On Constructive Models of Theories with Linear Rudin-Keisler Ordering
On Constructive Models of Theories with Linear Rudin-Keisler Ordering
Alexander N. Gavryushkin
[email protected]
Novosibirsk State University
De nition A model A is said to be decidable if the set {ϕ( 0 , . . . , n ) | A |= ϕ( 0 , . . . , n )} is computable. a
a
a
a
De nition A model A is said to be decidable if the set {ϕ( 0 , . . . , n ) | A |= ϕ( 0 , . . . , n )} is computable. a
a
a
a
De nition A model A is said to be computable if its domain, functions and predicates are uniformly computable.
De nition A model A is said to be decidable if the set {ϕ( 0 , . . . , n ) | A |= ϕ( 0 , . . . , n )} is computable. a
a
a
a
De nition A model A is said to be computable if its domain, functions and predicates are uniformly computable. De nition A model A has computable presentation (is said to be computably presentable) if it is isomorphic to a computable model.
Let be a countable complete theory. Denote by ω( ) the number of countable models of up to isomorphism. T
T
T
Let be a countable complete theory. Denote by ω( ) the number of countable models of up to isomorphism. De nition A theory is said to be Ehrenfeucht theory if 3 6 ω( ) < ω. T
T
T
T
T
Let be a countable complete theory. Denote by ω( ) the number of countable models of up to isomorphism. De nition A theory is said to be Ehrenfeucht theory if 3 6 ω( ) < ω. T
T
T
T
T
De nition A model M |= is quasi-prime if it is prime over some realization of some type of the theory . T
T
Let be a countable complete theory. Denote by ω( ) the number of countable models of up to isomorphism. De nition A theory is said to be Ehrenfeucht theory if 3 6 ω( ) < ω. T
T
T
T
T
De nition A model M |= is quasi-prime if it is prime over some realization of some type of the theory . Denote by Mp the set of all (isomorphic) prime models over realizations of , i. e. T
T
p
is a prime model of where M |= ( )}.
Mp = {Ma | hMa , ai
p a
Th
(M, a),
De nition A type does not exceed a type under the Rudin-Keisler pre-order ( is dominated by ) if Mq |= . Written 6RK . ∼RK ⇔ ( 6RK & 6RK ). Mp 6RK Mq ⇔ 6RK . p
q
p
p
q
q
p
q
p
q
q
p
p
p
q
De nition A type does not exceed a type under the Rudin-Keisler pre-order ( is dominated by ) if Mq |= . Written 6RK . ∼RK ⇔ ( 6RK & 6RK ). Mp 6RK Mq ⇔ 6RK . p
q
p
p
q
q
p
q
q
p
Mq |= p ⇔ Mp Mq
q
.
p
p
p
q
De nition A type does not exceed a type under the Rudin-Keisler pre-order ( is dominated by ) if Mq |= . Written 6RK . ∼RK ⇔ ( 6RK & 6RK ). Mp 6RK Mq ⇔ 6RK . p
q
p
p
q
q
p
q
q
p
Mq |= p ⇔ Mp Mq
p
p
p
q
.
Denote by ( ) the set of all types (over ∅) consistent with the theory . S T
T
q
De nition A type does not exceed a type under the Rudin-Keisler pre-order ( is dominated by ) if Mq |= . Written 6RK . ∼RK ⇔ ( 6RK & 6RK ). Mp 6RK Mq ⇔ 6RK . p
q
p
p
q
q
p
q
q
p
p
p
p
q
Mq |= p ⇔ Mp Mq
.
Denote by ( ) the set of all types (over ∅) consistent with the theory . S T
T
Denote by ( ) the set of all types of isomorphism of Mp , throughout all ∈ ( ). This set is pre-ordered by the relation 6RK . RK T p
S T
q
De nition A type of a theory is said to be powerful in the theory T if every model M of T, realizing , also realizes every type from ( ). p
T
p
S T
De nition A type of a theory is said to be powerful in the theory T if every model M of T, realizing , also realizes every type from ( ). p
T
p
S T
De nition A model sequence M0 M1 . . . is said to be elementary chain over a type if Mn ∼ = Mp , for every ∈ ω . p
n
De nition A type of a theory is said to be powerful in the theory T if every model M of T, realizing , also realizes every type from ( ). p
T
p
S T
De nition A model sequence M0 M1 . . . is said to be elementary chain over a type if Mn ∼ = Mp , for every ∈ ω . p
n
De nition A model M is said to be limit over a type if M = S Mn , for n∈ω some elementary chain (Mn )n∈ω over , and M 6∼= Mp . p
p
Lemma (S. Sudoplatov) Every model of an Ehrenfeucht theory either quasi-prime or limit.
Lemma (S. Sudoplatov) Every model of an Ehrenfeucht theory either quasi-prime or limit.
Consider e ∈ ( )/ ∼RK . Let e = {Mp0 , . . . , Mpn }. Denote by ( e ) the number of two by two non-isomorphic models each of which is limit over some type i . M
M
RK T
IL M
p
Lemma (S. Sudoplatov) Every model of an Ehrenfeucht theory either quasi-prime or limit.
Consider e ∈ ( )/ ∼RK . Let e = {Mp0 , . . . , Mpn }. Denote by ( e ) the number of two by two non-isomorphic models each of which is limit over some type i . Theorem (S. Sudoplatov) M
M
RK T
IL M
p
The following conditions are equivalent:
1
ω(T ) < ω ;
2
|S (T )| = ω , |RK (T )| < ω , e ∈ RK (T )/ ∼RK . M
( e ) < ω,
IL M
for any
De nition Let h ; 6i is nite pre-ordered set with the least element 0 and the greatest class en in ordered factor-set h ; 6i/∼ (where ∼ ⇔ 6 and 6 ), e0 6= en . Let : /∼ → ω is a function, satisfying next properties (e0) = 0, (en ) > 0, (e) > 0, when |e| > 1. The pair ( , ) is said to be e-parameters. At that, the set is said to be the rst e-parameter and the function the second e-parameter. X
x
x
x
y
x
y
X
y
x
x
x
f
y
X
X
f
x
X f
x
f
y
f
f
De nition Let h ; 6i is nite pre-ordered set with the least element 0 and the greatest class en in ordered factor-set h ; 6i/∼ (where ∼ ⇔ 6 and 6 ), e0 6= en . Let : /∼ → ω is a function, satisfying next properties (e0) = 0, (en ) > 0, (e) > 0, when |e| > 1. The pair ( , ) is said to be e-parameters. At that, the set is said to be the rst e-parameter and the function the second e-parameter. X
x
x
x
y
x
y
X
y
x
x
x
f
y
X
f
x
X f
x
f
y
f
X
f
De nition A theory is said to be Ehrenfeucht theory with e-parameters ( , ) if there exists an isomorphism ϕ : → ( ) and for any e ∈ / ∼, an equality (ϕ(e)) = (e) holds. T
X
x
f
X
X
IL
x
f
x
RK T
Let be an Ehrenfeucht theory with e-parameters ( T
X
,f )
.
Let be an Ehrenfeucht theory with e-parameters ( De nition T
X
,f )
.
Spectrum of decidable models of Ehrenfeucht theory SDM(T )
is a pair ( , ), where = { ∈ | element corresponds to a decidable model of the theory } (corresponds in terms of isomorphism ϕ form previous de nition); δ = δ (δ is domain of a function), ( ( ) = ⇔ there exist exactly decidable limit non-isomorphic models of over the model, corresponding to ). Y
g
Y
x
X
x
T
f
g x
m
g
m
T
x
Let be an Ehrenfeucht theory with e-parameters ( De nition T
X
,f )
.
Spectrum of decidable models of Ehrenfeucht theory SDM(T )
is a pair ( , ), where = { ∈ | element corresponds to a decidable model of the theory } (corresponds in terms of isomorphism ϕ form previous de nition); δ = δ (δ is domain of a function), ( ( ) = ⇔ there exist exactly decidable limit non-isomorphic models of over the model, corresponding to ). Y
g
Y
x
X
x
T
f
g x
m
g
m
T
x
De nition
Spectrum of computable models of Ehrenfeucht theory
is a pair ( , ), where = { ∈ | element corresponds to a computable model of the theory }; δ = δ , ( ( ) = ⇔ there exist exactly computable limit non-isomorphic models of over the model, corresponding to ).
SCM(T )
Y
g
Y
x
X
x
T
g x
m
f
g
m
T
x
Problem Describe sets theory T .
SDM(T )
and
SCM(T )
for arbitrary Ehrenfeucht
Denote by
a linear ordered set, composed of 1 < . . . < n }.
{x0 < x
L
n
x
n
+
1 elements:
Denote by
a linear ordered set, composed of { 0 < 1 < . . . < n }. Theorem 16 ∈ω x
Let
x
L
n
n
+
1 elements:
x
n
n
theory T
. There exists hereditary decidable Ehrenfeucht
( n) ∼ = Ln
for which RK T
holds.
Denote by
a linear ordered set, composed of { 0 < 1 < . . . < n }. Theorem 16 ∈ω x
Let
L
n
x
n
+
1 elements:
x
n
n
theory T
. There exists hereditary decidable Ehrenfeucht
( n) ∼ = Ln
for which RK T
Theorem 16
0
holds.
∈ ω , 6 k 6 n. There exists Ehrenfeucht theory Tn ( n) ∼ = Ln holds. At that, models, corresponding to elements x0 , x1 , . . . , xk from Ln , are decidable, models, corresponding to elements xk +1 , . . . , xn , have no computable
Let
n
which RK T
presentations.
for
Theorem 16
m ∈ ω , there exists Ehrenfeucht theory Tm , such that ( m) ∼ = Lm , every quasi-prime model of Tm is not computably
For all RK T
m.
presentable and there exists computably presentable model of T
Theorem 16
m ∈ ω , there exists Ehrenfeucht theory Tm , such that ( m) ∼ = Lm , every quasi-prime model of Tm is not computably
For all RK T
m.
presentable and there exists computably presentable model of T
Corollary 16
m ∈ ω , there exists Ehrenfeucht theory Tm , ∼ RK (Tm ) = Lm , such that a model M |= Tm have computable presentation if and only if M is limit model over powerful type For all
m
the theory T
.
of
Theorem 16
m ∈ ω , there exists Ehrenfeucht theory Tm , such that ( m) ∼ = Lm , every quasi-prime model of Tm is not computably
For all RK T
m.
presentable and there exists computably presentable model of T
Corollary 16
m ∈ ω , there exists Ehrenfeucht theory Tm , ∼ RK (Tm ) = Lm , such that a model M |= Tm have computable presentation if and only if M is limit model over powerful type For all
m
the theory T
of
.
Corollary 16
m ∈ ω , there exists Ehrenfeucht theory Tm , ∼ RK (Tm ) = Lm , such that every quasi-prime model of Tm For all
m,
computable presentation, every limit model of T computable presentation.
have
have no
Thank you for attention!
Alexander N. Gavryushkin
[email protected]
Novosibirsk State University
De nition A model A is said to be decidable if the set {ϕ( 0 , . . . , n ) | A |= ϕ( 0 , . . . , n )} is computable. a
a
a
a
De nition A model A is said to be decidable if the set {ϕ( 0 , . . . , n ) | A |= ϕ( 0 , . . . , n )} is computable. a
a
a
a
De nition A model A is said to be computable if its domain, functions and predicates are uniformly computable.
De nition A model A is said to be decidable if the set {ϕ( 0 , . . . , n ) | A |= ϕ( 0 , . . . , n )} is computable. a
a
a
a
De nition A model A is said to be computable if its domain, functions and predicates are uniformly computable. De nition A model A has computable presentation (is said to be computably presentable) if it is isomorphic to a computable model.
Let be a countable complete theory. Denote by ω( ) the number of countable models of up to isomorphism. T
T
T
Let be a countable complete theory. Denote by ω( ) the number of countable models of up to isomorphism. De nition A theory is said to be Ehrenfeucht theory if 3 6 ω( ) < ω. T
T
T
T
T
Let be a countable complete theory. Denote by ω( ) the number of countable models of up to isomorphism. De nition A theory is said to be Ehrenfeucht theory if 3 6 ω( ) < ω. T
T
T
T
T
De nition A model M |= is quasi-prime if it is prime over some realization of some type of the theory . T
T
Let be a countable complete theory. Denote by ω( ) the number of countable models of up to isomorphism. De nition A theory is said to be Ehrenfeucht theory if 3 6 ω( ) < ω. T
T
T
T
T
De nition A model M |= is quasi-prime if it is prime over some realization of some type of the theory . Denote by Mp the set of all (isomorphic) prime models over realizations of , i. e. T
T
p
is a prime model of where M |= ( )}.
Mp = {Ma | hMa , ai
p a
Th
(M, a),
De nition A type does not exceed a type under the Rudin-Keisler pre-order ( is dominated by ) if Mq |= . Written 6RK . ∼RK ⇔ ( 6RK & 6RK ). Mp 6RK Mq ⇔ 6RK . p
q
p
p
q
q
p
q
p
q
q
p
p
p
q
De nition A type does not exceed a type under the Rudin-Keisler pre-order ( is dominated by ) if Mq |= . Written 6RK . ∼RK ⇔ ( 6RK & 6RK ). Mp 6RK Mq ⇔ 6RK . p
q
p
p
q
q
p
q
q
p
Mq |= p ⇔ Mp Mq
q
.
p
p
p
q
De nition A type does not exceed a type under the Rudin-Keisler pre-order ( is dominated by ) if Mq |= . Written 6RK . ∼RK ⇔ ( 6RK & 6RK ). Mp 6RK Mq ⇔ 6RK . p
q
p
p
q
q
p
q
q
p
Mq |= p ⇔ Mp Mq
p
p
p
q
.
Denote by ( ) the set of all types (over ∅) consistent with the theory . S T
T
q
De nition A type does not exceed a type under the Rudin-Keisler pre-order ( is dominated by ) if Mq |= . Written 6RK . ∼RK ⇔ ( 6RK & 6RK ). Mp 6RK Mq ⇔ 6RK . p
q
p
p
q
q
p
q
q
p
p
p
p
q
Mq |= p ⇔ Mp Mq
.
Denote by ( ) the set of all types (over ∅) consistent with the theory . S T
T
Denote by ( ) the set of all types of isomorphism of Mp , throughout all ∈ ( ). This set is pre-ordered by the relation 6RK . RK T p
S T
q
De nition A type of a theory is said to be powerful in the theory T if every model M of T, realizing , also realizes every type from ( ). p
T
p
S T
De nition A type of a theory is said to be powerful in the theory T if every model M of T, realizing , also realizes every type from ( ). p
T
p
S T
De nition A model sequence M0 M1 . . . is said to be elementary chain over a type if Mn ∼ = Mp , for every ∈ ω . p
n
De nition A type of a theory is said to be powerful in the theory T if every model M of T, realizing , also realizes every type from ( ). p
T
p
S T
De nition A model sequence M0 M1 . . . is said to be elementary chain over a type if Mn ∼ = Mp , for every ∈ ω . p
n
De nition A model M is said to be limit over a type if M = S Mn , for n∈ω some elementary chain (Mn )n∈ω over , and M 6∼= Mp . p
p
Lemma (S. Sudoplatov) Every model of an Ehrenfeucht theory either quasi-prime or limit.
Lemma (S. Sudoplatov) Every model of an Ehrenfeucht theory either quasi-prime or limit.
Consider e ∈ ( )/ ∼RK . Let e = {Mp0 , . . . , Mpn }. Denote by ( e ) the number of two by two non-isomorphic models each of which is limit over some type i . M
M
RK T
IL M
p
Lemma (S. Sudoplatov) Every model of an Ehrenfeucht theory either quasi-prime or limit.
Consider e ∈ ( )/ ∼RK . Let e = {Mp0 , . . . , Mpn }. Denote by ( e ) the number of two by two non-isomorphic models each of which is limit over some type i . Theorem (S. Sudoplatov) M
M
RK T
IL M
p
The following conditions are equivalent:
1
ω(T ) < ω ;
2
|S (T )| = ω , |RK (T )| < ω , e ∈ RK (T )/ ∼RK . M
( e ) < ω,
IL M
for any
De nition Let h ; 6i is nite pre-ordered set with the least element 0 and the greatest class en in ordered factor-set h ; 6i/∼ (where ∼ ⇔ 6 and 6 ), e0 6= en . Let : /∼ → ω is a function, satisfying next properties (e0) = 0, (en ) > 0, (e) > 0, when |e| > 1. The pair ( , ) is said to be e-parameters. At that, the set is said to be the rst e-parameter and the function the second e-parameter. X
x
x
x
y
x
y
X
y
x
x
x
f
y
X
X
f
x
X f
x
f
y
f
f
De nition Let h ; 6i is nite pre-ordered set with the least element 0 and the greatest class en in ordered factor-set h ; 6i/∼ (where ∼ ⇔ 6 and 6 ), e0 6= en . Let : /∼ → ω is a function, satisfying next properties (e0) = 0, (en ) > 0, (e) > 0, when |e| > 1. The pair ( , ) is said to be e-parameters. At that, the set is said to be the rst e-parameter and the function the second e-parameter. X
x
x
x
y
x
y
X
y
x
x
x
f
y
X
f
x
X f
x
f
y
f
X
f
De nition A theory is said to be Ehrenfeucht theory with e-parameters ( , ) if there exists an isomorphism ϕ : → ( ) and for any e ∈ / ∼, an equality (ϕ(e)) = (e) holds. T
X
x
f
X
X
IL
x
f
x
RK T
Let be an Ehrenfeucht theory with e-parameters ( T
X
,f )
.
Let be an Ehrenfeucht theory with e-parameters ( De nition T
X
,f )
.
Spectrum of decidable models of Ehrenfeucht theory SDM(T )
is a pair ( , ), where = { ∈ | element corresponds to a decidable model of the theory } (corresponds in terms of isomorphism ϕ form previous de nition); δ = δ (δ is domain of a function), ( ( ) = ⇔ there exist exactly decidable limit non-isomorphic models of over the model, corresponding to ). Y
g
Y
x
X
x
T
f
g x
m
g
m
T
x
Let be an Ehrenfeucht theory with e-parameters ( De nition T
X
,f )
.
Spectrum of decidable models of Ehrenfeucht theory SDM(T )
is a pair ( , ), where = { ∈ | element corresponds to a decidable model of the theory } (corresponds in terms of isomorphism ϕ form previous de nition); δ = δ (δ is domain of a function), ( ( ) = ⇔ there exist exactly decidable limit non-isomorphic models of over the model, corresponding to ). Y
g
Y
x
X
x
T
f
g x
m
g
m
T
x
De nition
Spectrum of computable models of Ehrenfeucht theory
is a pair ( , ), where = { ∈ | element corresponds to a computable model of the theory }; δ = δ , ( ( ) = ⇔ there exist exactly computable limit non-isomorphic models of over the model, corresponding to ).
SCM(T )
Y
g
Y
x
X
x
T
g x
m
f
g
m
T
x
Problem Describe sets theory T .
SDM(T )
and
SCM(T )
for arbitrary Ehrenfeucht
Denote by
a linear ordered set, composed of 1 < . . . < n }.
{x0 < x
L
n
x
n
+
1 elements:
Denote by
a linear ordered set, composed of { 0 < 1 < . . . < n }. Theorem 16 ∈ω x
Let
x
L
n
n
+
1 elements:
x
n
n
theory T
. There exists hereditary decidable Ehrenfeucht
( n) ∼ = Ln
for which RK T
holds.
Denote by
a linear ordered set, composed of { 0 < 1 < . . . < n }. Theorem 16 ∈ω x
Let
L
n
x
n
+
1 elements:
x
n
n
theory T
. There exists hereditary decidable Ehrenfeucht
( n) ∼ = Ln
for which RK T
Theorem 16
0
holds.
∈ ω , 6 k 6 n. There exists Ehrenfeucht theory Tn ( n) ∼ = Ln holds. At that, models, corresponding to elements x0 , x1 , . . . , xk from Ln , are decidable, models, corresponding to elements xk +1 , . . . , xn , have no computable
Let
n
which RK T
presentations.
for
Theorem 16
m ∈ ω , there exists Ehrenfeucht theory Tm , such that ( m) ∼ = Lm , every quasi-prime model of Tm is not computably
For all RK T
m.
presentable and there exists computably presentable model of T
Theorem 16
m ∈ ω , there exists Ehrenfeucht theory Tm , such that ( m) ∼ = Lm , every quasi-prime model of Tm is not computably
For all RK T
m.
presentable and there exists computably presentable model of T
Corollary 16
m ∈ ω , there exists Ehrenfeucht theory Tm , ∼ RK (Tm ) = Lm , such that a model M |= Tm have computable presentation if and only if M is limit model over powerful type For all
m
the theory T
.
of
Theorem 16
m ∈ ω , there exists Ehrenfeucht theory Tm , such that ( m) ∼ = Lm , every quasi-prime model of Tm is not computably
For all RK T
m.
presentable and there exists computably presentable model of T
Corollary 16
m ∈ ω , there exists Ehrenfeucht theory Tm , ∼ RK (Tm ) = Lm , such that a model M |= Tm have computable presentation if and only if M is limit model over powerful type For all
m
the theory T
of
.
Corollary 16
m ∈ ω , there exists Ehrenfeucht theory Tm , ∼ RK (Tm ) = Lm , such that every quasi-prime model of Tm For all
m,
computable presentation, every limit model of T computable presentation.
have
have no
Thank you for attention!
