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1.3.
TYPE REDUCIBILITY
81
The predicateSconnects all possible triples of elementsa,b,
such that
Thus,
¬(y ≈ z))
The theory
T0in Example 1.2.3.5 to the language
result
Q
T
, and with the following axioms:
0
1)
|= R2(a, b) ∧ R2(a, c) ∧ ∃z(Q(z, b) ∧ Q(z, c)) ∧ ¬(b ≈ c)
S(x, y, z) ≡ (R2(x, y) ∧ R2(x, z) ∧ ∃u(Q(u, x) ∧ Q(u, y)) ∧
.
Q
T
of the theory
0
united with the
n∈ω
R2instead ofQin
R
2
T
of substitution
0
T1is axiomatized by the restriction
hQ, Colni
Q
Q
[T
]
of the symbol
0
R
2
∀x, y¡Q(x, y) →¡∃=1z1(Q(x, z1) ∧ R1(z1, y))∧
∧∃=1z2(Q(x, z2) ∧ R1(y, z2))¢¢;
µ
2)
n ≥ 1
3)
4)
n ∈ ω
5)
n
¬∃x, y0, . . . , y
V
n
Q(x, yi) ∧ (y0≈ yn) ∧
i=0
;
∀x, y(R1(x, y) → ∃z(Q(z, x) ∧ Q(z, y)))
∀x (Coln(x) → ∀y ((R1(x, y) ∨ R1(y, x)) → Coln(y)))
;
∀x, y, z(S(x, y, z) ↔ (R2(x, y) ∧ R2(x, z)∧
n−1
V
R1(yi, y
i=0
;
c
.
)¶,
i+1
,
∧∃u(Q(u, y) ∧ Q(u,z)) ∧ ¬(y ≈ z)));
6)
∀x, y(R2(x, y) → (∃=1zS(x, y, z)∧
∧∀z0((S(x, y, z0) ∧ Coln(y)) → Coln(z0)))), n ∈ ω;
7)
∀x, y(∃z(Q(x, z) ∧ R2(y, z)) → (∃≤2z(Q(x, z) ∧ R2(y, z))∧
∧∀z1, z2((Q(x, z1) ∧ Q(x,z2) ∧ R2(y, z1) ∧ R2(y, z2) ∧ Coln(z1)) →
→ Coln(z2)))), n ∈ ω;
8)
∀x, y, z¡(R2(x, y) ∧ R2(x, z) ∧ ¬S(x, y, z)) →
→ ¬(Q ∪ Q−1)n(y, z)¢, n ∈ ω;
82
Chapter
9)
∀x, y, z¡(R2(y, x) ∧ R2(z, x) ∧ ¬(y ≈ z)) →
1. CHARACTERIZATION OF EHRENFEUCHTNESS
→ ¬(Q ∪ Q−1)n(y, z)¢, n ∈ ω;
10)
∀x, y (R2(x, y) → ¬(Q ∪ Q−1)n(x, y)),n ∈ ω
ÃÃ
;
0
n−1
_
→
∃y0((S(x, y, y0) ∨ (y0≈ y)) ∧ (zi≈ x) ∧ (z
i=0
αi∈ {Q, Q−1, R2, R
¡
12)
∀y1, y
(∃x(Q(x, y1) ∧ Q(x,y2)) ∧ Coln(y1) ∧ Coln(y2)∧
2
∧¬(y1≈ y2)) → ∃≥ωz(Col0(z) ∧ S(z, y1, y2))¢, n ∈ ω;
ÃÃ
13)
∀y1, y
2
^
∧
−j<k<j
¬R
k
1
n
∧
−1
},0 ≤ i < n < ω
2
n−1
^
i=0
2
αi(zi, z
i+1
0
)!→
;
n
≈ y0))!,
i+1
∃x(Q(x, y1) ∧ Q(x,y2)) ∧ Coli(y1) ∧ Coli(y2)∧
(y1, y2)!→ ∃≥ωz(Colj(z) ∧ S(z, y1, y2))!,
0 < j ≤ i < ω
14)
∀x, y, zµ(Coli(x) ∧ S(x, y, z)) →
15)
0 ≤ i ≤ j < ω,i ≤ k < ω,k > 0
;
V
−i<k<i
∀x¡Coli(x) → ∃≥ωy, z¡Colj(y) ∧ R
.
k
¬R
(y, z)¶,
1
k
(y, z) ∧ S(x, y, z)¢¢,
1
Similar to Example 1.2.3.5 we shall state that all requirements
can be realized such that the theory
T1= Th(M)
of structure
i ∈ ω
;
1.3.
TYPE REDUCIBILITY
M
, satisfying the list of axioms, is∆-based, where∆is the least
83
closed (with respect to substitutions of variables) set of formu-
las without more then two free variables, containing the formula
(x ≈ y)
∃z (ϕ(x, z) ∧ <
and satisfying the following condition: if
δ
1
(z, y) ∧ Col
δ
2
(z)) ∈ ∆
n
, where
ϕ(x, y) ∈ ∆
then
δ1∈ {−1, 1},δ2∈
{0, 1},<1(x, y) = <(x, y),<−1(x, y) = <(y, x),< ∈ {Q, R1, R2}
1
Col
(z) = Coln(z)
n
,
0
Col
(z) = ¬Coln(z)
n
.
,
1.3.3.2. Theorem.
Proof.
the restriction
By Axiom (5) it suces to show the∆-baseness for
T
work with elements in aω-saturated model
The theory
0
of
T1to the language
1
T1is∆-based.
hQ, R1, R2, Colni
M1of
T
1
n∈ω
0
.
In view
and to
of Lemmas 1.2.1.5 and 1.2.1.7 it suces to prove that for any
tuple
a = ha0,
. . . , ani ∈
ements, the following
a
element
ha1, . . . , a
where
akin
k−1
|= Col
such that the type
and
, a
, . . . , ani
k+1
δ
(ak),|= χ
n
tp(ak/
property by induction on
a)
If
l(
= 1
then
tp(
M1,
consisting of pairwise distinct el-
choice condition
a set
δ
a)
Φ Φ(x, (
, of formulas of form
0
(ak, ai),δ, δ0∈ {0, 1},n ∈ ω,χ(x, y) ∈ ∆
a0)
is
isolated byΦ. We shall prove this
a)
.
l(
is
isolated by some set of formulas
is satised: there are an
a \
{ak}))
Col
a \
,
δ
(x),χ
n
{ak} =
0
δ
Col
(x, ai)
δ
(x)
n
since any two elements having same color are connected by an
automorphism. Whence
Let
a) ≥ 2
l(
.
a = ha0i
If there are elements
distinct connected components with respect to
a
tuple
of
aiand for the tuple
C(ai, Q ∪ R2)
united
do
So we can take for
for
satises
elements in
the choice condition. Indeed, for the tuple
a
lying
in the common connected component with
a2of
elements in
, the type
tp(
a)
with the set of formulas in∆\saying" that elements in
not connect with elements in
akan element satisfying the choice condition
a1existing
by the induction hypothesis.
Thus, below we assume that all elements in
mon
(Q ∪ R2)
be denoted by
-component. As above the tuple
a \
{an}
.
satises
the choice condition.
ai,
a
,
which do not belong to
is
isolated by the set
a2b
y
(Q ∪ Q−1∪ R2∪ R
a
ajin
b
elonging to
Q ∪ R2then the
a1) ∪ tp(a2)
tp(
−1
2
a
b
elong to a com-
ha0, . . . , a
)
-paths.
i
n−1
a
a
will
,
,
1
1
84
1.3.3.3. Lemma.
(1)
Chapter
for any
1. CHARACTERIZATION OF EHRENFEUCHTNESS
Let the tuple
ai,
ajin
consisting of elements in
(2)ifai∈ C(aj,
Q)
consisting of elements in
a)
Then
the type
tp(
is
isolated by the set
a
satisfy
a
ther
e is
(Q∪Q−1∪R2∪R
a
;
then there is a
a
.
the following conditions:
−1
2
(Q ∪ Q−1)
n
S
Θi∪ Λ
i=0
)
-path
, where
-path
r(ai,
r0(ai,
aj)
aj)
δ
Θi {Col
ni
(xi) | |= Col
n
Λ {Q(xi, xj) | |= Q(ai, aj)} ∪ {(R
|= ∃y(Q(y, ai) ∧ Q(y, aj)) ∧ (R
∪{∃y(R2(xi, y)∧R
s
(y, xj)) | |= R2(ai, aj)∧∃y(R2(ai, y)∧R
1
s ∈ Z} ∪ {¬∃y(R2(xi, y) ∧ R
|= R2(ai, aj) ∧ ¬∃y(R2(ai, y) ∧ R
δ
ni
(ai), δni∈ {0, 1}, n ∈ ω},
n
s
δ
ij
)
(xi, xj) |
1
s
δ
ij
)
(ai, aj), s ∈ ω, δij∈ {0, 1}}∪
1
s
(y, xj)) |
1
s
(y, aj)), s ∈ Z}∪
1
∪{¬∃y(Q(y, xi) ∧ Q(y, xj)) | |= ¬∃y(Q(y, ai) ∧ Q(y, aj))∧
∧∃z(R2(z, ai) ∧ R2(z, aj))} ∪ {¬(xi≈ xj) | i 6= j}.
a)
Proof.
We shall use induction on
was considered above. Let
a) ≥ 2
l(
.
.
l(
The case
We denote by
Θi∪ {ψ(xi, aj) | |= ψ(ai, aj), ψ(xi, xj) ∈ Λ},0 ≤ i ≤ n
nite tree
dened by predicate
and only if there are
R2(b,
end-vertex
of vertices being elements in the tuple
formed by elements in
in the tree
Γ1consisting of vertices
0
R
2
b ∈ C(ai, Q)
c) ∨ R2(c, b)
C(a
. Since the nite graph
, Q)
. Now we construct a nite tree
i
0
a
Γ2is dened by predicate
such that
b
elonging to
C(ai, Q),i = 0,. . . , n
Γ1|= R
and
0
(C(ai, Q), C(aj, Q))
2
c ∈ C(aj, Q)
Γ1is acyclic, there is its
b = hb0,
C(a
i
0
. . . , bti
, Q)
. The set of edges
Q0such that
s
(y, aj)),
1
l(a)= 1
Λithe set
. Consider a
, and edges
M1|=
with
Γ2consisting
, which is
if
Γ2|= Q0(bi, bj) ⇔
M1|= Q(bi,
Without loss of generality we assume that
a = ha0,
. . . , a
n−t−1
, b0, . . . , bti.
bj) ∨ Q(bj, bi).
1.3.
TYPE REDUCIBILITY
Let
t = 0
, i. e.,
Γ2be a trivial tree consisting of unique element
b0= an. Then one of the following disjoint cases holds:
a
in
(
a0)
some
aiis unique element of
element separates
b0and
a \
{an}
;
R2(M1,
b0)
85
, and this
(
b0)
some
aiis unique element of
ainR2(b0, M1)
is unique elementc(which does not belong to
R2(b0, c) ∧ ∃x(Q(x, ai) ∧ Q(x, c))
a \
and
aiand
b0and
{an}
;
(
c0)
the set
aj,
a \
R2(b0,
|= ∃x(Q(x, ai)∧ Q((x, aj))
{an}
.
For the case(a0)
(α0)
we have
|= ∃y(R2(ai, y) ∧ R
M1)
con
, one of the following conditions hold:
and the set
{ai, c}
tains exactly two elements in
, and the set
s
(y, b0))
1
for some
or
(β0)
we have
If
(α0)
s
M1))
R
(y,
1
Then since
|= ¬∃y(R2(ai, y) ∧ R
holds then for any
,
the equality
aiseparates
tp(d) = tp(d0)
b0and
d, d0∈ R2(ai,
a \
s
1
{an}
(y, b0))
M1) ∩
implies
, the type
for all
tp(d/ai) = tp(d0/ai)
is isolated by the set
Θn∪ {R2(ai, xn), ∃y(R2(ai, y) ∧ R
s
(y, xn))}∪
1
∪{¬∃z(Q(z, xn) ∧ Q(z, aj)) | |= R2(ai, aj), j < n}.
,
and there
a
)
such that
separates
{ai, aj}
separates
s ∈ Z \ {0}
s ∈ Z \ {0}
∃y(R2(ai, y) ∧
tp(b0/(
a \
|=
b
a
,
say
.
{an}))
0
,
.
If
(β0)
holds then for any
d, d0∈ R2(ai,
the equality
type
tp(b0/(
M1) \ (∪{∃y(R2(ai,
tp(d) = tp(d0)
a \
{an}))
implies
is isolated by the set
y) ∧ R
tp(d/ai) = tp(d0/ai)
Θn∪ {R2(ai, xn)} ∪ {¬∃y(R2(ai, y) ∧ R
∪{¬∃z(Q(z, xn) ∧ Q(z, aj)) | |= R2(ai, aj), j < n}.
Thus, for the case(a0)
by
Λn.
, the type
s
M1)) | s ∈ Z \
(y,
1
. Then the
s
(y, xn)) | s ∈ Z \ {0}}∪
1
tp(an/(
a \
{an}))
{0}}),
is isolated
86
Chapter
1. CHARACTERIZATION OF EHRENFEUCHTNESS
For the case(b0)
(γ0)
we have
, one of the following conditions hold:
|= ∃y(R2(b0, y) ∧ R
s
(y, c))
1
or
(δ0)
we have
Similar above,
|= ¬∃y(R2(b0, y) ∧ R
(γ0)
implies that the type
isolated by the type
Θn∪ {R2(xn, ai), ∃y(R2(xn, y) ∧ R
∪{¬(xn≈ aj) | 0 ≤ j < n},
and
(δ0)
implies that the type
tp(b0/(
set
Θn∪ {R2(xn, ai)} ∪ {¬∃y(R2(xn, y) ∧ R
∪{¬(xn≈ aj) | 0 ≤ j < n}.
Thus, for the case(b0)
, the type
tp(an/(
Λn.
If(c0)
the type
holds, having the separating set
tp(an/(
a \
{an}))
is isolated by the set
s
(y, c))
1
for some
for all
s ∈ Z \ {0}
s ∈ Z \ {0}
tp(b0/(
s
(y, ai))}∪
1
a \
{an}))
s
(y, ai)) | s ∈ Z \ {0}}∪
1
a \
is isolated by the
{an}))
{ai, aj}
is isolated by
we obtain that
a \
{an}))
,
.
is
Θn∪ {R2(xn, ai) ∧ R2(xn, aj)} ∪ {¬(xn≈ aj) | 0 ≤ j < n}
and so
tp(an/(
Thus, if
Now we assume that
tree
Γ2such that the tuple
a \
t = 0
{an}))
then
is isolated by
tp(an/(
a \
t ≥ 1
. We choose an end-vertex
ha0, . . . , a
{an}))
n−t−1
Λn.
is isolated by
, b0, . . . , b
l−1
, b
Λn.
l+1
blin the
, . . . , bti
satises the Lemma's conditions. Without loss of generality we
assume that
l = t
. There are following possibilities (and at least
one of them is satised):
b
(
at)
some element
M1)
Q(bt,
(
in
Q(
s
R
(bj, bt) ∧ Q(bi, bj)
1
,
and this element separates
bt)
some element
M1,
bt)
biin
is
unique element of the tuple
btand
b
biin
is
unique element of the tuple
, and there is unique element
for some
s ∈ Z \ {0}
;
a \
bjin
{an}
b
;
suc
h that
a
in
|=
a
1.3.
TYPE REDUCIBILITY
(
ct)
some element
M1,
Q(
Q(
|Q(bi,
some
unique
bt)
, and this element separates
(
dt)
some element
M1,
bt)
, some element
M1) ∩ R2(aj, M1)| =
s ∈ Z \ {0}
(
et)
the condition(át)
element of the tuple
unique element of
and there are no
biin
biin
ajis unique element of
;
ainR2(M1,
s ∈ Z\ {0}
b
is
unique element of the tuple
btand
b
is
unique element of the tuple
a \
{an}
;
ainR2(M1,
2
, and
|= ∃y(R2(aj, y) ∧ R
is not satised, some element
a
in
Q(M1,
bt),|Q(bi,
such that
bt)
, some element
M1) ∩ R2(aj, M1)| =
|= ∃y(R2(aj, y) ∧ R
s
(y, bt))
1
biin
s
(y, bt))
1
87
a
in
a
in
bt)
for
b
ajis
2
,
is
,
.
If(at)
holds, the separability of
that the type
tp(bt/(
a \
{an}))
btand
is isolated by the set
a \
{an}
by
biimplies
Θn∪ {Q(xn, bi)} ∪ {¬(xn≈ aj) | 0 ≤ j < n}.
For the case(bt)
k
R
(bj, xn)
1
set
{Q(bi, xn), R
(
bt)
we obtain in any case that the type
by the set
Since by the choice of
of Lemma, then by induction and Lemma 1.2.1.5 the type
isolated
by the set
Thus, if the tuple
1.3.3.3, then
Suppose that
We extend
, we have
bt∈ dcl({bi, bj})
and whence the type
k
(bj, xn)}
1
. Considering(ct){(
Λn.
btthe tuplea\{an}
n
S
Θi∪ Λ.¤
i=0
a
satises
a
a
till
satises
a
a tuple
the choice condition.
do
es not satisfy the conditions of Lemma 1.3.3.3.
b
of
least length such that the conditions of
tp(bt/(
by the formula
a \
{an}))
et)
similar to(at)
tp(an/(
a \
is isolated by the
{an}))
Q(bi, xn)∧
is isolated
satises the conditions
tp(
the conditions (1) and (2) of Lemma
Lemma 1.3.3.3 hold. Without loss of generality we assume that
hb0,
. . . , b
tree
Γ1consisting of vertices
dened by predicate
if and only if there are
M1|= R2(d1,
Now we construct a nite tree
i = ha0, . . . , an, b
n+r
d2) ∨ R2(d2, d1)
n+1
C(bi, Q),i = 0, . . . , n + r
0
R
such that
2
d1∈ C(bi, Q)
. We choose an end-vertex
, . . . , b
i
. We consider a nite
n+r
, and edges
Γ1|= R
0
(C(bi, Q), C(bj, Q))
2
and
d2∈ C(bj, Q)
C(b
i
Γ2consisting of vertices being el-
and
a)
b =
with
, Q)
0
is
.
88
Chapter
1. CHARACTERIZATION OF EHRENFEUCHTNESS
ements of a tuple
longing to
predicate
C(b
, Q)
i
0
Q0such that
Γ2|= Q0(ci, cj) ⇔
Since the tuple
a
akin
andinc
b
c = hc0,
. The set of edges in the tree
has
is
an end-vertex in the tree
. . . , cti
formed by all elements in
Γ2is dened by
M1|= Q(ci,
cj) ∨ Q(cj, ci).
least possible length, then a common element
Γ2, moreover, for the
b
b
e-
tuple
ha0, . . . , a
k−1
, a
k+1
, . . . , an, b
n+1
, . . . , b
n+r
i,
the conditions of Lemma 1.3.3.3 are satised. Without loss of
generality, we assume that
Lemma 1.3.3.3 dened for the tuple
Denote
by
X
(accordinglyY) the set of formulas which are
obtained from the formulas
s ∈ Z\{0},∃z(R2(x, z)∧R
k = n
. Let
Θi, Λ
are sets of formulas in
b
.
Q(x, y),∃x(Q(x, y)∧Q(x, z)),R
s
(z, y)),s ∈ Z
1
, (
Q(x, y),R2(x, y)
s
(x, y)
1
) by all
possible substitutions of free variables. We denote byZthe least
set of formulas containingYand such that if
then
∃z(ψ(x, z)∧χ(z, y)) ∈ Z
. Clearly each formula
ψ(x, z), χ(z, y) ∈ Z
η(x, y)inX∪Z
can be uniquely dened, up to its sequence of variables, by the tuple
σ = hσ0, . . . , σui
0 ≤ i ≤ u
, sequentially used for the construction of
of symbols
σi∈ {Q, Q−1, R2, R
−1
} ∪ {R
2
s
| s ∈ Z}
1
η(x, y)
. This
formula (as well as any formula corresponding to a sequenceσ) will
be denoted by
σ0. . . σu(x, y)
. If
a ∈
M1then
we denote by
τ(a/σ)
the formula
σ0. . . σu(a, x) ∧ ¬(a ≈ x)∧
0
∧ ∧ {¬σ
. . . σ
i
0
(a, x) | σ
i
0
∈ {Q±1, R
j
±1
}, 0 ≤ j ≤ i < u}.
2
,
,
By Lemmas 1.2.1.5 and 1.3.3.3 the type
isolated by the setΞconsisting of formulas
0
ξ
hϕ
,...ϕ
i,Λ
n+r
n+1
(x,
a \
{an}) ∃x
n+1
, . . . , x
∧ϕ(x) ∧ ∧{ψ(x, ai) | ψ(xn, xi) ∈ Λ0, i < n}∧
∧ ∧ {ψ(x, xi) | ψ(xn, xi) ∈ Λ0, i > n}∧
∧ ∧ {ψ(xi, xj) | ψ(xi, xj) ∈ Λ0, i, j > n}∧
tp(an/(
Ã
n+r
a \
n+r
^
i=n+1
{an}))
ϕi(xi)∧
is
1.3.
TYPE REDUCIBILITY
∧ ∧ {ψ(xi, aj) | ψ(xi, xj) ∈ Λ0, j < n < i}!,
89
where
the choice condition for the tuple
following condition
las
0, . . . , n − 1
ri(ai,
h{b0, . . . , b
|= α
such that if
If
and for
x
ϕi(xi) ∈ Θi,
for any formula
i
(χ
i
(x, ai))
0
δ
0
, . . . ,
, such that
Ã
n−1
^
`
i=0
We x a formula
d
n+r
i
j+1
i
, . . . , d
0
}; Q, R2i
)
holds,
an) =d
i
i
(d
,
j
j
|= ∃z(Q(z, d
i
d
= bv,
j
i
.
j
v 6= n
v > n
, then for
we set
Then for the formula
ϕ(xn) ∈ Θn,
(∗)
:
ξ(x,
¡
χ
|=¡χ
n
i
^
¡
i
χ
(x, ai)
j
j=0
®
i
be shortest
l
i
Λ ∩ X ⊆ Λ0⊂ Λ,|Λ0| < ω
a
will
be implied if we prove the
a \
i
(x, ai)
n
i
i
(an, ai)
j
{an}) ∈ Ξ
i
¢
δ
n
¢
δ
, there are formu-
i
i
,
δ
, . . . , δ
0
i
j
i
,
χ
(x, xi) ∈ ∆
j
!
i
¢
δ
j
→ ξ(x,
ξ
hϕ
,...ϕ
i,Λ
n+r
n+1
(ai, an)
a \
0
a \
(x,
-paths in the structure
{an}).
{an}) ∈ Ξ
being the restriction of the model
i
α
∈ {Q±1, R
j
i
) ∧ Q(z, d
j
i
ϕ
(x
j
ξ
v < n
hϕ
n+1
±1
2
i
j+2
we denote by
i
i
≈ x
)
j
; the variable
j
,...ϕ
n+r
},0 ≤ j ≤ li− 1,0 ≤ i < n
))
then
i
α
j
= Q−1,
ϕ
0
a \
i,Λ
(x,
{an})
. Then
i
∈ {0, 1},i =
n
i
and
. Let
M1and
i
α
= Q
i
the formula
j
j+1
xvis denoted by
the following
ϕv,
implication is forced, where the premise describes the existence
R
¡
i
α
(ai, z1) ∧ ϕ
0
, x) ∧ ϕ
i
βα
w−1
ri(ai,
an)
s
:
1
i
(z2)¢∧τ¡a
2
i
. . . α
w+1
s
1
as well as all possible shortest
i
(z1) ∧ α
1
i
li−1
| s ∈ Zª∪©R
±
i
(ai, x)
i
(z1, z2) ∧ ϕ
1
i
α
, . . . , α
0
¢
δ
i
| α
= R
w
s
−1
R
1
2
i
(z2) ∧ .. .
2
®¢
i
∧
li−1
±1
,
2
| s ∈ Zª,
of shortest
(ai, an)
-paths
transitions by relations
Ã
n−1
^
i=0
(β(x
∃z1, . . . , z
. . . ∧ α
∧∧©¡α
i
i
, x
w
w+1
li−1
i
(z
li−1
li−1
i
. . . α
0
))δ∈ Λ0, β ∈©R2R
,
.
δ ∈ {0, 1}, 0 ≤ w ≤ li− 1ª∧
90
Chapter
|= Q(d
1. CHARACTERIZATION OF EHRENFEUCHTNESS
∧∧©¡α
i
, d
w+1
i
. . . α
0
i
) ∧ Q(d
w
i
w−1
i
w+1
R
s
1
, d
α
i
w+2
i
w+2
. . . α
), (R
s
1
i
li−1
(x
(ai, x)
i
, x
w
i
))δ∈ Λ0,
w+2
¢
δ
|
δ ∈ {0, 1}, 0 ≤ w ≤ li− 2ª∧
i
|= R2(d
∧∧©¬α
i
, d
w+1
0
i
) ∧ R2(d
w
. . . α
i
w−1
i
w+1
Q−1Qα
i
, d
w+2
i
w+2
. . . α
i
li−1
) ∧ ¬∃z(Q(z, d
(ai, x) |
i
) ∧ Q(z, d
w
i
w+2
!
ª
0
0 ≤ w ≤ li− 2
Hence, the condition
→ ξ
hϕ
,...ϕ
n+1
(∗)
holds and the tuple
choice condition that we had to prove.
n+r
¤
i,Λ
(x,
a \
{an}).
a
satises
Similar to Example 1.2.3.5, using∆-baseness, a routine consid-
eration for cases of connections of elements in tuples shows that
isω-stabile.
The set of formulas
nonprincipal1-type. This type, denoted by
{¬Coln(x) | n ∈ ω }
isolates a unique, in
p∞(x)
, is realized by
elements of innite color.
For any elementaof innite color, the formula
a non-
p∞-principal
(2, p∞)
-type
q(x, y) ∈ S(T1)
S(a, x, y)
isolates
, that isolated by
the set
)),
the
T
T1,
1
{∃z (Q(z, x) ∧ Q(z, y) ∧ ¬Coln(z)) ∧ ¬R
Here for elements
anof color
n ∈ ω
, formulas
n
(x, y) | n ∈ Z}.
1
types approximating the description of the type
1.3.4. Local countable categoricity
1.3.4.1. Denition.
guage is called1-locally countably categorical
T
has nitely many nonprincipal1-types
any formulas
ordinates realize
eters in
ϕi(x) ∈ pi(x),i = 1, . . . , n
pi, the structure, dened by formulas with param-
a
,
on the set, dened by the formula
A countable theoryTof a predicate lan-
, or a
p1(x), . . . , pn(x)
, and any tuple
¬ϕ1(x)∧ . . . ∧ ¬ϕn(x)
isω-categorical.
S(an, x, y)
q(x, y).¤
LCC1
-theory
a
,
whose co-
isolate
, if
, and for
,
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