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1.5.
THE TSUBOI AND KIM THEOREMS
111
By the induction hypothesis,
A
. Thus, by non-forking transitivity, we have
a
(
tp(
2j+1)j
a
2j+1)j
<i
a
2i+1a2i+2
is independent overA,
(4) tp((
Since
also
independent overA. Again by
/A
<i
/
A ∪ {
a
2i+1a2i+2
a
2j+1
a
tp((
2j+1)j
) ⊃nftp((a
(4)
| j
< i}) ⊃nftp(
/A
<i
2i+1)j
(4)
shows that
,
a2i)
Thus we have
a
Since
tp(
2i+2
b
andcare
/
a
A ∪ {
| j ≤ i}) ⊃nftp(a
2j+1
independent overA,
a
2i+1
independent overA. Hence we have
a
2i+2
/
tp(
Thus the sequence
Now we obtain the independent set
by the transitivity ofR, each
a
2i+1
are
dependent overA,
and
A ∪ {
a
{
2j+1
a
| j ≤ i}) ⊃nftp(a
2j+1
| j ≤ i}
tp(
∪ {
a0a
i ∈ ω
a
}
is
2i+2
a
{
2i+1
)
b
2i+1
elongs toR. Hence
. This is a contradiction to
the assumption thatTis pseudo-supersimple.
do
es not fork over
/A).
<i
a
(
2j+1)j≤i
a
2i+1a2i+2
/Aa
2i+2
and
2i+2
/
2i+1
a
2i+2
A).
/
).
A).
are
also
independent overA.
| i ∈ ω}
o
verAand,
¤
is
a
0
1.5.2.10. Corollary.
T
,
n ∈ ω,T
n+1
R {tp(
Proof.
ab) | (a,b) ∈ SIpand
be the union of all
If we assume
Let
Tnbe pseudo-supersimple theories,
wR(p) ≥ 2
1.5.2.9, we obtain a sequence
the following conditions:
(1)
(2) {
both
a
2j+1
a2ia
tp(
| j ≤ i}
2i+1
)
∪ {
and
a
2i+2
Moreover, there exist formulas
symmetric and such that
(3) |= ϕj(
(4)
every type
b
elongs toR,
a2i, a
j = 1, 2
),j =
2i+j
x, y) ∈ S(T )
q(
.
1, 2
Tn, andpbe a type in
(b, a) 6∈ SIp}
6= ∅
, as in the proof of Proposition
ai)
(
tp(a2ia
}
ϕj(
of
i∈ω
2i+2
are
independent.
x, y)
,
realizations ofpsatisfying
)
b
elong toR;
witnessing that
;
,
containing
then
x) ∪
p(
Tn⊆
S(T)
wR(p) = 1
SIpis non-
x, y)}
{ϕj(
. If
.
,
112
Chapter
1. CHARACTERIZATION OF EHRENFEUCHTNESS
Notice that for
witness that
SIpis non-symmetric. By Proposition 1.5.2.6, this
x, y) ϕ1(x, y)∨ϕ2(x, y)
ψ(
implies that each element of the sequence
a0and
with
formula of
that all
this also true for the theory
Tn. But this is a contradiction, since we are assuming
Tnare pseudo-supersimple.
¤
,
all
ψk(
a
(
2i+1)i∈ω
Tn, where
x, y),k
is
dependent
x, y)
ψ(
> 1
is
Now we are ready to prove a generalization of both the Tsuboi
theorem for unions of pseudo-superstable theories [461] and the
Kim theorem for supersimple theories [257].
1.5.2.11. Theorem.
theories
I(T, ω) ≥ ω
Tn, where
Proof.
.
Suppose thatTis the Ehrenfeucht theory. By Lemma
1.1.1.2, there exists a powerful type
1.1.1.10 any nonempty relation
LetTbe a union of pseudo-supersimple
Tn⊆ T
n+1
,
n ∈ ω
p(
. Then
x) ∈ S(T )
I(T, ω) = 1
.
In view of Lemma
SIpis non-symmetric. By Corollary
or
1.5.2.8, the set
R {tp(
is
a transitive forking class on
wR(p) = 1
.
ab) | (a,b) ∈ SIpand
{p}
At the same time, there are independent realizations
ofp.
Since each type
q ∈ S(T )
is realized in the model
by Lemma 1.1.1.10, there exist realizations
a, d) ∈ Ipand
(
c
in
M(d)
d)
.
M(
Prop
c
are
Thus
osition 1.5.2.6 implies that
dependent. Hence we have
contradiction.
(d, a) 6∈ SIp,
and
then consider an elementary extension
a, b) ∈ Ip,
(
we can nd that independent
(b, a) 6∈ SIp,
a
andbare
wR(p) ≥ 2
¤
(b, a) 6∈ SIp}
. Corollary 1.5.2.10 implies
b
and
Mpand,
a,d
ofpsuc
(a, c) ∈ Ip,
dependent, and
(c, a) 6∈ SIp.
h that
M(
b
a
a)
and
of
and
, which leads to a
,
a
c
Chapter
2
GENERIC CONSTRUCTIONS
2.1. Semantic generic constructions
At rst we recall a semantic approach to generic constructions.
2.1.0.1. Denition
class of nite structures of at most countable predicate language,
endowed with the empty set and a partial order relation6which is
invariant under the transition to isomorphic structures, connoting
the property of being a
ture
. The class
tion
if it satises the following axioms:
(1) if
A 6 B
(2) if
A 6 C,B ∈ K0, and
(3)∅is the least element of the system
(4) (the
B,C ∈ K0, having embeddings
that
f0(A) 6 B
and embeddings
g1(C) 6 D
gamofB
With the partially ordered class
and with at most countably many isomorphism types, taking nite
structures in
structuresBandCover
the amalgamation property), we construct step-by-step a countable
(K0; 6)
for
-generic structureM, that is also called a
(K0; 6)
(K0; 6)
amalgamation property
and
f0◦ f1= g0◦ g1; the structureDis called the
andCover the structureAand the four
K
0
), i. e., a structure satisfying the following conditions:
[96, 206, 269, 471]. Let
self-sucient structure
is called
, then
A ⊆ B
and
g0(A) 6 C
f1:
B → D
and using the
A
generic,generative
;
A ⊆ B ⊆ C
and
amalgamation
in structuresDso as to comply with
K
be a nonempty
0
, or
strong substruc-
, or
, then
) for any structures
f0:
A → B
, there are a structure
g1:
C → D
(K0; 6)
A 6 B
(K0; 6)
and
for which
, being dened above
(i. e., embedding the
;
g0:
A → C
(f0, g0, f1, g1)
Hrushovski limit
amalgama-
;
A,
such
D ∈ K
f1(B) 6 D
amal-
.
0
,
114
Chapter
2. GENERIC CONSTRUCTIONS
(a) for any nite substructure
B ∈ K0,
structure
A ⊆ B ⊆ M
B0∈ K0with
, for which
B ⊆ B0⊆ M
(b) for any nite substructure
B ∈ K0such that
A 6 B
, there is a structure
A ⊆ M
B 6 M
;
A ⊆ M
, there is a structure
, i. e.,
B 6 B0for any
and any structure
B06 M
for which
B 'AB0.
Note that for particular cases, when a generic class
have the relation6which is equal to⊆(i. e.,
substructures), the class
Hrushovski limit is called the
(K0; 6)
Frasse limit
is called the
K
Frasse class
[17, 28]. In such a case
is closed under
0
(K0; 6)
and the
nite structures can be replaced by nitely generated structures.
By the denition of Hrushovski limit, the following theorem
holds (Theorem 2.12 in J. T. Baldwin and N. Shi [96]).
2.1.0.2. Theorem.
most countably many isomorphism types, there exists a
For any generative class
(K0; 6)
, having at
(K0; 6)
generic structure.
The scheme above represents a
constructing a generic structure
generic theory
Th(M)
.
semantic approach
M
and the corresponding
to
The utility of the semantic approach for realizations of
desired model-theoretic properties has conrmed by numerous ex-
amples (see the bibliography, reected in the historical survey) for
the cases when predicates are independent, i. e., are not denable
in terms of each other.
-
2.2. Syntactic generic constructions
In constructing generic structures in which some predicates are
a priori denable via other ones, it is more preferable (and some-
times inevitable) to use a
plete or incomplete types over nite sets containing some external
information on elements are treated rather than nite structures.
syntactic approach
, within which com-
2.2.
SYNTACTIC GENERIC CONSTRUCTIONS
115
The syntactic approach for creating generic theories, written
in this Section, generalizes the semantic approach, described above,
as well as syntactic interpretations for concrete semantic construc-
tions [100, 101, 111, 116, 118, 225, 226, 349, 350, 351] etc., and
also leads to creating generic structures. This approach will be
used below, in Chapters 3{7, for creating generic theories, repre-
senting all possible distributions of binary semi-isolating formulas,
all possible stable and unstable theories with respect to Rudin{
Keisler preorders and distribution functions of numbers of limit
models. Examples of this and following Sections show that the
syntactic approach forms a proper generalization of semantic one
for constructions of generic structures.
We begin by discussing collections of formulas in rst order
logic over a vocabularyΣ. Thus, as usual,`means proof from
no hypotheses deducing
` ϕ
for a formulaϕof vocabularyΣ,
which may contain function symbols and constants. If deducingϕ,
hypotheses in a setΦof formulas can be used, we write
Φ ` ϕ
.
UsuallyΣwill be xed in context and not mentioned explicitly.
Below in this Section, we write
ables, and denote by
A, B, C, . . .
X, Y, Z, . . .
for nite sets of vari-
nite sets of elements, as well as
nite sets in structures, or else the structures with nite universes
themselves.
In diagrams,
disjoint from the constant symbols inΣand
with the constants fromAadjoined.
Σ
-diagrams
Σ(A)-,Σ(B)-,Σ(C)
are called
universes
Below we assume that for any considered diagram
are distinct elements inAthen
A, B, C, . . .
denote nite sets of constant symbols
Σ(A)
Φ(A), Ψ(B), X(C)
(of setsA,B,C), that is,
-sentences
, respectively. These setsA,B,
of correspondent diagrams.
¬(a1≈ a2) ∈ Φ(A)
is the vocabulary
stand for
consistent sets of
C
Φ(A)
, if
a1, a
. This means
that ifcis a constant symbol inΣ, then there is at most one
element
2.2.0.1. Denition.
obtained by replacing a subset
disjoint fromΣand with
a ∈ A
such that
(a ≈ c) ∈ Φ(A)
We denote by
A0⊆ A
|A0| = |B0|
.
[Φ(A)]
by a set
, where
A
the diagram
B
B0⊆ B
of constants
A \ A0= B \ B0.
Φ(B)
2
116
Chapter
2. GENERIC CONSTRUCTIONS
Similarly we call the consistent set of formulas denoted by
the type
Φ(A)
that
denote the diagram
2.2.0.2. Remark.
then diagrams
Φ(X)
if it is the result of a bijective substitution into
of variables ofXfor the constants inA. In this case, we say
Φ(B)
is a
copyofΦ(A)
Φ(A)
by
and a
[Φ(X)]
representativeofΦ(X)
X
.
A
If the vocabulary contains functional symbols
Φ(A)
containing equalities and inequalities of terms
[Φ(A)]
. We also
A
X
can generate both nite and innite structures. The same eect is
observed for purely predicate vocabularies if it is written in
that the model for
Φ(A)
should be innite. For instance, diagrams
Φ(A)
containing axioms for nitely axiomatizable theories [38] have this
property. For diagrams of the predicate vocabularies considered,
in particular, in Chapters 4 and 5, each diagram is realized by
a nite structure. Thus, the required structures are locally nite
with respect to the considered diagrams.
By the denition, for any diagram
bol inΣappears in some formula of
considered as
Φ(A ∪ K)
, where
K
Φ(A)
, each constant sym-
Φ(A)
. Thus,
Φ(A)
can be
is the set of constant symbols
inΣ.
We now give conditions on a partial ordering of a collection of
diagrams which suce for it to determine a structure. We mod-
ify some of the conditions for structures bydto signify they are
conditions on diagrams, not structures.
2.2.0.3. Denition.
say that
Σ
-diagrams of nite sets so that
(D0; 6)
LetΣbe at most countable vocabulary. We
(or
D
) is
genericorgenerative
0
D
is partially ordered by a binary
0
if
D
0
is a class of
relation6such that6is preserved by bijective substitutions, i. e.,
,
[Ψ(B)]
0
A
= Φ(A0)
0
A
B
B
are in
0
and
D
if
Φ(A) 6 Ψ(B)
[Φ(A)]
1
Note
B
= Ψ(B0)
0
B
A
A
that
[Ψ(B)]
and
by bijective substitutions and6is reexive.
, and
A0⊆ B0such that
are dened, then
6 [Ψ(B)]
0
D0is closed under bijective substitutions since6is preserved
B
.1Furthermore:
0
B
[Φ(A)]
[Φ(A)]
A
A
0
2.2.
SYNTACTIC GENERIC CONSTRUCTIONS
117
(i) results of bijective substitutions
ables for constants inAinto diagrams
0
either
then
2 3
ϕ(a) ∈ Φ(A)or¬ϕ(a) ∈ Φ(A)
Φ ⊆ Ψ
5
;
Φ ⊆ Ψ ⊆ X
Φ0(∅)
is the least element of
form a countable set;
(ii) if
Φ(A) ∈ D
an
y tuple
(iii)
(iv) if
a ∈ A
if
Φ 6 Ψ
Φ 6 X,Ψ ∈ D0, and
(v) some diagram
D0\ {Φ0(∅)}
is nonempty;
(vi) (thed-amalgamation property
[Φ(A)]
Φ(A) ∈ D0(over all setsA)
) for any diagrams
Ψ(B), X(C) ∈ D0, if there exist injections
C
with
[Φ(A)]
a diagramΘ(D)∈
for which
A
f0(A)
[Ψ(B)]
6 Ψ(B)
D
B
f1(B)
g0◦ g1; the diagram
X(C)
(f0, g0, f1, g1)
∃x ϕ(x)
an element
over the diagram
;
(vii) (the
local realizability property
, then there are a diagram
b ∈ B
for which
and
[Φ(A)]
and injections
0
6 Θ(D),[X(C)]
Θ(D)
is called the
Φ(A)
and witnessed by the four maps
Ψ(B) ` ϕ(b)
A
g0(A)
f
1
C
g1(C)
) if
Ψ(B) ∈ D0,
A
of sets
X
, then
f0:
6 X(C)
Φ 6 Ψ
A → B
, then there are
:B→Dand
6 Θ(D)
amalgam
Φ(A) ∈ D0and
Φ(A) 6 Ψ(B)
;
X
ϕ(
4
;
;
(D0; 6)
and
g0:
:C→
g
1
and
f0◦ f1=
of
Ψ(B)
of vari-
x)
and
, and
Φ(A),
A →
D
and
Φ(A) `
, and
2
The
condition (i) (as well as countableΣ) is assumed to get at most
countable generic structure. Sometimes it can be omitted obtaining structures
of arbitrary cardinalities. At the same time, as K. Zh. Kudaibergenov [268]
showed, there are classes
such that correspondent generic classes of nitely generated structures do not
have Frasse limits.
3
The class
tives for
4
Assuming
x)
and
ϕ(
formula
vice
versa, where
substituting
w
e get
ϕ(
5
Note
Φ(A)
, so
whence
D0can be replaced by a set of diagrams containing representa-
D0. Having (i) this set can be chosen countable.
a) ∈ Φ(A)or¬ϕ(a) ∈ Φ(A)
ϕ(
a ∈ A
,
we have
x)
and
ϕ(
a) ∈ Φ(A)or¬ϕ(a) ∈ Φ(A)
that
Φ(A) 6 Ψ(B)
a ∈ B
a ∈ A ∪ K
c ∈ K
c ∈ K
Φ(A) 6 Ψ(B)
.
D0with uncountably many quantier-free diagrams
for
any quantier free formula
a) ∈ Φ(A)or¬ϕ(a) ∈ Φ(A)
ϕ(
since
.
Indeed, taking a quantier free formula
foryw
e have
ψ(
implies
implies
Φ(A) ⊆ Ψ(B)
x)
any
x, c)
,
i. e.,
A ⊆ B
can
ϕ(
,
therefore considering
a, c) ∈ Φ(A)or¬ψ(a, c) ∈ Φ(A)
ψ(
, since if
and we have
for
any quantier free
be considered as
ψ(
a ∈ A
then
(a ≈ a) ∈ Ψ(B)
x, c)
ψ(
ψ(
x, c)asϕ(x)
and
x, y)
and
(a ≈ a) ∈
,
.
,
118
Chapter
2. GENERIC CONSTRUCTIONS
(viii) (thed-uniqueness property
Ψ(B) ∈ D0if
Φ(A) = {ϕ(
2.2.0.4.
Φ(A), Ψ(A) ∈ D0, then
A ⊆ B
and the set
b) ∈ Ψ(B) | b ∈ A}
Remark.
Note that if
Φ(A) = Ψ(A)
) for any diagrams
Φ(A) ∪ Ψ(B)
.
M |= Φ(A)
. Indeed,
is consistent then
and
M |= Ψ(A)
Φ(A) ∪ Ψ(A)
Φ(A),
for
is
consistent (witnessed byM), so by thed-uniqueness property we
have
Φ(A) = {ϕ(
a) ∈ Ψ(A) | a ∈ A} =
Ψ(A).
The following examples, suggested by J. T. Baldwin, illustrate
and clarify the motivation for Denition 2.2.0.3.
2.2.0.5. Example.
<
is a linear order andfis a unary function. Taking a set
and assuming that
Φ(A)
, we get that
all
n ∈ ω
. Thus, the nite setAgenerates an innite structure.
1. Consider a vocabulary
∀x(f(x) > x)
Φ(A)
contains the formulas
and
(a ≈ a)
Σ = {<, f }
, where
A = {a}
belong to a diagram
n+1
f
(a) > fn(a)
for
2. Relational version of Ehrenfeucht's example: LetΣconsists
of unary predicate symbols
bol<, where
Pkare disjoint singletons and<is a dense linear order
without endpoints such that if
then
ak< am. Quantier free diagrams
Pk,
k ∈ ω
ak∈ Pkand
, and a binary predicate sym-
am∈ Pmfor
Φ(A)
describing relation-
k < m
ships between elements in arbitrarily large nite setsAallows one
to construct a structure whose theory has three countable models
3. At the same time, the vocabularies for original Ehrenfeucht's
examples
stants and can not be represented by generative classes
Tnwithncountable models contain innitely many con-
K
with
0
diagrams describing nite structures. This formulation with in-
nitely many constants require generative classes
D
of the form
0
in Denition 2.2.0.3.
A crucial point for the Ehrenfeucht's example is that anyΣ-
diagram
Φ(A)
consists of asserting that the set
{ci| i ∈ ω}
of
constants have order typeωand species how the nite setAis
interpolated in the
ci. Eventually a union of nite setsAform a
dense order and such a union can be taking as universe for the
required structure.
¤
,
2.2.
SYNTACTIC GENERIC CONSTRUCTIONS
119
2.2.0.6. Denition.
of a diagramΨif
embeddable
such that
f
, in this instance, is called a (
in diagram
A diagram
ture
MifΦ(A)
where
Ψ(B)
, in this event, is called a (
in structure
in a diagram
[Φ(A)]
A
f(A)
Ψ(B)
Φ(A)
is (strongly) embeddable in some diagram
M |= Ψ(B)
M
and is denoted byf:
If a (strong) embeddingfequals
tical
.
2.2.0.7. Denition.
of structures of some language, and
class
D
is
conal
0
there are a nite setB,
such that
M |= Φ(B)
conal in every structure of
all structures
M
byPa subclass of
A diagramΦis called a
Φ 6 Ψ
. A diagram
Ψ(B)
if there is an injectionf:
⊆ Ψ(B)([Φ(A)]
Φ(A)
A
f(A)
6 Ψ(B)
strong subdiagram
is said to be (
strong)embedding
and is denoted byf:
is said to be (
Φ(A) → Ψ(B)
strongly)embeddable
. The corresponding embeddingf:
strong)embedding
Φ(A) → M
.
idA, as usual it is called
Let
D
be a class of diagrams,
0
M
be a structure in
in the structure
A ⊆ B ⊆ M
. The class
P
with the condition that
K(D0)
such that each diagram
M
if for each nite set
, and a diagram
D
is
0
. We denote by
0
D
conal
K(D0)
is conal inM, and
0
for some structure inP.
Now we extend the relation6from the generative class
to a class of subsets of structures in the class
Let
M
be a structure in
with
A ⊆ B
M
), and write
for which
. We callAa
A 6 B
Φ(A) 6 Ψ(B)
, if there exist diagrams
A nite setAis called a
the structureM), where
such that
A ⊆ B ⊆ M0and
Φ(A), Ψ(B) ∈ D0with
then, as above, we write
toAas a
self-sucient set
K(D0),A
strong subset
and
M |= Ψ(B)
strong subset
A ⊆ M0, if
Φ(A) ⊆ Ψ(B)
M |= Ψ(B)
A 6 M0. If
(inM).
andBbe nite sets in
of the setB(in the structure
A 6 B
. IfAis a strong subset of
A 6 MinM
K(D0)
Φ(A), Ψ(B) ∈ D0,
.
of a set
for any nite set
for some diagrams
strongly
A → B
). The injection
of diagram
Φ(A)
.
in a struc-
Ψ(B)
Φ(A) →
of diagram
Φ(A)
iden-
P
be a class
0
P
. The
0
A ⊆ M
Φ(B) ∈ D
in
P
if
D
0
0
the class of
Φ ∈ D0is true
(D0; 6)
.
M
M0⊆ M
(in
M
then we refer
)
,
,
0
is
B
0
120
Chapter
2. GENERIC CONSTRUCTIONS
Notice that, by thed-uniqueness property, the diagrams
and
Ψ(B)
dened uniquely. A diagram
sucient setAinM, is said to be a
2.2.0.8. Denition.
its subdiagram
D
of diagrams
0
Φ1(A) ` Φ(A)
in terms of cardinalities
that
|Φ1(A)| ≤ f (|A|)
that the
determines
specied in the denition of strong subsets are
Φ(A) ∈ D0, corresponding to a self-
self-sucient diagram
We say that a diagram
Φ1(A) ⊆ Φ(A)ifΦ1(A) ` Φ(A)
Φ(A)
, the cardinalities
and their nite subdiagrams
|Φ1(A)|
|A|
if there is a function
Φ(A)isdeduced
holds. For a class
are
uniformly bounded
for any nite setA. In such a case, we say
uniformly locally nite class
D
.
0
0
D
of subdiagrams
0
Φ1(A)
f ∈ ωωsuch
Φ(A)
(inM).
from
with
Φ1(A)
The following proposition, generalizing Lemma 2.8 in
J. T. Baldwin and N. Shi [96], shows that for nite sets corre-
sponding to generative classes whose diagrams are generated by
uniformly nite diagrams, the condition of being self-sucient is
type-denable.
2.2.0.9. Proposition.
termined by a uniformly locally nite class
a structure in the class
there exists a type
ΓA(X)
A0⊆ M,M |= ΓA(A0)
Let
D
K(D0)
such that
implies
be a generative class which is de-
0
. Then for each nite set
M |= ΓA(A)
A06 M
.
0
D
, and let
0
M
A 6 M
, and for each set
be
,
Proof.
in
D
0
grams
nite
[[Ψ(B)]
,
Ψ0(B) ∈ D
LetAbe a self-sucient set inM,
M |= Φ(A)
Ψ(B) ∈ D0with
B\A
A
]
, where
X
Y
. For any setB,
Φ(A) 66 Ψ(B)
0
. Then the formula
0
X ∩ Y = ∅
. The required type is
Φ(A)
A ⊆ B ⊆ M
, let
Ψ(B)
be implied by the
∧Ψ0(X, Y )
be a diagram
, and any dia-
implies the type
ΓA(X) Φ(X) ∪ {∀Y ¬ ∧ Ψ0(X, Y ) | A ⊆ B, Φ(A) 66 Ψ(B)}. ¤
2.2.0.10. Denition.
ding property
is a diagram
embeddable in
(JEP) if for any diagrams
X(C) ∈ D0such that
X(C)
.
A class
(D0; 6)
Φ(A)
possesses the
joint embed-
Φ(A), Ψ(B) ∈ D0, there
and
Ψ(B)
are strongly
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