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arXiv:2302.01609v1 [math.LO] 3 Feb 2023
EMBEDDING THE PRIME MODEL OF REAL
EXPONENTIATION INTO O-MINIMAL EXPONENTIAL
FIELDS
LOTHAR SEBASTIAN KRAPP
Abstract. Motivated by the decidability question for the theory of
real exponentiation and by the Transfer Conjecture for o-minimal expo-
nential fields, we show that, under the assumption of Schanuel’s Con-
jecture, the prime model of real exponentiation is embeddable into any
o-minimal exponential field. This is deduced from a more general un-
conditional result on the embeddability of exponential algebraic closures
in o-minimal exponential fields.
1. Introduction
In his highly influential work [18], Tarski proved that the complete the-
ory Trcf of the real closed field Rin the language of ordered rings Lor =
{+,−,·,0,1, <}is decidable, by presenting an explicit quantifier-elimination
algorithm for Trcf . As a result, he noted that the Lor-structure (R,+,−,·,0,
1, <) is – in modern terminology – o-minimal, i.e. any unary definable sub-
set of this structure is a finite union of points and open intervals. In the
same work, Tarski asked whether decidability can also be obtained for the
complete theory Texp of the real exponential field Rexp = (R,+,−,·,0,1,
<, exp), where exp denotes the standard exponential function x7→ exon
R. While this question is open to the date, considerable progress has been
made since the 1990s: In [19], Wilkie proved that Rexp is model complete,
and thus this structure is o-minimal (cf. also van den Dries, Macintyre and
Marker [4]). Bulding on this result, Macintyre and Wilkie [13] showed that
under the assumption of the real version of Schanuel’s Conjecture below,
Texp is decidable.
Schanuel’s Conjecture. Let n∈Nand let α1,...,αn∈Rbe Q-linearly
independent. Then the transcendence degree of Q(α1,...,αn,eα1,...,eαn)
over Qis at least n.
In general, an ordered exponential field K= (K, +,−,·,0,1, <, E) is an
expansion of an ordered field (K, +,−,·,0,1, <) by an exponential E, i.e. an
order-preserving isomorphism from the ordered additive group (K, +,0, <)
Date: 6 February 2023.
2020 Mathematics Subject Classification: 03C64 (03C60 12J15 12L12 12J10)
1
2 L. S. KRAPP
to the ordered multiplicative group (K>0,·,1, <). We denote the corres-
ponding first-order language Lor ∪ {E}, where Eis a unary function sym-
bol, by Lexp. Following the terminology of Krapp [10], we call an ordered
exponential field Kan EXP-field if its exponential satisfies the first-order
Lexp-sentence expressing the differential equation E′=E. While Rexp
and, more generally, any model of Texp is an o-minimal EXP-field, it is
not known whether every o-minimal EXP-field is already a model of Texp.
In [5, page 153 f.], van den Dries stated a transfer conjecture for o-minimal
structures, which specialises to EXP-fields as follows (cf. Krapp [11, Obser-
vation 4.54] for details):
Transfer Conjecture. Any o-minimal EXP-field is elementarily equivalent
to Rexp.
Berarducci and Servi [1] showed that the Transfer Conjecture would imply
that Texp is decidable. This result motivated the study of o-minimal EXP-
fields in connection to the Transfer Conjecture.
One approach towards proving the Transfer Conjecture is to show that
the (unique) prime model of real exponentiation, i.e. the prime model of
Texp, elementarily embeds into any o-minimal EXP-field, which is also the
motivating question for this line of research.
Question 1.1 (Main question).Does the prime model of Texp elementarily
embed into any o-minimal EXP-field?
Note that Question 1.1 can be answered positively if and only if the Trans-
fer Conjecture holds: given an o-minimal EXP-field K, if Kis a model of
Texp, then the prime model Pof Texp elementarily embeds into K, and if P
elementarily embeds into K, then Kis already a model of Texp . It is thus
natural to attempt answering Question 1.1 affirmatively only under the as-
sumption of Schanuel’s Conjecture, as a positive answer would imply the
decidability of Texp via the Transfer Conjecture.
While this note does not provide a complete answer to Question 1.1, we
show in Theorem 3.4 that under the assumption of Schanuel’s Conjecture,
the prime model of Texp embeds into any o-minimal EXP-field (however,
this embedding might not necessarily be elementary). This result is deduced
from the technical heart of this paper, Theorem 2.6, which is a more general
unconditional embeddability result for exponential algebraic closures of Z
within one o-minimal EXP-field into another. Finally, we complete this
note by discussing in Section 4 how Question 1.1 relates to another approach
towards the Transfer Conjecture via elementary embeddability of the residue
exponential field (cf. Krapp [10, Proposition 3.6]).
1.1. Notation and terminology. More background on the model the-
oretic notation and terminology we use can be found in Marker [15]. If
it is clear from the context, then the Lor -structure of an ordered field
(K, +,−,·,0,1, <) is simply denoted by Kand the Lexp-structure of an
ordered exponential field (K, +,−,·,0,1, <, E) simply by (K, E ). Given an
EMBEDDING THE PRIME MODEL OF REAL EXPONENTIATION 3
ordered exponential field (K, E), we say that a subfield F⊆Kis exponen-
tially closed in (K, E ) if the restriction E|Fis an exponential on F. In
this case, if no confusion is likely to arise, then we also denote E|Fsimply
by Eand write (F, E)⊆(K, E) for the corresponding embedding of ordered
exponential fields. For any structure M= (M,...), its complete theory is
denoted by Th(M) and its existential theory by Th∃(M). For instance,
Texp = Th(Rexp) denotes the theory of real exponentiation and Th∃(Rexp)
denotes the existential theory of real exponentiation. The definable closure
of a set Cin Mis denoted by dcl(C;M). We say that a set is definable if it
is definable with parameters, and we say that it is A-definable if we wish to
specify the set of possible parameters A. Variable tuples are denoted by x,
and we only specify their length if it is of importance. If x= (x1,...,xn),
we write E(x) for the tuple (E(x1),...,E(xn)). We denote by Nthe set
of natural numbers without 0. Throughout the rest of this note, let
(K, E)denote an o-minimal EXP-field.
Remark 1.2. The Transfer Conjecture motivates us to entirely focus on
o-minimal ordered exponential fields (K, E) whose exponential satisfies the
differential equation E′=E. For a general o-minimal ordered exponen-
tial field (F, e), it is ensured by o-minimality that eis differentiable in all
but finitely many points in F(cf. van den Dries [6, Chapter 7, Proposi-
tion 2.5]). Due to a general property of exponentials, this implies that esat-
isfies the differential equation e′=ae for some positive a∈F(cf. Krapp [11,
Corollary 2.14]). We can thus change the base of exponentiation, i.e. set
E:F→F>0, x 7→ e(a−1x), to obtain an exponential Eon Fwith E′=E,
that is, (F, E) becomes an o-minimal EXP-field. By this base change ar-
gument, most results in this note, in particular Theorem 2.6, can also be
generalised to any o-minimal ordered exponential field, e.g. o-minimal ex-
ponential fields (F, e) with the property e(1) = 2 (base-2 exponentiation as
done in Carl and Krapp [2, Section 4.2]).
2. Exponential algebraic closures
In the following, we mostly follow the terminology of Macintyre [14] and
Kirby [9] adjusted to our context. For a subring A⊆K, we say that Ais
an E-ring if it is closed under E, i.e. for any a∈Aalso E(a)∈A. The
smallest E-ring in Kis denoted by ZE. If Ais an E-ring, then we denote by
A[x1,...,xn]Ethe ring of exponential polynomials in nvariables, that
is, the smallest ring containing the polynomial ring A[x1,...,xn] such that
for any y∈A[x1,...,xn]Ealso E(y)∈A[x1,...,xn]E. (We refer the reader
to [14, Section 1.7] for an explicit construction of A[x1,...,xn]E.)
4 L. S. KRAPP
Definition 2.1. (i) Let n∈Nand let A⊆Kbe an E-ring. Moreover,
let f1,...,fn∈A[x1,...,xn]Eand let |Jf1,...,fn(x)|denote the de-
terminant of the Jacobian matrix
Jf1,...,fn(x) =
∂f1
∂x1(x)... ∂ f1
∂xn(x)
.
.
..
.
.
∂fn
∂x1(x)... ∂ fn
∂xn(x)
.
Then the Khovanskii system S(x) of f1,...,fnover Ais the sys-
tem of equations and inequations
f1(x) = ...=fn(x) = 0 and |Jf1,...,fn(x)| 6= 0.
(ii) Let Bbe a subset of Kand let A⊆Kbe the smallest E-ring
containing B. An element a1∈Kis said to be exponentially
algebraic over Bif for some n∈Nthere exist a Khovanskii system
S(x1,...,xn) over Aand a2,...,an∈Ksuch that (a1,...,an) solves
this system. The set of all elements of Kthat are exponentially
algebraic over Bis called the exponential algebraic closure of B
in (K, E) and is denoted by CLE
K(B).
Note that the condition on the determinant of the Jacobian to be non-
zero implies that Khovanskii systems only have isolated solutions. This is
a consequence of the Implicit Function Theorem in o-minimal structures
(cf. van den Dries [6, page 113]).
Lemma 2.2. Let Bbe a subset of K. Then CLE
K(B)is a subfield of Kthat
is exponentially closed in (K, E).
Proof. Let F= CLE
K(B) and let A⊆Kbe the smallest E-ring containing B.
By Kirby [9, Lemma 3.3] and Macintyre [14, Lemma 21], Fis an E-ring and a
field. Thus, E|Fdefines an order-preserving homomorphism from (F, +,0, <
) to (F>0,·,1, <). Since Eis injective, so is its restriction to F. To verify
that E|Fis also surjective, let a1∈F>0and fix n∈N,f1(x),...,fn(x)∈
A[x1,...,xn]Eand a2,...,an∈Ksuch that a= (a1,...,an) solves the
Khovanskii system S(x) of f1,...,fn. Set f0(y, x1) = E(y)−x1. Then there
is some d∈Kwith E(d) = a1, i.e. f0(d, a1) = 0. Note that |Jf0,...,fn(y, x)|=
E(y)|Jf1,...,fn(x)|. Hence, (d, a1,...,an) solves the Khovanskii system e
S(y,
x) of f0,...,fn. This yields y∈F, as required.
We now investigate the connection between the definable closure and
the exponential algebraic closure in o-minimal EXP-fields. In particular,
we show that for any model of real exponentiation (K, E) we have that
(CLE
K(∅), E) is the prime model of Texp (see Proposition 2.5).
Lemma 2.3. Let A⊆Kbe an E-ring and let S(x)be a Khovanskii system
over A. Then Sonly has finitely many solutions in K.
Proof. Since (K, E ) is o-minimal, S(x) = S(x1,...,xn) defines finitely many
connected components in Kn(cf. van den Dries [6, §3.2]). By the condition
EMBEDDING THE PRIME MODEL OF REAL EXPONENTIATION 5
that the Jacobian matrix of the system of exponential polynomial equations
in Shas to be non-singular, all solutions of S(x) in Kare isolated. Thus,
S(x) defines a set consisting of finitely many isolated points in Kn. In
particular, S(x) only has finitely many solutions.
Lemma 2.4. Let Bbe a subset of K. Then CLE
K(B)⊆dcl(B; (K, E)).
Proof. Let A⊆Kbe the smallest E-ring containing B, let a1∈CLE
K(A) and
let S(x) be a Khovanskii system over Asuch that for some a2,...,an∈K
the tuple (a1,...,an) solves S. Let C⊆Knbe the A-definable set of all
c∈Knwith (K, E)|=S(c). Lemma 2.3 shows that Cis finite.
Let C′⊆Kbe the projection of Conto the first coordinate. Then C′
is A-definable as well as finite and a1∈C′. Suppose that a1is the k-th
element of C′in the ordering on C′induced by the ordering on K. Then
there exists an Lexp-formula ϕ(x) with parameters in Aexpressing that xis
the k-th element of C′, and (K, E)|=ϕ(a1)∧ ∃!x ϕ(x). Hence, a1∈dcl(A;
(K, E)). It remains to note that any element of Ais B-definable in (K, E)
and thus dcl(A; (K, E)) = dcl(B; (K, E)).
Recall that due to definable Skolem functions in o-minimal expansions of
ordered groups, for any subset Bof Kwe have that (dcl(B; (K, E)), E)
(K, E), i.e. dcl(B; (K, E )) is the domain of an elementary substructure of
(K, E). Hence, (dcl(∅; (K, E )), E) is the unique prime model of Th(K, E )
(cf. Krapp [11, Proposition 4.75] for further details).
If (K, E) is not only assumed to be an o-minimal EXP-field but already
a model of real exponentiation, then we can strengthen the conclusion of
Lemma 2.4 to CLE
K(B) = dcl(B; (K, E)). The following result is mentioned
by Macintyre in [14, Theorem 22], who attributes it to Wilkie [19]. For the
convenience of the reader, we give a proof based on Macintyre and Wilkie [13,
Theorem 2.1].
Proposition 2.5. Let (K, E)|=Texp and let Bbe a subset of K. Then
dcl(B; (K, E)) = CLE
K(B).
In particular, (CLE
K(B), E)(K, E )and (CLE
K(∅), E)is the prime model
of Texp.
Proof. By Lemma 2.4, we only have to show dcl(B; (K, E)) ⊆CLE
K(B). Let
A⊆Kbe the smallest E-ring containing B, and let b∈dcl(A; (K, E )) =
dcl(B; (K, E)). Then there exists a function gthat is ∅-definable in (K, E)
such that g(a) = bfor some a∈A. Since (K, E) is model complete, there
is some atomic Lexp-formula ϕ(v, w) such that ∃w ϕ(v, w) defines g. After
replacing the variables, this defining formula is equivalent to one of the form
∃zp(x, y, z) = 0,
where p(x, y, z) = q(x, y, z , E(x), E(y), E (z)) for some q∈Z[x, y, z , x′, y′, z′]
(cf. e.g. Servi [17, Proposition 4.5.4] for details). Thus,
(K, E)|=∀x∀y(g(x) = y↔ ∃z p(x, y, z) = 0).
6 L. S. KRAPP
In particular, we have
(K, E)|=∃z p(a, b, z ) = 0 ∧ ∃!y∃z p(a, y, z) = 0.(2.1)
Let f(y, z) = p(a, y, z )∈A[y, z ]E=A[y, z1,...,zℓ]E. We now apply [13,
Theorem 2.1] to the setting n=r=ℓ+ 1, k= CLE
K(A) and S=V={α∈
Kn|f(α) = 0}: There exist f0,...,fℓ∈k[y, z]Eand c, d ∈Ksuch that
f(c, d) = f0(c, d) = ...=fℓ(c, d) = 0 and |Jf0,...,fℓ(c, d)| 6= 0.
This means that (c, d) solves the Khovanskii system of f0,...,fℓ, whence
c∈CLE
K(k). By (2.1) we already have c=b. In conclusion, b∈CLE
K(k) =
CLE
K(CLE
K(A)) = CLE
K(A) = CLE
K(B) (by Kirby [9, Lemma 3.3 (3)]).
The aim of this note is to show that, under the assumption of Schanuel’s
Conjecture, the prime model of Texp is embeddable into any o-minimal EXP-
field (see Theorem 3.4). This result will be deduced from the more general
Theorem 2.6. The proof of Theorem 2.6 relies on K˝onig’s Infinity Lemma,
which we introduce in the following.
Let Vbe a set and let E ⊆ P(V) such that |e| ∈ {1,2}for any e∈E.
We call the pair C= (V , E) a graph on V. The elements of Vare called
the vertices of Cand the elements of Eare called the edges of C. For any
x, y ∈Vwe say that xand yare neighbours if {x, y} ∈ E. Suppose that
Vis infinite and let Cbe a graph on V. Let (vn)n∈N∪{0}be a sequence of
pairwise distinct elements in V. We say that v0v1... is a ray in Cif for any
n∈N0we have {vn, vn+1} ∈ E. The proof of the following can be found in
Diestel [3, Lemma 8.1.2].
K˝onig’s Infinity Lemma. Let (Vn)n∈N∪{0}be an infinite family of disjoint
non-empty finite sets and let Cbe a graph on their union. Suppose that for
any n∈N, any vertex x∈Vnhas a neighbour y∈Vn−1. Then Ccontains
a ray v0v1... with vn∈Vnfor any n∈N∪ {0}.
Theorem 2.6. Let (K1, E1)and (K2, E2)be o-minimal EXP-fields. Sup-
pose that (K2, E2)|= Th∃(K1, E1). Then there exists an Lexp-embedding of
the ordered exponential field (CLE1
K1(∅), E1)into (K2, E2).
Proof. Let A= CLE1
K1
(∅). Lemma 2.2 shows that (F, E1) is an ordered ex-
ponential field. Since there are only countably many Khovanskii systems
with coefficients in ZEand by Lemma 2.3 each Khovanskii system only has
finitely many solutions in K1, the set Ais countable. Let A={a1, a2,...}
be an enumeration of A. Let (ck)k∈Nbe new constant symbols and expand
Lexp to L=Lexp ∪ {ck|k∈N}. Now (K1, E1) expands to an L-structure
K1by interpreting cK1
k=akfor any k∈N. Consider the atomic diagram
Diag(A, E1) of (A, E1) given by the set of all atomic L-formulas and neg-
ations of such satisfied in K1. By the Diagram Lemma (cf. Marker [15,
Lemma 2.3.3 i)]), we need to interpret each ckin (K2, E2) such that the
L-expansion
K2=K2, E2,cK2
kk∈N
EMBEDDING THE PRIME MODEL OF REAL EXPONENTIATION 7
satisfies K2|= Diag(A, E1), in order to obtain an Lexp-embedding of (A, E1)
into (K2, E2).
For convenience, we introduce the convention that any quantifier-free
Lexp-formula αis written as α(x), where x= (x1, x2,...) and substitut-
ing for a variable that does not appear in α(x) does not change the for-
mula. Moreover, an expression of the form α(c) stands for the L-sentence
α(c1, c2,...) and ∃xα(x) stands for existential quantification over all free
variables in α(x).
Since Diag(A, E1) is countable, we can enumerate it by
Diag(A, E1) = {ϕ1(c), ϕ2(c),...},
where each ϕi(x) is a quantifier-free Lexp-formula. For each i∈N, set
ψ′
i(x) =
i
^
ℓ=1
ϕℓ(x) and ψi=ψ′
i(c) =
i
^
ℓ=1
ϕℓ(c).
Note that any model of ψisatisfies the first isentences in the enumeration
of Diag(A, E1). Thus, for the L-theory Σ = {ψi|i∈N}, we have
Σ⊢ {ϕ1(c), ϕ2(c),...}.
Hence, it suffices to interpret each ckin K2such that K2|= Σ.
Let k∈Nbe arbitrary. Since ak∈CLE1
K1
(∅), there is a Khovanskii system
Sk(y1,...,yn) over ZEsuch that
(K1, E1)|=∃ySk(ak, y2,...,yn).
Since (K2, E2)|= Th∃(K1, E1), the set
Bk=b∈K2|(K2, E2)|=∃y Sk(b, y2,...,yn)
is non-empty. Moreover, Bkis finite by Lemma 2.3.
For any n∈N, let
Xn=n(b1,...,bn)∈Kn
2
(K2, E2)|=∃x∃yψ′
n(b1,...,bn, xn+1,...)∧
n
^
ℓ=1
Sℓ(bℓ, y)o.
Since
(K1, E1)|=∃x∃yψ′
n(a1,...,an, xn+1,...)∧
n
^
ℓ=1
Sℓaℓ, y,
we can argue as above to obtain that Xn6=∅. Moreover, since Xn⊆
B1×...×Bn, we have that Xnis finite.
Let i∈N, let πidenote the projection of a tuple onto the i-th coordinate
and let eπidenote the projection of a tuple of length at least ionto the first
icoordinates. Since for any j∈Nwe have that {ψ′
j+1} ⊢ ψ′
j, we obtain
Xi=eπi(Xi)⊇eπi(Xi+1)⊇. . . .(2.2)
8 L. S. KRAPP
Let
X=n(bk)k∈N∈KN
2
(b1,...,bn)∈Xnfor any n∈No.
We construct an infinite graph Cand then apply K˝onig’s Infinity Lemma to
show that X6=∅. The set of vertices of Cis given by the disjoint union of
all Xifor i∈Nplus some element v0not contained in any Xi. We construct
the set of edges Eof Cas follows. All vertices in X1are neighbours of v0.
Moreover, for any i∈N, a vertex w∈Xi+1 is a neighbour of a vertex v∈Xi
if eπi(w) = vor, in other words, if the tuples vand wcoincide in the first i
components. By (2.2), for any n∈Nwith n≥2, any vertex w∈Xnhas
a neighbour v∈Xn−1. Moreover, any vertex in X1has v0as a neighbour.
Thus, all conditions for K˝onig’s Infinity Lemma are satisfied, yielding that
Ccontains an infinite ray v0v1... with vi∈Xifor any i∈N. In particular,
for any i∈Nwe have
eπi(vi+1) = vi.
Finally, we can interpret
cK2
i=πi(vi)
for any i∈N. We can now show that K2|= Σ. Let i∈Nand consider
ψi∈Σ. Let N∈Nwith N≥isuch that for any j > N the constant cj
does not appear in ψi. By construction of v0v1..., we have that
K2|=∃xψ′
N(c1,...,cN, xN+1,...).
But ψ′
N(x) is of the form ψ′
i(x)∧ϕi+1(x)∧...∧ϕN(x). Thus, we have
K2|=ψ′
i(c1,...,cN)∧ ∃x(ϕi+1(x)∧...∧ϕN(x)) .
Hence, K2|=ψi, as required.
3. Prime model of real exponentiation
For the rest of this note, we denote by (P, exp) the prime model
of Texp.Since (P, exp) |=Texp, any structure containing (P, exp) as a
substructure already satisfies the existential theory Th∃(Rexp). Thus, by
Proposition 2.5 and Theorem 2.6 we obtain the following.
Corollary 3.1. There is an emebedding of (P, exp) into (K, E )if and only
if (K, E)|= Th∃(Rexp ).
Corollary 3.1 shows that o-minimal EXP-fields satisfying the existential
theory of real exponentiation contain the prime model of real exponentiation
as a substructure. Under the assumption of Schanuel’s Conjecture, any o-
minimal EXP-field satisfies Th∃(Rexp). This fact is basically due to Servi [17,
page 104] (cf. also Krapp [11, Proposition 4.57]).
Fact 3.2. Assume Schanuel’s Conjecture. Then any o-minimal EXP-field
satisfies Th∃(Rexp).
EMBEDDING THE PRIME MODEL OF REAL EXPONENTIATION 9
Remark 3.3. In light of the Transfer Conjecture, one would hope that
Schanuel’s Conjecture implies that any o-minimal EXP-fields already satis-
fies the complete theory Texp of real exponentiation. However, already for
the weaker conclusion of Fact 3.2 it is sensible to assume Schanuel’s Conjec-
ture: As discussed in [17, Section 4.5], if the existential theory Th∃(Rexp)
is decidable, then so is the complete theory Texp. Due to Fornasiero and
Servi [7] as well as Hieronymi [8], there exists a recursive axiomatisation of
the class of all o-minimal EXP-fields (cf. also [11, Theorem 4.30]). Hence,
already the conclusion of Fact 3.2 would imply the decidability of Texp .
As a result of Corollary 3.1 and Fact 3.2, we obtain the desired main
result of this note.
Theorem 3.4. Assume Schanuel’s Conjecture. Then (P, exp) embeds into
any o-minimal EXP-field.
4. Residue exponential field
In this final section, we discuss how our results above relate to another
approach towards the Transfer Conjecture via exponential residue fields.
Following the notation of Krapp [10], we denote by Rvthe valuation ring of
the natural valuation on K, by Ivits valuation ideal and by K=Rv/Ivthe
residue field. In terms of the ordering on K, we have Rv={a∈K| |a|<
nfor some n∈N}(the convex hull of Zin K) and Iv={a∈K| |a|<
1
nfor all n∈N}. The ordering on Kis given by a > 0 if and only if a > 0
and a6= 0, where a∈Rvand a=a+Iv. Due to [10, Proposition 3.2], the
residue exponential E:K→K>0, a 7→ E(a) is a well-defined exponential
on K. Moreover, [10, Theorem 3.3] shows that the residue exponential field
(K, E) elementarily embeds into Rexp. (Note that this Lexp -embedding is
unique, as any Lor-embedding of an archimedean field into Ris already
unique.) In particular, (K, E ) is a model of Texp. Hence, an affirmative
answer to the following question is equivalent to the Transfer Conjecture
(cf. [10, Proposition 3.6]).
Question 4.1. Let (K, E)be an o-minimal EXP-field. Is (K , E)element-
arily embeddable into (K, E )?
Just as for Question 1.1, the first approach towards Question 4.1 would
be to show that the residue exponential field of (K, E ) is embeddable into
(K, E). Under the assumption of Schanuel’s Conjecture, this is indeed the
case for sufficiently saturated o-minimal EXP-fields.
Proposition 4.2. Suppose that (K, E)is 2ℵ0-saturated. Then (K, E) =
Rexp. Moreover, under the assumtion of Schanuel’s Conjecture, (K, E)em-
beds into (K, E ).
10 L. S. KRAPP
Proof. Since Kis ω-saturated as an Lor -structure, the residue field Kis iso-
morphic to Rvia a unique isomorphism (see e.g. S. Kuhlmann, F.-V. Kuhl-
mann, Marshall and Zekavat [12, Theorem 6.2]). By [10, Theorem 3.3], we
already obtain (K, E ) = Rexp.
Assuming Schanuel’s Conjecture, Fact 3.2 implies that (K, E)|= Th∃(
Rexp). Equivalently, Rexp satisfies the universal theory of (K, E ). Thus,
Rexp can be embedded into some (K1, E1)|= Th(K, E) (see e.g. Sarbadhi-
kari and Srivastava [16, Proposition 2.4.2]), where K1can be chosen of car-
dinality |K1|= max{|R|,|Lexp|} = 2ℵ0by the L¨owenheim–Skolem Theorem.
Now (K, E) is 2ℵ0+
-universal, and thus (K1, E1) elementarily embeds into
(K, E). Hence, Rexp embeds into (K, E).
Let (F, E) be a 2ℵ0-saturated elementary extension of (K, E). We obtain
the following chain of substructures (where each elementary embedding is
unique):
(P, exp) (K , E)Rexp = (F , E )
Assuming Schanuel’s Conjecture, Theorem 3.4 and Proposition 4.2 respect-
ively give us Lexp-embeddings
ϕ: (P, exp) ֒→(K, E) and ψ:Rexp ֒→(F, E ).
In conclusion, we obtain the following picture:
(K, E) (F, E)
(P, exp) (K , E) (F , E)
ϕ
ψ
Acknowledgements. Several results of this work were part of my doctoral
thesis [11]. In this regard, I wish to thank my supervisor Salma Kuhlmann
for her continuous help and support. I also thank Alessandro Berarducci for
an insightful discussion about [11, Corollaries 4.59 and 4.90], which encour-
aged me to compose this note.
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Fachbereich Mathematik und Statistik, Universit¨
at Konstanz, 78457 Kon-
stanz, Germany
Email address:sebastian.krapp@uni-konstanz.de
