we suppose that both K and k have characteristic 0. We start by defining a three-sorted language of such fields.

**Definition 2.112.** We define the Denef–Pas language as *L*Pas := (*L*K*,L**Γ**,L*k*,*val,ac), where
*L*K is the language of rings associated withK,*L*K := (0_{K}*,*1_{K}*,*+_{K}*,·*K),*L*k is the language of
rings associated withk,*L*k := (0_{k}*,*1k*,*+k*,·*k),*L**Γ* := (0_{Γ}*,−∞**Γ**,*+*Γ**,*⩽*Γ*)denotes the language
of ordered groups with bottom element associated with*Γ*, andval,acare two function symbols.^{6}
In this way, any valued field with angular component is an *L*Pas-structure. We now define
the associated theory.

**Definition 2.113.** The theory of henselian valued fields with angular component in
equicharac-teristic0, denotedTh_{Pas}, is a theory in the Denef–Pas language*L*Pas. It consists of the axioms
of fields (forK andk), axioms of ordered abelian groups with bottom element (for*Γ∪{−∞}),*
the axioms specifying that val is a valuation (see Definition 2.60), the axioms specifying that
ac is an angular component (see Definition 2.66), and infinite sequences of axioms describing
the facts that K and k have characteristic 0, and an infinite sequence of axioms describing
henselianity (see Definition 2.72).

*Remark* 2.114. We point out that the axioms mentioned in the definition above can be written
as sentences in Th_{Pas}. We leave this as an exercise. The most problematic are the axioms of
henselianity. However, as noted in Remark 2.67, the definition of the residue map is already
implied in the definition of angular component, because the map res(x) defined as ac(x) if
val(x) = 0 and res(x) = 0 otherwise, is a residue map. Therefore, one can write the infinite
sequence of axioms describing henselianity using only the symbols forac and val.

6The types of symbols are given in a natural way: we put*I* =*{*1,2,3*}*,*C*1 :=*{*0K*,*1K*}*,*C*2 :=*{*0*Γ**,**−∞**Γ**}*,
*C*3:=*{*0k*,*1k*}*. Then,valhas type(1,2),achas type(1,3),+khas type(3,3,3)and so on.

2.6. Model theory 41
Pas [Pas89] has shown that ThPas admits elimination of K-quantifiers, i.e., the quantifiers
that act on the variables associated with K. To state this theorem, we use the following
notation. If*θ*is a*L*Pas-formula, then we denote it by*θ(X, Y, Z), whereX*= (x_{1}*, . . . , x*_{n}_{1})(resp.

*Y* = (y1*, . . . , y**n*2), *Z* = (z1*, . . . , z**n*3)) is the sequence of free variables of *ϕ* associated with the
sort K (resp. *Γ,*k). We have the following theorem.

**Theorem 2.115** ([Pas89]). *Any* *L*Pas*-formulaθ(X, Y, Z)* *is equivalent in*Th_{Pas} *to a formula of*
*the form*

∨*m*
*i=1*

(

*ϕ*_{i}^{(}val(f* _{i1}*(X)), . . . ,val(f

_{ik}*(X)), Y*

_{i}^{)}

*∧ψ*

_{i}^{(}ac(f

_{i(k}

_{i}_{+1)}(X)), . . . ,ac(f

_{il}*(X)), Z*

_{i}^{))}

*,*(2.11)

*where, for every*

*i*= 1, . . . , m,

*f*

*i1*

*, . . . , f*

*il*

*i*

*∈*Z[X]

*are polynomials with integer coeﬃcients,*

*ϕ*

*i*

*is an* *L**Γ**-formula, and* *ψ*_{i}*is an* *L*k*-formula.*

*Remark* 2.116. We point out that the original statement of this theorem given by Pas [Pas89,
Theorem 4.1] is slightly weaker. We took our formulation from [CLR06, Theorem 4.2].
Never-theless, the proof of Pas actually proves the theorem stated above.

We use Theorem 2.115 to prove a quantifier elimination result in real closed valued fields.

Let (K*, Γ,*k*,*val,ac) be valued field such that K is real closed, val is nontrivial and convex,
and residue field and angular component are fixed. We point out that, by Theorem 2.75, k is
real closed. As previously, we start by defining the associated language and theory.

**Definition 2.117.** We define the language of real closed valued fields as
*L*rcvf := (*L*K*,L**Γ**,L*k*,*val,ac)*,*

where *L*K is the language of ordered rings associated with K, *L*K := (0_{K}*,*1_{K}*,*+_{K}*,·*K*,*⩽K),
*L*k is the language of ordered rings associated with k, *L*k := (0_{k}*,*1_{k}*,*+_{k}*,·*k*,*⩽k), *L**Γ* :=

(0_{Γ}*,−∞**Γ**,*+_{Γ}*,*⩽*Γ*) denotes the language of ordered groups with bottom element associated
with*Γ*, andval,acare two function symbols.^{7}

**Definition 2.118.** The theory of real closed valued fields, denoted Thrcvf, is a theory in the
language *L*rcvf. It consists of the axioms of real closed fields for K, the axioms of ordered
abelian groups with bottom element for *Γ* *∪ {−∞}*, the axioms of ordered fields for k, the
axioms specifying thatval is a nontrivial and convex valuation, and the axioms specifying that
ac is an angular component.

*Remark* 2.119. We point out that the axioms of Th_{rcvf} imply thatkis real closed. Indeed, the
axioms of acimply that k is a residue field and hence it is real closed by Theorem 2.75.

Using the quantifier elimination results presented in this section, we can prove that the theory of real closed valued fields admits quantifier elimination and is complete.

**Theorem 2.120.** *The theory* Thrcvf *admits quantifier elimination and is complete. Moreover,*
*anyL*rcvf*-formulaθ(X, Y, Z)* *is equivalent in* Th_{rcvf} *to a formula of the form*

∨*m*
*i=1*

(
*ϕ**i*

(val(f*i1*(X)), . . . ,val(f_{ik}* _{i}*(X)), Y

^{)}

*∧ψ*

*i*

(ac(f_{i,k}_{i}_{+1}(X)), . . . ,ac(f_{il}* _{i}*(X)), Z

^{))}

*,*

*where, for every* *i*= 1, . . . , m, *f*_{i1}*, . . . , f*_{il}_{i}*∈*Z[X] *are polynomials with integer coeﬃcients,* *ϕ*_{i}*is a quantifier-free* *L**Γ**-formula, and* *ψ**i* *is a quantifier-free* *L*k*-formula.*

7The types of symbols are given in a natural way as in Definition 2.112.

42 Chapter 2. Preliminaries
*Sketch of the proof.* Let *θ(X, Y, Z)* denote any*L*rcvf-formula. We recall (Lemma 2.12) that the
order⩽in any real closed field can be defined as*x*_{1} ⩽*x*_{2} *⇐⇒ ∃x*_{3}*, x*_{2} =*x*^{2}_{3}+*x*_{1}. This enables
us to eliminate the symbols ⩽. More precisely, by a simple induction, *θ(X, Y, Z*) is equivalent
inThrcvf to some*L*Pas-formula*θ(X, Y, Z*ˆ ). Moreover, by Theorem 2.75,(K*, Γ,*val)is henselian.

Furthermore, since K and k are ordered, they have characteristic zero. This enables us to
apply Theorem 2.115. As a result, *θ(X, Y, Z)*ˆ is equivalent in Th_{rcvf} to a formula of the form
(2.11). Then, we apply Theorem 2.106 and Theorem 2.110 to eliminate the quantifiers in the
formulas*ϕ**i* and*ψ**i*. This shows the last part of the statement. In the case where*θ*is a sentence,
the formulas *ϕ** _{i}* and

*ψ*

*are also sentences. The completeness results in Theorem 2.106 and Theorem 2.110 applied to each subformula*

_{i}*ϕ*

*i*and

*ψ*

*i*allow to prove that either

*θ*or

*¬θ*is a logical consequence of Th

_{rcvf}.

*Remark* 2.121. We note that the proof of Theorem 2.120 is constructive, in the sense that there
exists an algorithm that, given a *L*rcvf-formula, outputs an equivalent quantifier-free formula
of the form stated in Theorem 2.120. Indeed, the proof first uses the Denef–Pas quantifier
elimination to eliminate quantifiers acting over K and the algorithm to do so is given in the
proof of Pas [Pas89]. Second, we eliminate the quantifiers in the residue field k by using
quantifier elimination in real closed fields (see [BPR06, Chapter 14] for a detailed discussion
about the algorithms for this task). Third, we eliminate the quantifiers in the value group
*Γ∪ {−∞}*. An algorithm for quantifier elimination in*Γ* is given in [FR75] and it can be easily
extended to *Γ* *∪ {−∞}*(using the proof of Theorem 2.106).