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1.1.
SYNTACTIC CHARACTERIZATION OF THEORIES
41
Proof.
ofToverp, where
b ∈ p(M0),c ∈ M0the
Suppose that there exists a limit model
Mn' Mp,
type
tp(
M0= M(
c/b)
is
a),|= p(a)
principal. Then models
M =
n∈ω
,
and for any
M
M
(and hence alsoM) realize just principal types over any realiza-
S
tions of typeplying in
over a realization ofp, that contradicts the assumption that
Mn(inM). Hence the model
M
is prime
M
is
limit.
a
Conversely, assume that for some tuple
tuples
b ∈ p(M0)
and
c ∈ M0suc
h that
,
realizingp, there are
q(
x, b)
= tp(
c/b)
is
nonprincipal type. Our goal is to construct an elementary chain
an))
(M(
a1= a
S
n∈ω
M(
,
an)
to prove that
for
some tuple
x, y)
ϕ(
are
principal, and so
o
verpsatisfying the following conditions:
n∈ω
and
a
tp(
is
not isomorphic to models
M
a0and
suc
h that
)
n+1an
and
= tp(
Mpare non-isomorphic, since if
a0∈ M(an)
M |= ϕ(
M = M(
ab)
a0, an)
a0) '
.
We argue to show that
Mr,
r ∈ S(∅)
then
there exists a formula
,
the formulas
an)
M(
= Mpby Proposition
a0, y),ϕ(x, an)
ϕ(
a0= b
M =
. It suces
M = M(
a0)
1.1.2.3.
By way of contradiction, i. e., assuming
d ∈ p(M(an))
however, the type
the types
the type
q(
tp(
x, an)
suc
h that
q(
anc0/d)
= tp(
x, an)
and
c0/an)
M = M(
is
realized in
tp(d/an)
is
principal too in spite of assumption.
d)
are
.
That contradiction shows that the model
M ' Mp, nd a tuple
By the construction ofM,
M
by some tuple
c0.
Since
principal, by Lemma 1.1.3.2,
M
is limit overp.
¤
n
n
a
,
1.1.3.5. Corollary.
If
Mpand
Mqare domination-equivalent
non-isomorphic models then there exists a model that is limit over
the typepand a model that is limit over the typeq.
Proof.
Let
tion ofqsuch that
of
Mp∼RKMq). Since
andbb
b) ≺
M(
Mp6' Mq, the type
e realizations ofpand
M(
c) ≺
M(
a)
c
b
(whic
h exist in view
a/c)
tp(
e a realiza-
is
nonprin-
a
cipal by Proposition 1.1.2.3. Hence a limit model overpexists by
Proposition 1.1.3.4. The existence of a limit model overqis proved
analogously.
¤
42
Chapter
1. CHARACTERIZATION OF EHRENFEUCHTNESS
1.1.3.6. Proposition.
If types
p1and
p2are domination-equi-
valent, and there exists a limit model over
exists a model that is limit both over
Proof.
an))
(M(
(a) models
oddn;
(b) the model
We construct inductively an elementary chain
of
n∈ω
models such that:
an)
M(
are
prime over
S
an)
M(
n∈ω
is
p1and over
p1for evenn, and over
prime neither over
By Proposition 1.1.3.4, there exists a type
a0)
whic
h is not realizable in the model
M(
b) Â
some
over a realization
b)
of
M(
(suc
At even steps
to
a model
odd steps, we extend
M(
S
the model
n∈ω
1.1.3.7. Lemma
a
tuple
isolates
not semi-isolate
(2)If(
a, b) ∈ Ipand
a0),b |= p1.
M(
a1of
type
h exists since
2n + 2
M(
, we extend the model
a
),a
2n+2
an)
2n+2
a
M(
2n+2
is
limit over
(B. Kim [257], P. Tanovic [451, 452]).
b
a tuple
a
.
,
whereas
(b, a) ∈ SIpthen
M(
Denote by
p2, which is an elementary extension
p1and
|= p1,
p2are domination-equivalent).
which realizes type
)toM(a
2n+3
p1and over
b
do
es not isolate
but
a1)
M(
M(
),a
p2.
(b, a) ∈ Ip.
p1, then there
p2.
p1nor over
x, a0),a0|= p1,
q(
is realizable in
the
prime model
a
),a
2n+1
2n+3
2n+1
x, a2n)
q(
|= p2.
Clearly,
¤
(1)
a
,
then
p2for
p2.
|= p2,
.
If a
b
do
At
es
a, y)
Pr
oof.
the contrary (i. e.,
witnessing
there exists a formula
and
x, b) ∧ψ(x, b) ∧
ϕ(
formulas imply
are
both consistent. This contradicts the fact that
(1) Suppose that
b
semi-isolatesa)
b
that
semi-isolatesa.
x, y)
χ(
¬χ(
a)
.
tp(
Hence
x, b)
suc
are
ϕ(
ϕ(
h that
a, y)∧χ(a, y)
isolates
and take a formula
Now as
tp(
x, b) ∧ ψ(x, b) ∧ χ(x, b)
ϕ(
both consistent. Moreover, both
principal formula.
(2) follows immediately from (1).
1.1.3.8. Denition
if for any realizations
i.
e., the relation
(A. Pillay [338]). A type
a
andbofp,
Ipis symmetric.
¤
(a, b) ∈ Ipimplies
Lemma 1.1.3.7 immediately implies
tp(b/a)
a/b)
is
nonisolated,
and
ϕ(a, y)∧
ϕ(
x) ∈ S(T )isgo
p(
.
Assume
x, b)
ψ(
a, y)
¬χ(
a, y)
is
od
(b, a) ∈ Ip,
a
,
1.1.
SYNTACTIC CHARACTERIZATION OF THEORIES
1.1.3.9. Corollary
(P. Tanovic [451, 452]).Ifp(
x)
type of a theoryTand for any model ofTthe relation
metric, then the typepis good.
is
a complete
SIpis sym-
43
P. Tanovic [451, 452] noticed that there exist good types
with non-symmetric
1.1.3.10. Example.
type
p ∈ S1(∅)
Indeed, if
|= p(a)
SIp:
In
Th(hω; <i)
is good and
SIpis non-symmetric.
then any principal formula
, the unique non-algebraic1-
ϕ(a, y)
describes a
nite number of steps for taking nitely many successors or prede-
cessors to come to the (unique) realizationbof this formula. The
existence of a converse way frombtoameans that there exists a
principal formula
At the same time, ifaandbare realizations ofp,
ψ(x, b)
for which
|= ψ(a, b)
.
a < b
, having
innitely many intermediate elements, thenasemi-isolatesbby
means of the formula
a < y
, butbdoes not semi-isolatea.
¤
The following theorem is obtained independently by
B. S. Baizhanov, V. V. Verbovskiy, and S. V. Sudoplatov [87]
and by E. Casanovas [139].
x)
1.1.3.11. Theorem.
T
. The following conditions are equivalent:
(1)
there exists a limit model overp;
(2)
the isolation relation
(
any)model
(3)
a
tions
not semi-isolate
M |= T
in some(any)model
andbofpsuch
a
Let
realizingpis non-symmetric;
that the type
and,
in particular,
b
p(
e a complete type of a small theory
Ipon a set of realizations ofpin a
M |= T
realizingp, there exist realiza-
b/a)
tp(
is
principal and
SIpis non-symmetric on the
b
do
es
set of realizations ofpinM.
p
Proof.
At rst we consider the conditions
modelM.
(1) ⇒ (2)
. Assume that the theoryThas a limit model over
and the nonempty relation
ofpinM.
Consider realizations
c ∈ M
,
which exist by Proposition 1.1.3.4, such that
(2)
and
(3)
for
some
Ipis symmetric on the set of realizations
a
andbofpin
the model
M
and a tuple
b, c ∈ M(a)
p
,
44
Chapter
1. CHARACTERIZATION OF EHRENFEUCHTNESS
a) 4 M
M(
b
,
and
pal, by the hypothesis, the type
Lemma 1.1.3.2, the type
c
,
the type
,
and
tp(
tp(
c/b)
bc/a)
is
nonprincipal. By choice of tuples
tp(
is
principal. Since
a/b)
tp(
c/b)
is
principal in spite of the assump-
is
also principal. So, by
tp(
b/a)
is
princi-
a
tion. The obtained contradiction shows that, having the symmetric
nonempty relation
Ipon the set of realizations ofpinM, we have
no limit models overp.
The implication
(2) ⇒ (3)
follows immediately from Lemma
1.1.3.7.
a
(3) ⇒ (1)
such that
do
es not semi-isolate
a) 4 M
M(
. Let
a
semi-isolatesbb
and
andbb
a
.
the type
e realizations ofpin the model
y some principal formula
b
Then the tuple
a/b)
tp(
is
nonprincipal. By Proposition
b
elongs to some model
θ(
a, y)
M
and
1.1.3.4, the theoryThas a limit model overp.
The equivalence of the existence and of the generality pointed
out in
the considered properties are reduced to the model
|= p(
model which realizes the typep.
a)
,
(2)
and
and
M(
(3)
for models realizing the typepis true since
M(
a)
is
isomorphic to an elementary submodel of any
¤
a)
,
where
Proposition 1.1.1.10 and Theorem 1.1.3.11 imply the following
,
b
1.1.3.12. Corollary.Ifp ∈ S(T )
is a nonprincipal powerful type
then there exists a limit model overp.
Another argument for Corollary 1.1.3.12 is that any countable
saturated model ofTis limit over a (any) nonprincipal powerful
type, ifThas these types.
1.1.3.13. Denition.
said to be
chains
tions:
(1)
equivalent
(Mn)
M =
n∈ω
S
n∈ω
and
Mn,
(2) there exist constant expansions
and
M
phism
0
N
= hN
0
n+1
n+1
' N
n+1
0
by an isomorphism
n+1
fnbetween structures
Limit models
(written
(Nn)
N =
, ci
c∈N
n
M
andNover a typepare
M ∼ N
overpsatisfying the following condi-
n∈ω
) if there exist elementary
S
Nn;
n∈ω
,
n ∈ ω,M
0
M
n
and
0
= M0,
0
f
n+1
N
0
M
= hM
n+1
0
N
= N0, such that
0
n+1
extending some isomor-
0
,
n ∈ ω
n
.
, ci
c∈M
n
1.1.
SYNTACTIC CHARACTERIZATION OF THEORIES
The following proposition is obvious.
45
1.1.3.14. Proposition.
typepthen
M ' N
If
M
if and only if
and
N
M ∼ N
are limit models over a
.
1.1.4. Characterizations for the classes of theories with
nitely many countable models
LetfM ∈ RK(T )/∼RKbe the class consisting of isomorphism
types of domination-equivalent models
IL(fM)
is limit over some type
the number of equivalence classes of models each of which
pi.
M
, . . . , M
p
1
. Denote by
p
n
Propositions 1.1.2.5, 1.1.3.14, Corollaries 1.1.3.5, 1.1.3.12, as
well as Lemma 1.1.1.10 and Theorems 1.1.3.1, 1.1.3.11 (for limit
models over powerful types) can be combined to yield
1.1.4.1. Theorem.
For any countable complete theoryT, the fol-
lowing conditions are equivalent:
(1)
I(T, ω) < ω
(2)
T
is small,
f
M ∈ RK(T)/∼RK.
If
(1)or(2)
(a)
RK(T )
prime model)and
(b)
RK(T )
;
|RK(T )| < ω
and
IL(fM) < ω
for any
holds thenTpossesses the following properties:
has a least element
IL(gM0) = 0
has a greatest
∼RK-classgM1(
M
(
an isomorphism type of a
0
;
a class of isomorphism
types of all prime models over realizations of powerful types)and
|RK(T )| > 1
(c)if|fM| > 1
Moreover, the following
implies
then
IL(gM1) ≥ 1
IL(fM) ≥ 1
decomposition formula
;
.
holds:
where
g
M0, . . . ,
dered set
I(T, ω) = |RK(T)| +
^
M
|RK(T )/∼RK|−1
RK(T )/∼RK.
|RK(T )/∼RK|−1
X
IL(fMi),
i=0
are all elements of the partially or-
46
Chapter
1. CHARACTERIZATION OF EHRENFEUCHTNESS
Note that, by Propositions 1.1.2.3 and 1.1.3.14, the conditions
specied in item (2) of Theorem 1.1.4.1 admit of a syntactic repre-
sentation; so, this theorem is similar to the Ryll-Nardzewski the-
orem providing a syntactic characterization ofω-categoricity. Be-
sides, since
Mp≤RKMq⇔ Mp4 M
q
(for some models
Mp,
Mqwith arbitrary types
p, q ∈ S(∅)
), Theo-
rem 1.1.4.1 admits an obvious reformulation in terms of elementary
embedding.
In Figure 1.1,aandb, possible variants for Hasse diagrams
of Rudin{Keisler preorders
tionsILof numbers of limit models on
represented for the cases
ure 1.2, corresponding congurations for
≤RKand values of distribution func-
∼RK-equivalence classes are
I(T, ω) = 3
and
I(T, ω) = 4
I(T, ω) = 5
. In Fig-
are shown.
Note that the diagram in Figure 1.1,a, the second diagram in
Figure 1.1,b, and the sixth diagram in Figure 1.2 are realized by
Ehrenfeucht examples. The diagram in Figure 1.1,ais also real-
ized by Peretya'kin example. The following example, as above, is
included into the list of \natural" examples of Ehrenfeucht theo-
ries.
1.1.4.2. Example.
We set
T Th((Q; <, cn, c
is an ordinary strict order on the setQof rationals, constants
form a strictly increasing sequence, and constants
decreasing sequence,
cn< c
0
,
n
n ∈ ω
. The theoryThas six pairwise
0
)
)
, where
n∈ω
n
0
c
form a strictly
n
<
c
non-isomorphic countable models:
n
f
•
1
f
•
0
a
f
•
f
•
Figure 1.1
¨
2
0
•
§
L
L
L
L
L
L
•
b
¥
•
1
¦
¯
¯
¯
¯
¯
¯
f
0
f
•
1
f
•
0
f
•
0
1.1.
SYNTACTIC CHARACTERIZATION OF THEORIES
47
h
•
3
h
•
0
£
h
©
•
2
ª
£
£
£
£
£
£
0
®
•
B
B
B
B
B
B
B
•
•
a prime model with empty set of realizations of type
isolated by the set
•
a prime model over a realization of
®
h
•
1
h
•
1
h
•
0
•
B
B
B
B
B
B
B
•
••
£
£
£
£
£
£
£
h
0
Figure 1.2
©
1
ª
®
h
•
2
h
•
0
h
•
0
•
@
@
¡
@
h
•
0
h
•
0
{cn< x | n ∈ ω} ∪ {x < c
p(x)
•
•
•
•
•
¡
¡
0
| n ∈ ω}
n
©
1
ª
•
·
·
·
·
h h
•
0 0
T
T
T
T
•
;
, with a unique real-
ization of this type;
•
a prime model over a realization of type
the set
forms a closed interval
of realizations of
p(x) ∪ p(y) ∪ {x < y}
•
three limit models over the type
q(x, y)
; here the set of realizations of
[a, b]
;
are intervals of forms
q(x, y)
q(x, y)
isolated by
, in which the sets
(a, b],[a, b),(a, b)
respectively.
In Figure 1.3 we represent the Hasse diagram of Rudin{Keisler
preorders
of limit models on
≤RKand values of distribution functionsILof numbers
∼RK-equivalence classes for the theoryT.
h
1
h
0
h
0
h
0
h
1
T
T
T
T
·
·
·
·
h
0
q(x, y)
¤
•
p(x)
Let
p1, . . . , pn∈ S(T )
be types the prime models over which
are representatives of all isomorphism types in a nite preordered
set
RK(T )ofT
extension property of chains of models prime over tuples
for any type
. We say that the theoryTpossesses the
pi, every two limit models over
piare equivalent.
consistent
(CEP)
if
48
Chapter
1. CHARACTERIZATION OF EHRENFEUCHTNESS
h
•
3
h
•
0
h
•
0
Figur
e 1.3
Proposition 1.1.3.6 implies that if
IL(fM) ≤ 1
for anyfM ∈ RK(T)/∼RK. Since there exists no model
that is limit over a principal type, if
T
satises
|RK(T )/∼RK| = 2
(CEP)
then
(CEP)
implies the existence of a unique (up to isomorphism) model
the isomorphism type of which does not lie in
RK(T )
(andM,
in this event, is saturated).
Thus, the following theorem, based on Theorem 1.1.4.1, is valid.
1.1.4.3. Theorem.
LetTsatisfy
(CEP)
. Then the following con-
ditions are equivalent:
(1)
I(T, ω) < ω
(2)Tis small and
;
|RK(T )| < ω
.
In this event, we have the inequality
I(T, ω) ≤ |RK(T)| + |RK(T )/∼RK| − 1,
which turns into equality for
|RK(T )/∼RK| ≤ 2
.
Theorem 1.1.4.1 implies
1.1.4.4. Corollary.
For any complete theoryT, the following are
equivalent:
(1)
I(T, ω) = 3
(2)Tis small, possesses
;
(CEP)
, and
|RK(T )| = 2.¤
then
M
1.1.
SYNTACTIC CHARACTERIZATION OF THEORIES
1.1.5. About Ehrenfeucht theories on dense orders
49
Recall [343] that a linearly ordered structure
Miso
-minimal
if each formula denable subset ofMis a nite union of singletons
and open intervals
A theoryTiso-minimal
(a, b)
, where
a ∈ M ∪ {−∞},b ∈ M ∪ {+∞}
if each model ofTiso-minimal.
As examples of Ehrenfeuchto-minimal theories, we mention the
theories
T1 Th((Q; <, cn)
n∈ω
and
T2 Th((Q; <, cn, c
0
)
n∈ω
n
where<is an ordinary strict order on the setQof rationals, con-
stants
cnform a strictly increasing sequence, and constants
a strictly decreasing sequence,
As shown above, the theory
ple with
I(T1, ω) = 3
, and the theory
cn< c
0
,
n
n ∈ ω
.
T1is the Ehrenfeucht's exam-
T2has six pairwise non-
c
0
n
form
isomorphic countable models.
1.1.5.1. Denition.
characteristically equivalent
RK(T1)
is isomorphic to the structure
sponding replacement of isomorphism types in
phism types in
RK(T2)
of limit models of
numbers of limit models of
We say that small theories
and write
T1∼chT2if the structure
RK(T2)
T1and
T2are
and, by the corre-
RK(T1)
to isomor-
, the distribution function IL for numbers
T1is transformed to the distribution function for
T2.
F
1.1.5.2. Denition
of pairwise disjoint structures
languages
n ∈ ω}
Σn,
n ∈ ω
with the universe
of predicate symbols in
Mn,
n ∈ ω
. The
languages
Σnaccordingly,
(R. Woodrow [60]). The
Mnfor pairwise disjoint predicate
, is the structure of language
F
Mn,
Pn= Mn, and interpretations
n∈ω
Σncoinciding with their interpretations in
disjoint union of theories
n ∈ ω
, is the theory
Ã
G
Tn Th
n∈ω
G
n∈ω
M
n
disjoint union
S
Σn∪ {P
n∈ω
n∈ω
(1)
n
M
Tnfor pairwise disjoint
!
,
.
,
n
|
where
Mn|= Tn,
n ∈ ω
.
50
Chapter
1. CHARACTERIZATION OF EHRENFEUCHTNESS
Obviously, the theory
els
Mn|= Tn,
n ∈ ω
.
Tndoes not depend on choice of mod-
n∈ω
The following theorem is a reformulation of theorem proved by
L. Mayer [299].
F
1.1.5.3. Theorem.
Any model of ano-minimal Ehrenfeucht the-
oryTis densely ordered besides, possibly, nitely many elements
with successors or predecessors laying in the denable closure of
empty set. The theoryTis characteristically equivalent to some -
nite disjoint union of theories of form
where
1
T
are similar to
i
T1and
2
T
are similar to
j
T1,
T2(
T ∼
ch
T2)
k
F
i=1
and has
l
F
1
T
t
i
j=1
2
T
j
3k·6
pairwise non-isomorphic countable models.
By Theorem 1.1.5.3, there is a structural description of model
of Ehrenfeucht theories formed by constant expansions of dense
linearly ordered sets.
Since dense linear orders are dened on dense cyclically ordered
sets [18, 292], the condition ofo-minimality for these sets implies
the same structural description of models of corresponding Ehren-
feucht theories.
As known [451], theories with three countable models, in which
denable closures of empty set are innite, has a structure of dense
linear order or a structure of dense branching tree
T
forming a
lower semilattice. Constant expansions ofTextends possibilities
for characteristic representations of Ehrenfeucht theories observed
foro-minimal Ehrenfeucht theories.
,
l
1.1.5.4. Example.
00
(c
)
of constants, where the rst one strictly increases, and two
n∈ω
n
Having three sequences
(cn)
n∈ω
others strictly decrease with respect to<on the treeT,
cn< c
00
,
n ∈ ω,c
n
0
i
and
00
c
are incomparable,
j
i, j ∈ ω
, we get the
following countable models:
•
a prime model with empty set of realizations of type
isolated by set
ω}
;
{cn< x | n ∈ ω} ∪ {x < c
0
| n ∈ ω} ∪ {x < c
n
,
(c
cn< c
0
n
00
n
)
n∈ω
0
n
p(x)
| n ∈
,
,
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