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Файл:Classification of countable models of complete theories. Р.1. Monograph in two parts
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1.1.
SYNTACTIC CHARACTERIZATION OF THEORIES
1.1.1.11. Denition
(may be incomplete)n-type over a set
a theoryT,Ba set in the modelM. A
inpis the set
a tuple
M
b ∈ B
b, c)
|= ϕ(
A
quasi-neighb
QV
and
.
QV
A
quasi-neighbourhood ofBin
(B. S. Baizhanov [85]). Let
(B)
p,M
a
of all tuples
(tp(b/A), p)
ourhood ofBin
n
(B)
A,M
p∈Sn(A)
c ∈ M
-preserving formula
Sn(A)
[
QV
S(A)
[
QV
A,M
(B)
QV
n∈ω
A ⊆ M
quasi-neighbourhood of
is a set
p,M
is a set
n
A,M
x)
p(
in a model
suc
h that there exist
ϕ(
(B).
(B).
b
e some
x, y)
M
with
31
of
B
,
a = ha1,
QV
A,M
. . . , ani
(a)
)
, we write
instead of
({a1, . . . , an})
A,M
For tuple
n
QV
A,M
n
(
QV
A,M
a)
(
({a1, . . . , an}),QV
Obviously, any quasi-neighbourhood of form
a)
p
is an-type and
a ∈ QV
n
A,M
(a)
M |= p(
.
At the same time the set
,
is nonempty:
QV
QV
a)
(
p,M
QV
p,M
).
QV
a ∈ QV
(B)
p,M
(resp
ectively
({a1, . . . , an})
a)
,
(
(a)
where
.
Thus,
p,M
p,M
can be empty
(for instance, one can take the empty set forBand a nonprincipal
type forp).
a
Notice that for any tuples
b
isolates
if
and only if
b ∈ QV
andbinM,
(
a)
b),M
tp(
the tuple
.
In particular, the relation
SIpon the set of realizations of typepin the model
a, b)
with the set of pairs
The
reexivity and the transitivity of the semi-isolation corre-
suc
(
h that
M |= p(
a)
and
b ∈ QV
M
a
semi-
coincides
(a)
p,M
spond to the following properties:
QV
and
1. Let
p,M
2.
c ∈ QV
Th
a ∈ M
a)
.
(
Letq,rbe types in
r
,M
us, the relation
b
e a realization of type
S(A)
b)
.
(
Then
c ∈ QV
a ∈ QV
,
p,M
r
a
,M
(b)
a
a)
(
p ∈ S(A)
tuple inM,
. Then
b ∈ QV
q
.
is
a preorder a preorder on
,M
a ∈
a)
(
the set of realizations ofpinM. By the same way we get that the
.
,
32
Chapter
1. CHARACTERIZATION OF EHRENFEUCHTNESS
relation
a
preorder on the set of all tuples inM.
a ∈ QV
n
A,M
(b)
is
a preorder on
Mn, and
a ∈ QV
The transitivity property above implies that if
a)
QV
p,M
The
then
(
non-symmetry of relation
the typepin the model
for
some tuple
QV
p,M
b ∈ QV
(b) ⊆ QV
M
means that
(a)
p,M
.
(a)
p,M
.
SIpon the set of realizations of
p,M
(
b) ⊂ QV
QV
Thus, Lemma 1.1.1.10 admits the
following reformulation:
1.1.1.12. Lemma.Ifp
a model
such
1.1.1.13.
n
-type over a set
inM. The
M
that
via some tuple
p,M
b) ⊂ QV
(
QV
Denition
A ⊆ M
neighbourhood ofBin the typepis the set
consisting of all tuples
a tuple
that
M |= ϕ(
The
b ∈ B
set
b, c)
and
.
is a nonprincipal powerful type realized in
a
then
there exists a tuple
(a)
p,M
[87]. Let
c ∈ M
a
(tp(b/A) ↔ tp(c/A))
n
V
(B)
A,M
.
x)
p(
in a model
suc
h that
[
p∈Sn(A)
b
e some (may be incomplete)
M
of a theoryT,Ba set
c)
M |= p(
and
-formula
V
(B)
p,M
A,M
b ∈ QV
V
there exist
x, y)
ϕ(
p,M
p,M
p,M
(b)
suc
is
b ∈
(a)
(a)
(B)
h
is the
is the
neighbourhood of setBin
The set
V
A,M
neighbourhood of setBin
(B)
Sn(A)
[
V
n∈ω
S(A)
.
n
A,M
.
(B)
Note the following easy properties of neighbourhoods.
a
1. Letp,qbe types in
M
of a theoryT,
exists a
a theoryT,
2.
(p ↔ q)
Let
-formula
a
b
e a realization of a type
A ⊆ M
A ⊆ M
. Then
Sn(A)
. Then
x, y)
ϕ(
a ∈ V
,
b ∈ V
suc
p,M
a
realization ofpin a model
a)
if
(
q
,M
h that
M |= ϕ(
p ∈ Sn(A)
(a)
and
a ∈ V
and only if there
a, b)
.
in a model
n
(a)
A,M
M
of
.
1.1.
SYNTACTIC CHARACTERIZATION OF THEORIES
a
3. Let
realization of a type
A ⊆ M
then
M
c ∈ V
. If
b ∈ V
4.
Letpandqbe types in
of a theoryT,
q
,M
Thus, the relation
set of realizations of typepin the modelM, as well as
is
an equivalence relation on the set
b
e a realization of a type
a ∈ V
n
A,M
(a)
q ∈ Sn(A)
(b)
p,M
.
then
, both in a model
b ∈ V
q
,M
S(A)
A ⊆ M
(B)
. A similar property is satised for
a ∈ V
. If
p,M
b ∈ V
p,M
(b)
is
an equivalence relation on the
p ∈ Sn(A)
a)
.
(
Besides, if
M
and
of a theoryT,
a ∈ V
andBbe a set in a model
(B)
and
V
n
A,M
c ∈ V
and for
q
,M
b)
(
a ∈ V
Mn, and
a ∈ V
A,M
(b)
equivalence relation on the set of all tuples in the modelM.
Finally we obtain that, for instance, the relation
mo
dulo the equivalence relation
a ∈ V
A,M
(b)
forms
a ∈ QV
a partial order
on the set of equivalence classes of tuples inM.
If a theory has a nonprincipal powerful typepthen that pre-
order contains innite chains.
1.1.2. Rudin{Keisler preorders
b
n
A,M
V
n
A,M
is
A,M
33
b
e a
(b)
then
A,M
(b)
an
(b)
.
In what follows in this Section, unless otherwise stated, we deal
with a class of small theories only.
1.1.2.1. Denition.
Letpandqbe types in
S(T)
. We say that
the typepis dominated by a typeq, orpdoes not exceedqunder
the Rudin{Keisler preorder
is,
Mpis an elementary submodel of
Besides, we say that a model
or
Mpdoes not exceed
write
Mp≤RKMq.
Syntactically, the condition
Mp≤RKMq) is expressed thus: there exists a formula
that the set
x)
.
p(
Since we deal with a small theory (there are only count-
q(
y) ∪
{ϕ(
ably many types over any tuple
a
with parameters in
parameters in
a),ϕ(x, y)
is
(written
p ≤RKq
), if
Mq(written
Mq|= p
Mp¹ Mq).
Mpis dominated by a model
Mqunder the Rudin{Keisler preorder
x, y)}
p ≤RKq
is
consistent and
a
and
so any consistent formula
(and hence also
x, y)
ϕ(
q(
y) ∪
{ϕ(
x, y)}
deducible from a principal formula with
can
be chosen so that for any formula
, that
Mq,
, and
suc
h
`
34
x, y)
ψ(
that
q(
is
said to be
Chapter
,
the set
y) ∪
{ϕ(
(q, p)
1. CHARACTERIZATION OF EHRENFEUCHTNESS
y) ∪
q(
x, y)}
-principal
x, y),ψ(x, y)}
{ϕ(
x, y)
` ψ(
.
b
eing consistent implies
.
In this event the formula
ϕ(
x, y)
1.1.2.2. Denition.
equivalent,realization-equivalent,Rudin{Keisler equivalent
equivalent
Mpand
equivalent
(written
Mqare said to be
, or
RK
As in [451], typespandqare said to be
Typespandqare said to be
p ∼RKq
) if
p ≤RKq
and
domination-equivalent,Rudin{Keisler
-equivalent
(written
Mp∼RKMq).
domination-
, or
RK
q ≤RKp
. Models
strongly domination-
equivalent,strongly realization-equivalent,strongly Rudin{Keisler
equivalent
some realizations
and
tp(
, or
a/b)
strongly
are
principal. Models
RK
a
andbofpandqresp
-equivalent
(written
Mpand
p ≡
ectively, both
RK
q
) if for
tp(
Mqare said to be
b/a)
strongly domination-equivalent,strongly Rudin{Keisler equivalent
or
strongly
RK
-equivalent
(written
Mp≡RKMq).
Clearly, domination relations form preorders, and (strong) do-
mination-equivalence relations are equivalence relations. Here,
Mp≡RKMqimplies
If
Mpand
Mqare not domination-equivalent then they are
Mp∼RKMq.
non-isomorphic. Moreover, non-isomorphic models may be found
among domination-equivalent ones.
In Ehrenfeucht examples, models
M
n
, . . . , M
p
0
n
p
n−3
are
domination-equivalent but pairwise non-isomorphic.
A syntactic characterization for the model isomorphism be-
tween
serts that the existence of an isomorphism between
Mpand
Mqis given by the following proposition. It as-
Mpand
Mqis
equivalent to the strong domination-equivalence of these models.
-
,
x)
1.1.2.3. Proposition.
For any types
p(
and
theoryT, the following conditions are equivalent:
(1)
(2)
(3)
x, y)
ϕ
(
q
,p
is
consistent;
the models
the models
there exist
r
espectively, such that the set
Mpand
Mpand
(p, q)
- and
x) ∪ q(y) ∪
p(
Mqare isomorphic;
Mqare strongly domination-equivalent;
(q, p)
-principal formulas
{ϕ
p,q
(y,
x),
x, y)}
ϕ
(
q,p
q(y)
ϕ
of
p,q
a small
x)
(y,
and
1.1.
SYNTACTIC CHARACTERIZATION OF THEORIES
(4)
there exists a
such
that the set
is
consistent.
Proof.
realizations
(1) ⇒ (3)
a
andbof
If there is an isomorphism between
tence of
satisfying
(p, q)
- and
the condition that
(p, q)
- and
x) ∪ q(y) ∪
p(
. Let
M(
types
(q, p)
p(
-principal formulas
a)
x)
(q, p)
-principal formula
x, y)}
{ϕ(
and
M(b)
and
q(y)
M(
b
e prime models over
,
respectively.
a)
and
M(b)
,
the exis-
x)
and
ϕ
(y,
p,q
ϕ
q
ϕ(
,p
x, y)
x, y)
(
35
,
,
x) ∪ q(y) ∪
p(
is
consistent, follows from the facts that
just
principal types over
for
some tuple
(3) ⇒ (1)
formulas
is
consistent. We argue to show that
where
is
{ϕ
q(
As
r2(y,
Mp= M(
(p,
q)
-principal and
y,x)}
(
p,q
y)∪
{ϕ
q,p
x)∪
p(
x)
.
Let
{ϕ
It follows by that
is
a prime model of
T ∪ q(
c2),(Mq, a0, b)
b0realizing
.
Suppose that there exist
x)
ϕ
(y,
p,q
and
x) ∪ q(y) ∪
p(
a),Mq= M(b),|= p(a),|= q(b)
ϕ
implies
x, y)}
(
(y,
p,q
|= r1(
Mp= M
all formulas of some type
implies
x)}
∪q(
ab0),|= r2(ba0)
= M(
r
1
(Mp,
T ∪ r1(
is
{ϕ
a
andb,
type
ϕ
x, y)
(
q
,p
{ϕ
x, y)is(q
(
q,p
all formulas of some type
y)∪
{ϕ
ab0) '
a)
is
a prime model of
c1, c2),(Mq, b)
a prime model of
x),
p,q
(y,
ϕ
q,p
M(
respectively, and
y)
.
q(
(p, q)
suc
h that the set
x),
p,q
(y,
ϕ
q,p
Mpand
, p)
-principal, the set
q,p
x, y)}
(
,
where
M(
is
consistent, so
b0∈ Mq,
ba0)
= M
T ∪ r1(
x, y)}
(
a)
and
M(b)
realize
M(a)= M(
- and
x, y)}
(
(q, p)
-principal
Mqare isomorphic,
.
Since
ϕ
p,q
p(
x, y) ∈ S(∅)
r1(
x) ∈ S(∅).
r2(y,
x, y)
r1(
a0∈ Mp,
= Mq.
r
2
c1),(Mp, a, b0)
T ∪p(
is
a prime model of
c1, c2)
then
,
and that
(y,
x) ∪
and
b0)
x)
=
any constant expansion of prime model is a prime model of new
theory.
36
Chapter
1. CHARACTERIZATION OF EHRENFEUCHTNESS
(3) ⇒ (4)
ϕ
w
ϕ
x)
(y,
p,q
e get a required
y,x) ∧ ϕ
(
p,q
The
directions
Clearly, any two principal types in
. Having
and
ϕ
p(
x, y).
(
q
,p
x, y)
,
(
q
,p
x) ∪ q(y) ∪
(p, q)
- and
(4) ⇒ (3)
(p, q)
- and
(q, p)
and consistent set
{ϕ
and
x),
(y,
p,q
(q, p)
ϕ
q,p
-principal formula
(4) ⇔ (2)
S(T)
-principal formulas
x, y)},
(
x, y)
ϕ(
are obvious.
¤
are domination-equi-
valent. Moreover, in view of Proposition 1.1.2.3, for small theories,
these types are strongly domination-equivalent. Besides, by the
denition, any principal type is dominated by any nonprincipal
type of the given theory.
1.1.2.4. Denition.
morphism types of models
of domination is induced by
among
phism types
M1∼RKM
Mp, that is,
M1, M2∈ PM
) if so are their representatives.
2
Clearly, the preordered set
Denote by
Mp,
RK(T )
p ∈ S(T )
≤RK, a relation deciding domination
RK(T ) = hPM; ≤RKi
are
domination-equivalent
RK(T )
has a least element, which
the set
PM
of iso-
, on which the relation
. We say that isomor-
(written
is an isomorphism type of a prime model.
1.1.2.5. Proposition.IfI(T, ω) < ω
ordered set whose factor set
tion-equivalence
∼RK, forms a partially ordered set with a greatest
RK(T )/∼RK, with respect to domina-
then
RK(T )
is a nite pre-
element.
Proof.
That
PM
is a nite set is obvious, and the fact that
RK(T )/∼RKcontains a greatest element follows from the existence
of a powerful type, which dominates any type in
S(T).¤
Below are two obvious remarks.
1.1.2.6. Remark.
only if
|RK(T )| = 1
A small theory
.
T
isω-categorical if and
1.1.
SYNTACTIC CHARACTERIZATION OF THEORIES
37
1.1.2.7. Remark.IfT
is a small theory and
|RK(T )| = 2
any nonprincipal type is powerful.
In the above-given Ehrenfeucht examples of theories
I(Tn, ω) = n
and
(n − 2)
models
, each preordered set
RK(Tn)
consists of least element
domination-equivalent elements corresponding to the
M
n
, . . . , M
p
0
n
. Thus, all ordered sets
p
n−3
RK(Tn)/∼RKare
two-element and linearly ordered.
Now we consider the relation
S(T)
hS(T); ≤RKi
of complete types of small theoryT. Denote the structure
by
RKT(T )
.
P. Tanovic noticed that the quotient of
alence relation
RK(T )
.
≡RKforms a structure, which is isomorphic to
≤RK, being dened on the set
RKT(T )
Indeed, in view of Proposition 1.1.2.3, for any type
the set of types, that are strongly
to the model
RK
-equivalent,
Mp. And for typespandqin
p ≤RKq
if and only if
RK
-equivalent top, corresponds
S(T)
, being not strongly
Mp≤RKMq.
then
Tnwith
by the equiv-
p ∈ S(T )
,
1.1.3. Limit models over a type
Recall that a sequence
(Mn)
of structures is called an
n∈ω
ementary chainifMnis an elementary substructure of
n ∈ ω
.
An elementary chain
a type
p ∈ S(T )ifMn' Mpfor any
1.1.3.1. Theorem.
is represented as the union of an elementary chain
of
prime models over tuples
(2)IfI(T
T
there exists a type
overpsuch that
, ω) < ω
M =
(Mn)
(1)
Any countable model
is said to be
n∈ω
n ∈ ω
.
M
ai.
then for any countable model
p ∈ S(T )
S
and an elementary chain
Mn.
n∈ω
elementary over
of small theory
(M(
M
of a theory
M
ai))
(Mn)
n+1
i∈ω
n∈ω
el-
,
T
38
Chapter
1. CHARACTERIZATION OF EHRENFEUCHTNESS
Proof.
small theoryT. We construct an elementary chainCof prime
models
this purpose, we enumerate all elements ofM:
and also all formulas of the form
m ∈ ω}
some nite sequence of tuples
such a tuple will be connected to a nite set
2
(1) Let
M
be an arbitrary countable model of a
S
ai)
M(
o
ver tuples
ai,
i ∈ ω
,
such that
M =
M(
i∈ω
M = {bk| k ∈ ω}
c),c ∈ M:Φ
ϕ(x,
.
We shall constructCinductively and, at any stepk,
a0,
. . . ,
anwill
be dened, and each
X
k
,
i
= {ϕm(x,
0 ≤ i ≤ n
ai)
, such
.
For
cm) |
that unions of these sets by allkwith respect to xediwill dene
ai)
universes of models
stepkthen sets
X
M(
l
are supposed to be empty for any
i
.
If a tuple
At the initial step, we x the tuple
ϕm(x, b0)
∃x ϕm(x, b0)
q0(x, b0)
fromΦhaving the minimal number and satisfying
we nd a realization
containing
ϕm(x, b0)
. Now we set
Suppose that at stepkwe have already found tuples
anand
have formed nite sets
X
k
0
aiis
not dened before the
l < k
.
a0 hb0i
and
for the formula
M |=
dmof a principal complete type
0
X
, . . . , X
{b0, dm}
0
k
satisfying the following
n
.
a0,
. . . ,
conditions:
(1) all elements of
i
< n
, and belong to
(2)
{b0, . . . , bk} ⊆ X
k
(3)
X
⊂ X
i
k
i+1
aiare
k
X
i
,
i < n − 1
(4) for the formula
minimal with respect to
contained in the set of elements of
;
k
;
n
;
cm)
ϕm(x,
m
and not considered before, contains
w
e chose at stepk, which is
a
i+1
,
,
2
The
initial proof of Theorem, represented in [401], has the following short
form: \Obviously, any countable modelMof a small theoryTcan be written
as a union of an elementary chain
type
p ∈ S(T )
innite subsequence of models
to some model
a detailed proof, since extending a set, over which a prime model is taken, the
latter model can be a non-elementary extension of the previous one and even
can do not contain it. J. Baldwin in the letter to the author pointed out an
example when a prime model can be not extensible to a prime model over the
extended set, see [110, Example 2.4]. In the present proof published in [413],
we give a detailed algorithmic procedure of step-by-step construction for an
elementary chain of prime, over tuples, models, whose union is equal to the
given countable model of a small theory. A modication of the construction
below for the proof is suggested in [88].
such that
Mp". A. Tsuboi in the letter to the author suggested to write
Mn' Mp. If
(M
(Mn)
)
n
m
, where for any
n∈ω
I(T, ω) < ω
, whose all elements are isomorphic
m∈ω
n ∈ ω
, the sequence has an
there is a
1.1.
SYNTACTIC CHARACTERIZATION OF THEORIES
39
only elements of the maximal nonempty set
cm)
M |= ∃x ϕm(x,
dm∈ M
ϕm(x,
tuple
of a principal complete type
cm)
so
that for any tuple
d ∈ X
k−1
i
tion is added in the minimal set
A
t step
k−1
i
.
k + 1
cm∈ X
then the sequence
X
k+1
by adding to
i
sets
tions (3) and (4) for
If
b ∈ X
a0,
. . . ,
b
k+1
k
∪
{b
i
anand
k
6∈ X
n
}
k+1
add the element
consequent sets
adding an element
k + 1
instead ofk.
If some type
to the sequence
of the set
k
set
X
n+1
k
X
∪{b
n
and form sets
dmof a principal complete type
the minimal (with respect tom) formula
,
we have found a realization a realization
qk(x, X
aiwith
∪
{dm}
the type
d/ai)
tp(
k
X
with respect toisuch that
i
, we consider the element
a0,
. . . ,
X
k + 1
anremains
k
some element
i
instead ofk.
the same and we construct
and, starting from some
k−1
j
cm∈ X
is
principal; this realiza-
b
. If it belongs to
k+1
dmsatisfying the condi-
i0≤ n
, are principal, we again do not extend the sequence
b
to the set
k
X
,
i0≤ i ≤ n
i
k+1
. Then we obtain sets
dmsatisfying the conditions (3) and (4) for
b/an),b ∈ X
tp(
a0,
k+1
anthe
. . . ,
}
. Then we add this set to the (initially empty)
k+1
X
i
k
∪
{b
k+1
n
a
tuple
,
0 ≤ i ≤ n+1
q
k+1
n+1
(x, X
}
, is not principal, we add
consisting
, by adding a realization
k
∪ {b
n
ϕm(x,
considered before and contains only elements of
cm)
M |= ∃x ϕm(x,
for
any tuple
add the element
,
such that for any tuple
d ∈ X
k
∪
{dm}
i
, the type
dmto the minimal (with respect toi) set
also to the consequent sets such that
k+1
set
X
n+1
By construction, the sets
models
and
model
M(
M =
M
mentary chain of the models
the model
k
X
n
∪ {b
k+1
, dm}
.
Xi
k∈ω
ai)
o
ver tuples
S
M(
i
ai)
ai.
Moreover, we have
.
If the number of indexesiis nite, the
is prime over the greatest tuple
ai)
M(
M
countably many times.
cm∈ X
S
X
to
aiwith
d/ai)
tp(
k
,
j
k
are the universes of prime
i
aiand
the countable chain taking
k−1
X
, and satises
j
∪ {bk})
k−1
i
containing
and
, all types
k
X
as well as to all
i
0
of all elements
})
k+1
cm)
X
containing
whic
k
n
h was not
and satisfying
cm∈ X
is
principal. We
i ≤ j ≤ n
ai) 4 M(a
M(
we add the ele-
for any
b/ai)
tp(
k+1
X
i
k
i
k
X
, and
i
.
Now we
X
by
and
i+1
k
n
,
)
40
Chapter
1. CHARACTERIZATION OF EHRENFEUCHTNESS
(2) Since
the sequence
(Mn)
n∈ω
I(T, ω) < ω
(Mi)
= (M
i
n
)
morphic to a model
1.1.3.2. Lemma.
c/a)
tp(
and
ar
Proof.
a, z)
ψ(
e principal then
By assumption, there are principal formulas
suc
h that
χ(
Ob
viously
a, y) ∈ tp(b/a)
χ(
principal.
Indeed,
holds.
a, c00).
ψ(
Take tuples
Since
b0and
let
the formula
morphismfxing
is
principal and satises
and thusTis small, we can choose, from
ai)
, where
i∈ω
of models such that all its elements are iso-
n∈ω
Mi= M(
Mp. This sequence is required.
a,b
If
,
|= ϕ(
c
and
tp(
ar
e tuples such that
b/a)
is
a, b, c) ∧ ψ(a, c)
,
an innite subsequence
principal.
.
Consider the formula
¤
tp(
a, y) ∃z (ϕ(a, y,z) ∧ ψ(a, z)).
.
It suces to show that
b00b
c0and
a
and
c00with
taking
|= ϕ(a,f(
e tuples for which
|= ϕ(a, b0, c0)∧ψ(a, c0)∧ϕ(a, b00, c00)∧
a, z)
ψ(
is
principal, there exists an auto-
c0to
c00.
As the formula
b0), c00) ∧ ϕ(a,b00, c00),
a, b0) ∧ χ(a, b00)
|= χ(
ab/c)
x, y,c)
ϕ(
a, y)
χ(
a, y,c00)
ϕ(
and
is
maps
b/a)
tp(
a,c00and
b0to
b00.
is
isolated.
S
Mnfor some
n∈ω
there
exists an automorphismgxing the tuples
b0)tob00.
f(
the formula
Then the
a, y)
χ(
1.1.3.3. Denition.
limit over a type
elementary chain
a
-automorphism
is
principal and the type
A model
p ∈ S(T )
(Mn)
n∈ω
M
orp-limitifM =
overpand
f ◦ g
of a theoryTis said to be
M 6' Mp.
Below, in this Section, we shall assume that the theoryTis
small.
1.1.3.4. Proposition.
type
p ∈ S(T )
if and only if for any(some)realization
there are a realization
c/b)
that
tp(
is
a nonprincipal type.
A small theoryThas a limit model over a
a
of
bofpinM(a)
and
a tuple
c ∈ M (a)
taking
Thus,
¤
type
such
p
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