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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
•
a prime model over a realization of
p(x)
with a unique real-
ization of this type;
•
a prime model over a realization of type
set
p(x) ∪ p(y ) ∪ {x < y} ∪ {cn< x ∧ y < c
here, the set of realizations of
where
tions of
set
c
forms a closed interval
lim
n→∞
izations in
a = lim
•
three limit models over
q(x, y)
•
a prime model over a realization of type
n→∞
cnand
b = lim
are intervals of forms
p(x) ∪ p(y) ∪ {x < y, y < z} ∪ {cn< x ∧ y < c
0
n
∧ ¬c
0
≤ z | n ∈ ω}
n
; here, the set of realizations in
[a, b]
0
c
∈ (a, b)
n
•
seven limit models over
;
r1(x, y, z)
satises one of the following conditions:
(a) it forms an interval
belongs to
(a, b)
or the limit does not exist;
(b) it forms an interval
belongs to
(a, b)
or the limit does not exist;
(c) it forms an interval
belongs to
(a, b)
or the limit does not exist;
(d) it forms an interval
•
a prime model over a realization of type
set
p(x) ∪ p(y) ∪ {x < z, z < y} ∪ {cn< x ∧ y < c
00
c
n
forms a closed interval
lim
n→∞
00
∧ ¬c
•
≤ y | n ∈ ω}
n
00
c
∈ (a, b)
n
;
seven limit models over
; here, the set of realizations of
[a, b]
q(x, y)
c
n→∞
q(x, y)
, where
r1(x, y, z)
(a, b]
[a, b)
(a, b)
[a, b]
and the limit
, where
r2(x, y, z)
forms a closed interval
0
= lim
n
n→∞
c
, for which the sets of realiza-
(a, b],[a, b),(a, b)
a = lim
n→∞
, for which each set of real-
and the limit
and the limit
and the limit
a = lim
n→∞
similar to limit models over
r1and obtained by transpositions of constants
•
a prime model over a realization of type
set
p(x)∪ p(y)∪ {x < y, x < z, ¬(y ≤ z), ¬(z ≤ y), x = inf{y, z}} ∪
{cn< x ∧ y < c
r3(x, y, z)
where
a = lim
0
∧ z < c
n
00
| n ∈ ω}
n
forms two closed intervals
n→∞
cn,
b = lim
n→∞
0
c
n
, and
; here, the set of realizations of
[a, b]
d = lim
n→∞
q(x, y)
00
;
n
0
∧ y < c
n
00
n
respectively;
r1(x, y, z)
0
∧ z < c
n
cn,
b = lim
n→∞
00
lim
c
n
n→∞
00
lim
c
n
n→∞
00
lim
c
n
n→∞
00
lim
c
does not exist;
n
n→∞
r2(x, y, z)
0
∧ z < c
n
cn,
b = lim
n→∞
0
c
and
n
c
r3(x, y, z)
and
[a, d],a = inf{b, d}
00
c
∈ (a, b)
n
;
isolated by
| n ∈ ω}
isolated by
00
n
r1(x, y, z)
exists and
exists and
exists and
isolated by
00
n
r2(x, y, z)
00
,
n
isolated by
[a, b]
∧ ¬z ≤
00
c
, and
n
∧ ¬y ≤
0
c
, and
n
n ∈ ω
51
;
,
;
,
52
•
tions of
endpointb,d, or with removed endpoints of
•
by set
inf{y, z}} ∪ {cn< x ∧ y < c
realizations of
and
[e, d],e = inf{b, d}
lim
n→∞
•
alizations of
Chapter
three limit models over
r3(x, y, z)
a prime model over a realization of type
1. CHARACTERIZATION OF EHRENFEUCHTNESS
r3(x, y, z)
consist of intervals
, in which the sets of realiza-
[a, b]
and
[a, d]
with removed
{b, d}
;
s(x, y, z, u)
isolated
p(x) ∪ p(y) ∪ {x < u, u < y, u < z, 6= y ≤ z, ¬z ≤ y, u =
s(x, y, z, u)
, where
00
c
∈ (a, b)
n
;
seven limit models over
s(x, y, z, u)
consist of intervals
0
∧ z < c
n
forms three closed intervals
a = lim
s(x, y, z, u)
00
| n ∈ ω}
n
; here, the set of
[a, e],[e, b]
n→∞
cn,
b = lim
n→∞
0
c
n
, in which the sets of re-
[a, e],[e, b]
, and
, and
d =
[e, d]
with removed endpointa,b, ord, or with removed endpoints in
{a, b},{a, d},{b, d}
Finally, the considered theory
and
27
limit models,
Hasse diagram for Rudin{Keisler preorder
function
IL
of distribution for numbers of limit models on
classes of theory
, or
{a, b, d}
.
T3has7prime models over tuples
I(T3, ω) = 34
. Figure 1.4,arepresents the
≤RKand values for the
∼RK-
T3; we also pointed out types over which models
are prime or limit.
0
Considering unary predicates
0
00
c
, c
,
n ∈ ω
n
n
, for the corresponding theory
Pn= {c
00
, c
}
n
instead of constants
n
T4, the types
r1and
r
become equal. Whence the number of prime models over tuples is
reduced by one, and the number of limit models is reduced by 7.
Thus,
T3, is represented on Figure 1.4,b.
I(T4, ω) = 26
. The diagram for
¤
T4, similar to diagram for
,
2
Note that additional expansions by strictly decreasing sequences
of constants preserve the Ehrenfeuchtness of theory. In this case,
the number of possibilities is dened, as above, by links between
limits of sequences.
Examples above demonstrate possibilities for complications of
characterizing pair (Rudin{Keisler preorder, distribution function
for numbers of limit models) and quite rapid increase of number of
limit model relative to constant expansions in the class of Ehren-
feucht theories.
1.2.
INESSENTIAL COMBINATIONS AND COLORINGS
53
s
•
¡
¡
r
2
¡
r
1
q
•
• •
7
¡
¡
f
¡
•
3
@
@
@
p
•
•
a b
f
7
@
@
@
ff f
r
3
7
¢
¢
¢
¢
¢
f
¢
0
f
0
r
•
3
Figure 1.4
1
q
•
¡
f
@
7
3
¡
@
s
¡
@
p
f
•
7
@
@
ff
@
r
•
3
3
¢
¢
¢
¢
¢
f
¢
•
0
f
•
0
1.2. Inessential combinations
and colorings of structures
In the rst paragraph of this Section, we dene operations
of inessential and almost inessential combinations of structures over
a set as well as of theories. A bases of almost inessential combi-
nations of theories are states, and also preservations of properties
of smallness andλ-stability at transformation to almost inessen-
tial combinations of theories are shown. A sucient condition is
resulted for the inessentiality of combinations of theories in as-
sumption of inessentiality of combinations of their models.
In the second paragraph, concepts of coloring of structure, of
colored structure and of colored theory are dened, and the results
of the rst item are transferred for almost inessential colorings. An
example is given showing that innerly inessential colorings of struc-
tures do not imply that corresponding theories have an inessential
colorings.
In the third paragraph, a concept of ordered coloring is de-
ned, the role of such colorings in constructions of Ehrenfeucht
theories is investigated, and an example is given ofω-stable theory
with an ordered coloring, inducing a continuum of pairwise non-
isomorphic limit models over a type. There we also consider colors
for neighbourhoods forcing the semi-isolation.
54
Chapter
1. CHARACTERIZATION OF EHRENFEUCHTNESS
1.2.1. Combinations of structures and theories
1.2.1.1. Denition
[330]). A theoryTis said to be∆-based
(E. A. Palyutin, J. Sae, and S. S. Starchenko
, where
∆
is some set
of formulas without parameters, if any formula ofTis equivalent
inTto a Boolean combination of formulas of∆.
For∆-based theoriesT, it is also said thatThas
eliminationorquantier reduction
up to∆.
quantier
The following denition generalizes the notion of∆-baseness.
1.2.1.2. Denition.3A theoryTis called∆-based with respect
to a set
of formulas without parameters, if for any tuple
pairwise distinct variables and having length
U ⊆ ω
, or simply∆-based overU, where∆is some set
x
,
consisting of
x) ∈ U
l(
,
any formula
^
x) ∧
ϕ(
xi,xj∈
x,
i6=j
¬(xi≈ xj)
ofTis equivalent inTto a Boolean combination of formulas of∆.
A theoryTis said to be
an innite set
U ⊆ ω
.
almost∆-basedifTis∆
-based over
Clearly, if a theoryTis∆-based over a setU, thenTis∆-
based over every set
of a theoryTmeans thatTid∆-based over
∆
-based theory is almost∆-based.
U0⊆ U
. By the denition, the∆-baseness
U = ω
. Hence, any
1.2.1.3. Denition.
complete) types over a setAis denoted by⊆S(A)
A type
x) ∈⊆S(A)
q(
of formulas ofqif
forced
from the denitions in the book [51] and in the paper [400] since old denitions
do not correspond completely to the constructions below and lead to some
wrong assertions. P. E. Alaev pointed out on facts of these assertions and the
author is thankful to him.
by some nite set of formulas in
3
The
denition of the almost∆-baseness and some further denitions dier
Recall that the set of all (complete and un-
.
is
isolatedoris dened
Φ(x,A) ` q(
x)
,
i. e., each formula in
by a set
Φ(x,A)
Φ(x,A) ∪ T ∪ tp(A)
q(
.
x)
is
1.2.
INESSENTIAL COMBINATIONS AND COLORINGS
The following two Lemmas are obvious.
55
1.2.1.4. Lemma.
Φ(x,A)
is
and the type
isolated by
Ψ(x,A)
If a type
Φ(x,A)
.
1.2.1.5. Lemma.If|= Φ(
x, y)
Φ(
type
(
if
and only if the type
a)
Ã
n
^
i=0
is
ϕi(x, y)
isolated by
!
¯
¯
¯
¯
ϕ0(x, y),
tp(
y
∃
Recall that a structure
T = Th(M)
S(T)
.
Note that a models
over the empty set is realizable inM, that is,
Mpis weakly saturated if and only ifpis
x) ∈⊆S(A)
q(
is isolated by a set
a, b)
then
the type
b/a)
tp(
is
isolated by
. . . , ϕn(
Misweakly saturated
is
isolated by a set
Ψ(x,A)
ab)
Φ(
is
a, y)
tp(
x, y) ∈ Φ(x,y),
if any type of
then
isolated by
and
n ∈ ω).
M |=
a powerful type.
1.2.1.6. Denition.
x)
p(
a
type ofTlying in
x)
is
isolated by a set of formulas
.
and
basedifp(
δ ∈ {0, 1}
Let∆be a set of formulas of a theoryT,
x)
S(T)
. The type
ϕδ∈ p
p(
is
said to be∆-
, where
ϕ ∈ ∆
The following lemma, being a corollary of Compactness Theo-
rem, noticed in E. A. Palyutin, J. Sae, S. S. Starchenko [330].
q(
x)
the
,
1.2.1.7. Lemma.
a
-b
of
ased.
any tuple
a)is∆
tp(
A theory
T
is∆-based if and only if, for
any(some)weakly saturated model ofT, the type
Similarly, by Compactness we have
1.2.1.8. Lemma.
only if for any(some)weakly saturated model
a ∈ M
tuple
length
a) ∈ U
l(
Since∆-baseness means the∆-baseness over the set
A theoryTis∆-based over a set
MofT
c
onsisting of pairwise distinct coordinates and with
,
the type
tp(
a)is∆
-b
ased.
U ⊆ ω
if and
and for any
U = ω
Lemma 1.2.1.8 is an immediate generalization of Lemma 1.2.1.7.
,
56
Chapter
1. CHARACTERIZATION OF EHRENFEUCHTNESS
1.2.1.9. Denition.
Σ1and
(Σ1∩ Σ2)
a
of
Σ2respectively such that
. A structure
combination of structures
M1and
M2over
Let
M1and
M
of the language
M1and
M0, or a
M2be structures of languages
M0 M1¹ (Σ1∩ Σ2) = M2¹
Σ1∪ Σ2is said to be
union of
M2over
M1and
M0, or a
M2over
M = M1and interpretations of language symbols of
with corresponding interpretations in
M ¹ Σi,
to be the
and it is denoted by
i = 1,2
If
Σ1∩Σ2= ∅
combination
. We denote
MbyComb
then the combination
or the
pairing
Comb(M1, M2)
M1and
(M1, M2)
M
0
Comb
M
of structures
.
M2, i. e.,
(M1, M2)
0
Note that by the denition any combination
M = Comb
can be represented as a combination
M2¹ (Σ(M2) \ Σ(M1))
A theoryTis said to be a
and
T2over models
.
combination
Mi|= Ti,
M1¹ Σ(M2) = M2¹ Σ(M1)
Comb
theories
combination
M1and
tpM(
and
of tuples
of
tp
be
if any tuple
with length
inessential over a set
model
(M1, M2) = Comb(M1, M2)
M
0
T1and
T2over the models
or the
pairing
M2.
a
b
Let
A combination
inessential over a set
A combination
e a tuple in a structure
a)
is
said to be an
a)iftpM(a)
tp
(
M
2
M
a)
(
1
a ∈ M
and
,
such that
tp
M
M = Comb
a ∈ M
l(
,
a) ∈ U
inessential combination of types
is
isolated by
(a)
,
will be denoted by
2
U ⊆ ω
consisting of pairwise distinct coordinates and
,
belongs to
T
of theories
U ⊆ ωifM = IEC
M |= T
, where
Miis a restriction of
of the theories
tpM(
(M1, M2)
M
0
Comb(M1, M
i = 0,1, 2
, if
, where
T = Th (Comb
M0,
M = Comb
tp
M
a)
is
an inessential combination
(M1, M2)
M
0
(written
IECTM.
T1and
0
)
2
or a
pairing
T0= T1∩T2,
(M1, M2))
M
0
then the pairingTof the
M1,
M2is called the
T1and
1
T2over the models
(M1, M2)
M
0
a) ∪ tp
(
M
IECTM.
of structures is said to
M = IEC
T2is said to be
U
(M1, M2)
M
0
M
to the language
Σ(Ti),i = 0,1, 2,T0= T1∩ T2.
pairing
M
coincide
.
M1and
, where
of theories
. A type
tp
(a)
.
The set
2
U
(M1, M2)
M
0
for any
M0, if
Mi=
is said
M2,
0
M
=
2
T
M0=
. If
a)
(
M
1
1
)
1.2.
INESSENTIAL COMBINATIONS AND COLORINGS
An inessential combination of structures (theories) over the set
U = ω
will be simply called an
inessential combination
structures (theories).
A combination
be
almost inessential
U
IEC
(M1, M2)
M
0
A combinationTof theories
inessential
we have
if for some innite setUand for any model
M = IEC
to the language
As above, we shall omit the index
also omit the index
M = Comb
(written
(M1, M2)
M
0
M = AIEC
of structures is said to
M
0
for some innite setU.
U
(M1, M2)
M
0
T1and
, where
T2is said to be
Miis a restriction of
Σ(Ti),i = 0,1, 2,T0= T1∩ T2.
·
if
Σ1∩ Σ2= ∅
M
0
·Uif
U = ω
.
(M1, M2)
) if
M |= T
. We shall
By the denition, any inessential combination of structures
(theories) is almost inessential.
57
of these
M =
almost
,
M
1.2.1.10. Proposition.
and
T2,
M
a weakly saturated model ofT,
Let
T
be a combination of theories
following conditions are equivalent:
(1)
the combinationTis inessential overU;
(2)
M = IEC
to the language
Proof
is obvious.
U
(M1, M2)
M
0
, where
Σ(Ti),i = 0,1, 2,T0= T1∩ T2.
In particular, we have
1.2.1.11. Corollary.
and
T2,
M
a weakly saturated model ofT,
Let
T
be a combination of theories
following conditions are equivalent:
(1)
the combinationTis(almost)inessential;
(2)
M = IEC
innite setU)
, where
(M1, M2)(M = IEC
M
0
Miis the restriction of
Σ(Ti),i = 0,1, 2,T0= T1∩ T2.
1.2.1.12. Theorem.
ories
Ti,
i = 1,2
, over a set
Let
T
be a combination of
U ⊆ ω
are equivalent:
(1)
the combinationTis inessential overU;
(2)Tis
(∆1∪ ∆2)
-based overU.
T
U ⊆ ω
Miis the restriction of
. Then the
M
T
U ⊆ ω
U
(M1, M2)
M
0
M
. Then the
for some
to the language
∆i-based the-
. Then the following conditions
1
1
58
Chapter
1. CHARACTERIZATION OF EHRENFEUCHTNESS
Proof.
bination of theories
suces to show that for any tuple
(1) ⇒ (2)
T1and
. Suppose that
T2overU. By Lemma 1.2.1.8, it
a
T
is an inessential com-
in
a model
MofT
, con-
sisting of pairwise distinct coordinates and having some length
a) ∈ U
l(
the restriction of
T1∩ T2. Since
by
tp
1.2.1.8, the type
las and negations of formulas in
1.2.1.4, the type
type
(2) ⇒ (1)
type
distinct coordinates and having some length
by some set
formulas in
of formulas in
Lemma 1.2.1.8 we can assume that the set
tp
M
Since
As
tp(
of
T1and
M
tpM(
tpM(
a)
(
i
Φi(
a)
,
its type
M
M = IEC
a) ∪ tp
(
1
M
2
(a)
tp
tpM(
a)is(∆1∪ ∆2)
.
LetTbe a
a)
of
tuple
x) ⊆ tpM(a)
Φ(
∆1∪∆2, where
x)
Φ(
,
where
x) ⊆ tp
is
Miis the restriction of
M
an arbitrary type, thenTis an inessential combination
T2overU.
a)is(∆1∪ ∆2)
tpM(
to the language
U
(M1, M2)
M
0
.
As
Tiis
a)
is
M
(
i
a)
isolated by some set
is
isolated by
-based.
(∆1∪ ∆2)
a
in
a model
Φ(x)= Φ1(
of
language
(a)
then
i
tp
¤
-based.
Denote by
Σ(Ti),i = 0, 1, 2,T0=
a)
, the type
tpM(
is
∆i-based overUthen, by Lemma
x)
Φi(
∆i,
i = 1, 2
Φ1(
. Then, by Lemma
x) ∪ Φ2(x)
.
Thus, the
-based theory overU, i. e., any
MofT
of
formulas and negations of
, consisting of pairwise
a) ∈ U
l(
,
is isolated
x)∪ Φ2(x),Φi(x)
Σ(Ti),i = 1,2
M
(a) ∪ tp
M
1
. By hypothesis and
x)
Φi(
isolates
to the language
(a)
2
isolates
M
isolated
of
formu-
is
the set
the type
Σ(Ti)
tpM(a)
M
i
.
.
1.2.1.13. Corollary.
theories
Ti,
i = 1,2
(1)
the combinationTis(almost)inessential;
(2)Tis(almost)(∆1∪ ∆2)
1.2.1.14. Denition
beλ-stable
, whereλis an innite cardinality if for any setAof
Let
T
be a combination of
∆i-based
. Then the following conditions are equivalent:
-based.
[23, 47]. Recall that a theoryTis said to
powerλ, the number of types overAis not more thanλ, that is,
|S(A)| ≤ λ
1.2.1.15. Theorem.
tion of theories
only if
.
T1and
If
T
is an almost inessential combina-
T1and
T2then
T
T2areλ-stable(small).
isλ-stable(small)if and
1.2.
INESSENTIAL COMBINATIONS AND COLORINGS
59
Proof.
(smallness) ofTimpliesλ-stability (smallness) of
Suppose that
having the cardinalityλ. Denote by
language
On the other hand, since the setUis innite, each type
S(M)
, which is not realized inM, is isolated by a union of types
b),b ∈ M
q(x,
pairwise distinct coordinates and satisfying the condition
SinceTis
type
p(x)
in
S(Mi)
As
T1and
Σ(Ti),i = 1, 2
,
where
T2are restrictions ofT, then theλ-stability
T1and
T2areλ-stable. Consider a model
. As
Tiareλ-stable, we have
y)
q(x,
is
a type of some tuple
the inessential combination of
is isolated by union
isolated by union of types
p1(x) ∪ p2(x)
q(x,
T1and
Mia restriction of
|S(Mi)| ≤ λ
a
consisting
T1and
, where
T2overU, the
pi(x)
b) ¹ Σ(Ti),i =
T2.
MofT
M
l(
is a type
1, 2
. Hence,
to the
p(x) ∈
of
a) ∈ U
|S(M)| ≤ |M| + |S(M1)| · |S(M2)| = λ + λ · λ = λ.
As considered model
Suppose now that
|S(T2)| = ω
is
equal to a subtype of a tuple
. Since the setUis innite, any type
M
is arbitrary, the theoryTisλ-stable.
T1and
T2are small, i. e.,
a
consisting
of pairwise distinct
|S(T1)| =
x) ∈ S(T )
p(
coordinates, where the number of these coordinated belongs toU.
a
As for any tuple
complete subtypes and the type
a) ¹ Σ(T1) ∈ S(T1)
tp(
of
described form there are only nitely many
a)
tp(
and
tp(a) ¹ Σ(T2) ∈ S(T2)
is
isolated by union of types
,
then
,
.
.
|S(T)| ≤ ω · |S(T1)| · |S(T2)| = ω · ω · ω = ω,
that is, the theoryTis small too.
1.2.1.16. Denition.
a
set of formulas
x)
is
p(
an)
(
if
said
n∈ω
|= ϕn(
to be
isolated by
of
tuples in
an)
regular
for
ϕn(
any
Let
x),n ∈ ω
x)
.
Φ(
M
is said to be
n ∈ ω
in
M
p(
Consider a model
. A dening sequence of type
if the type
Otherwise a dening sequence is said to be
¤
x)
b
e a type in
,
such that
` ϕ
dening
x)
has
p(
S(T),Φ(
x) → ϕn(x)
(
n+1
MofT
for
p(
x) ⊂ p(x)
. A sequence
x)
(o
ver
p(
a realization inM.
irregular inM.
Φ(
and
x)
x)
is
Clearly, any sequence of tuples may be dening for a unique
type. So it is possible not to indicate types corresponding to regular
sequences.
)
60
Chapter
1. CHARACTERIZATION OF EHRENFEUCHTNESS
an)
If
(
p(
n∈ω
x)inM
p(
x)
in
LimM(
type
an)
(
the set
of
1.2.1.17.
of weakly saturated structures
sequence
Then
the structure
tial combination of theories
Proof.
Th(M)
b
e a dening sequence of type
in
and show that
M1and in
a regularity of this sequence in
there exists a tuple
As
the combination of structures is inessential, the set
tp
so
M
a)
(
M
2
x)
p(
= tpM(
is weakly saturated.
is
n∈ω
and
a regular, in the modelM, dening sequence of
M
M
and
an)
n∈ω
a)
|= p(
,
then we say that
it will be denoted by
coincides
.
Proposition.
Let
M1and
(
an)
n∈ω
in
Lim
Miis
M
M
regular in
an)
(
1
n∈ω
∩ Lim
is weakly saturated and
Th(M1)
Consider an arbitrary type
x)
is
p(
realizable inM. Indeed, let
p(
M2. The weak saturation of
a
isolates
the type
a)
andais
suc
h that
a ∈ Lim
tpM(
a realization of type
a
a ∈ LimM(an)
with the set
M
be an inessential combination
p(M)
M2such that any regular
M
,
i = 0,1
2−i
M
(an)
2
n∈ω
6= ∅.
Th(M)
and
Th(M2)
x) ∈ S(∅)
p(
x)inM
.
Then
M1and
M1, and also in
(an)
a)
.
But
M
tp
1
M
n∈ω
a) ∪ tp
(
1
An inessentiality of the combination of theories
Th(M2)
1.2.1.11.
follows from the weak saturation of
¤
is
a
limit point
.
n∈ω
of realizations
, and, moreover,
is an inessen-
.
of
the theory
an)
(
an)
(
n∈ω
is
dening
M2implies
M2. Moreover,
∩Lim
x)inM
p(
M
2
tp
(a) ⊂ p(x)
M
2
(an)
M
1
.
Th(M1)
M
by Corollary
of
Here,
n∈ω
n∈ω
a) ∪
(
Thus
and
.
,
1.2.2. Colored structures
1.2.2.1. Denition.
Col:
M → λ ∪ {∞}
innity, is said to be a
a ∈ M
hM,
, the value Col
Coliis said to be a
Let
M
, whereλis a power and
coloring of structure
(a)
is said to be a
colored structure
Below, colored structures
pansions of
M
by disjoint unary predicates Colµ= {a ∈ M |
be a structure. Any function
∞
is a symbol of
M
. Here, for any
color of elementa. A pair
.
hM,
Coliwill be identied with ex-
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