Definable families of theories,
related calculi and ranks111Mathematics
Subject Classification: 03C30, 03C15, 03C50, 03F03, 54A05.
This research was partially
supported by Committee of Science in Education and Science
Ministry of the Republic of Kazakhstan (Grants No. AP05132349,
AP05132546) and Russian Foundation for Basic Researches (Project
No. 17-01-00531-a).
Nurlan D.
Markhabatov, Sergey V.
Sudoplatov222nur [email protected],
[email protected]
Abstract
We consider sentence-definable and diagram-definable subfamilies
of given families of theories, calculi for these subfamilies, as
well dynamics and characteristics of these subfamilies with
respect to rank and degree.
Key words: family of theories, definable subfamily, rank,
degree.
The rank for families of theories was introduced and studied in
general context in [1]. All possible values of the ranks
and degrees for families of all theories in given languages were
described in [2]. In the present paper we consider
sentence-definable and diagram-definable subfamilies of given
families of theories, calculi for these subfamilies, as well as
dynamics and characteristics of these subfamilies with respect to
rank and degree.
The paper is organized as follows. In Section 1, preliminary
notions, notations and results are collected. In Section 2, we
consider calculi subfamilies of families of theories as well as
links for sentence-definable and diagram-definable subfamilies.
Compactness and E-closeness for definable subfamilies are
studied in Section 3. In Section 4 we consider dynamics of ranks
with respect to definable subfamilies of theories and prove the
existence of subfamilies of given rank.
1 Preliminaries
Throughout we consider families T of complete
first-order theories of a language Σ=Σ(T).
Throughout the paper we consider complete first-order theories T
in predicate languages Σ(T) and use the following
terminology in [1, 3, 4, 5, 6, 7, 8, 9].
Let P=(Pi)i∈I, be a family of nonempty unary predicates,
(Ai)i∈I be a family of structures such that
Pi is the universe of Ai, i∈I, and the
symbols Pi are disjoint with languages for the structures
Aj, j∈I. The structure
AP⇌i∈I⋃Ai expanded by the predicates
Pi is the P-union of the structures
Ai, and the operator mapping (Ai)i∈I to AP is the P-operator. The structure AP
is called the P-combination of the
structures Ai and denoted by CombP(Ai)i∈I if
Ai=(AP↾Ai)↾Σ(Ai), i∈I. Structures
A′, which are elementary equivalent to CombP(Ai)i∈I, will be also considered as
P-combinations.
Clearly, all structures A′≡CombP(Ai)i∈I are represented as unions of
their restrictions Ai′=(A′↾Pi)↾Σ(Ai) if and only if the set
p∞(x)={¬Pi(x)∣i∈I} is inconsistent. If
A′=CombP(Ai′)i∈I, we write
A′=CombP(Ai′)i∈I∪{∞}, where
A∞′=A′↾i∈I⋂Pi, maybe applying
Morleyzation. Moreover, we write CombP(Ai)i∈I∪{∞} for CombP(Ai)i∈I with the empty structure
A∞.
Note that if all predicates Pi are disjoint, a structure
AP is a P-combination and a disjoint union of
structures Ai. In this case the P-combination
AP is called disjoint. Clearly, for any
disjoint P-combination AP, Th(AP)=Th(AP′), where
AP′ is obtained from AP replacing
Ai by pairwise disjoint
Ai′≡Ai, i∈I. Thus, in this case,
similar to structures the P-operator works for the theories
Ti=Th(Ai) producing the theory TP=Th(AP), being P-combination of Ti, which is denoted
by CombP(Ti)i∈I.
Notice that P-combinations are represented by generalized
products of structures [10].
For an equivalence relation E replacing disjoint predicates
Pi by E-classes we get the structure
AE being the E-union of the structures Ai. In
this case the operator mapping (Ai)i∈I to
AE is the E-operator. The
structure AE is also called the E-combination of the structures
Ai and denoted by CombE(Ai)i∈I; here
Ai=(AE↾Ai)↾Σ(Ai), i∈I. Similar
above, structures A′, which are elementary equivalent
to AE, are denoted by CombE(Aj′)j∈J, where Aj′ are
restrictions of A′ to its E-classes. The
E-operator works for the theories Ti=Th(Ai)
producing the theory TE=Th(AE),
being E-combination of Ti, which
is denoted by CombE(Ti)i∈I or by CombE(T), where
T={Ti∣i∈I}.
Clearly, A′≡AP realizing p∞(x)
is not elementary embeddable into AP and can not be
represented as a disjoint P-combination of
Ai′≡Ai, i∈I. At the same time,
there are E-combinations such that all
A′≡AE can be represented as
E-combinations of some Aj′≡Ai. We
call this representability of A′ to be the E-representability.
If there is A′≡AE which is not
E-representable, we have the E′-representability replacing E
by E′ such that E′ is obtained from E adding equivalence
classes with models for all theories T, where T is a theory of
a restriction B of a structure
A′≡AE to some E-class and
B is not elementary equivalent to the structures
Ai. The resulting structure AE′ (with
the E′-representability) is a e-completion, or a e-saturation, of AE. The
structure AE′ itself is called e-complete, or e-saturated, or e-universal, or e-largest.
For a structure AE the number of new structures with respect to the
structures Ai, i. e., of the structures B
which are pairwise elementary non-equivalent and elementary
non-equivalent to the structures Ai, is called the
e-spectrum of AE and
denoted by e-Sp(AE). The value sup{e-Sp(A′))∣A′≡AE} is called
the e-spectrum of the theory Th(AE) and denoted by e-Sp(Th(AE)). If structures Ai represent
theories Ti of a family T, consisting of Ti,
i∈I, then the e-spectrum e-Sp(AE) is
denoted by e-Sp(T).
If AE does not have E-classes Ai,
which can be removed, with all E-classes
Aj≡Ai, preserving the theory Th(AE), then AE is called e-prime, or e-minimal.
For a structure A′≡AE we denote by
TH(A′) the set of all theories Th(Ai) of
E-classes Ai in A′.
By the definition, an e-minimal structure A′
consists of E-classes with a minimal set TH(A′). If TH(A′) is the least for
models of Th(A′) then A′ is called
e-least.
Definition [4]. Let TΣ be
the set of all complete elementary theories of a relational
language Σ. For a set
T⊂TΣ we denote by
ClE(T) the set of all theories Th(A), where A is a structure of some
E-class in A′≡AE,
AE=CombE(Ai)i∈I, Th(Ai)∈T. As usual, if T=ClE(T) then T is said to be E-closed.
The operator ClE of E-closure can be naturally extended
to the classes T⊂T, where
T is the union of all
TΣ as follows: ClE(T) is the union of all ClE(T0) for subsets
T0⊆T, where new language symbols
with respect to the theories in T0 are empty.
For a set T⊂T of theories
in a language Σ and for a sentence φ with
Σ(φ)⊆Σ we denote by
Tφ the set
{T∈T∣φ∈T}. Any set
Tφ is called the φ-neighbourhood,
or simply a neighbourhood, for T, or the
(φ-)definable subset of T. The set
Tφ is also called (formula- or sentence-)definable (by the sentence φ) with
respect to T, or (sentence-)T-definable, or simply s-definable.
Proposition 1.1 [4]. If
T⊂T is an infinite set and
T∈T∖T then T∈ClE(T) (i.e., T is an accumulation
point for T with respect to E-closure ClE) if and only if for any formula φ∈T the
set Tφ is infinite.
If T is an accumulation point for T then we also say
that T is an accumulation point for ClE(T).
Theorem 1.2 [4]. For any sets
T0,T1⊂T, ClE(T0∪T1)=ClE(T0)∪ClE(T1).
Definition [4]. Let T0 be a closed set in
a topological space (T,OE(T)),
where OE(T)={T∖ClE(T′)∣T′⊆T}. A
subset T0′⊆T0 is said to be generating if T0=ClE(T0′). The generating set T0′ (for
T0) is minimal if
T0′ does not contain proper generating subsets. A
minimal generating set T0′ is least if T0′ is
contained in each generating set for T0.
Theorem 1.3 [4]. If T0′ is a
generating set for a E-closed set T0 then the
following conditions are equivalent:
(1)* T0′ is the least generating set for
T0;*
(2)* T0′ is a minimal generating set for
T0;*
(3)* any theory in T0′ is isolated by some set
(T0′)φ, i.e., for any T∈T0′
there is φ∈T such that
(T0′)φ={T};*
(4)* any theory in T0′ is isolated by some set
(T0)φ, i.e., for any T∈T0′
there is φ∈T such that
(T0)φ={T}.*
Definition [9]. Let T be a family of
theories and T be a theory, T∈/T. The theory T
is called T-approximated, or approximated
by T, or T-approximable, or a pseudo-T-theory, if for any formula φ∈T
there is T′∈T such that φ∈T′.
If T is T-approximated then T is called
an approximating family for T, theories T′∈T
are approximations for T, and T is an accumulation
point for T.
An approximating family T is called e-minimal
if for any sentence φ∈Σ(T), Tφ
is finite or T¬φ is finite.
It was shown in [9] that any e-minimal family
T has unique accumulation point T with respect to
neighbourhoods Tφ, and T∪{T}
is also called e-minimal.
Following [1] we define the rank RS(⋅)
for the families of theories, similar to Morley rank
[11], and a hierarchy with respect to these ranks in the
following way.
For the empty family T we put the rank RS(T)=−1, for finite nonempty families T
we put RS(T)=0, and for infinite families
T — RS(T)≥1.
For a family T and an ordinal α=β+1 we put
RS(T)≥α if there are pairwise
inconsistent Σ(T)-sentences φn,
n∈ω, such that RS(Tφn)≥β, n∈ω.
If α is a limit ordinal then RS(T)≥α if RS(T)≥β
for any β<α.
We set RS(T)=α if RS(T)≥α and RS(T)≥α+1.
If RS(T)≥α for any α, we put
RS(T)=∞.
A family T is called e-totally transcendental,
or totally transcendental, if RS(T) is an
ordinal.
Similarly [11], for a nonempty family T, we
denote by B(T) the Boolean algebra
consisting of all subfamilies Tφ, where
φ are sentences in the language Σ(T).
Theorem 1.4 [1, 11]. A nonempty family
T is e-totally transcendental if and only if the
Boolean algebra B(T) is superatomic.
Proposition 1.5 [1]. If an infinite family
T does not have e-minimal subfamilies
Tφ then T is not e-totally
transcendental.
If T is e-totally transcendental, with RS(T)=α≥0, we define the degree ds(T) of T as the maximal number of
pairwise inconsistent sentences φi such that RS(Tφi)=α.
Proposition 1.6 [1]. A family T
is e-minimal if and only if RS(T)=1 and ds(T)=1.
Proposition 1.7 [1]. For any family
T, RS(T)=RS(ClE(T)), and if T is nonempty and
e-totally transcendental then ds(T)=ds(ClE(T)).
Recall the definition of the Cantor–Bendixson rank. It is defined
on the elements of a topological space X by induction: CBX(p)≥0 for all p∈X; CBX(p)≥α if and
only if for any β<α, p is an accumulation point of
the points of CBX-rank at least β. CBX(p)=α if and only if both CBX(p)≥α and
CBX(p)≱α+1 hold; if such an ordinal α
does not exist then CBX(p)=∞. Isolated points of X
are precisely those having rank [math], points of rank 1 are those
which are isolated in the subspace of all non-isolated points, and
so on. For a non-empty C⊆X we define CBX(C)=sup{CBX(p)∣p∈C}; in this way CBX(X) is defined and CBX({p})=CBX(p) holds.
If X is compact and C is closed in X then the sup is
achieved: CBX(C) is the maximum value of CBX(p)
for p∈C; there are finitely many points of maximum rank in
C and the number of such points is the CBX-degree of C, denoted by nX(C).
If X is countable and compact then CBX(X) is a
countable ordinal and every closed subset has ordinal-valued rank
and finite CBX-degree nX(X)∈ω∖{0}.
For any ordinal α the set {p∈X∣CBX(p)≥α} is called the α-th CB-derivative Xα of X.
Elements p∈X with CBX(p)=∞ form the perfect kernel X∞ of X.
Clearly, Xα⊇Xα+1, α∈Ord,
and X∞=α∈Ord⋂Xα.
It is noticed in [1] that any e-totally transcendental
family T defines a superatomic Boolean algebra
B(T) with RS(T)=CBB(T)(B(T)), ds(T)=nB(T)(B(T)),
i.e., the pair (RS(T),ds(T))
consists of Cantor–Bendixson invariants for
B(T) [12].
By the definition for any e-totally transcendental family
T each theory T∈T obtains the CB-rank CBT(T) starting with
T-isolated points T0, of CBT(T0)=0. We will denote the values CBT(T) by RST(T) as the rank for
the point T in the topological space on T which is
defined with respect to Σ(T)-sentences.
Definition [1]. Let α be an ordinal. A
family T of rank α is called α-minimal if for any sentence φ∈Σ(T), RS(Tφ)<α or RS(T¬φ)<α.
Proposition 1.8 [1]. (1)* A family
T is [math]-minimal if and only if T is a
singleton.*
(2)* A family T is 1-minimal if and only if
T is e-minimal.*
(3)* For any ordinal α a family T is
α-minimal if and only if RS(T)=α and
ds(T)=1. *
Proposition 1.9 [1]. For any family
T, RS(T)=α, with ds(T)=n, if and only if T is represented
as a disjoint union of subfamilies
Tφ1,…,Tφn, for some
pairwise inconsistent sentences φ1,…,φn, such
that each Tφi is α-minimal.
2 Calculi for families of theories. Links for sentence-definable and diagram-definable families
In this section we define calculi for families of theories,
similar to first-order calculi for sentences, as well as discuss
properties and links for these calculi.
For a family T and sentences φ and ψ we
say that φ T-forces ψ, written
φ⊢Tψ if Tφ⊆Tψ.
We put ⊢Tψ if Tψ=T,
and φ⊢T if
Tφ=∅. For ⊢Tψ we
say that ψ is T-provable, and if
φ⊢T then we say that φ is T-contradictory or T-inconsistent.
By the definition the relation ⊢Tψ is
equivalent to χ⊢Tψ for any identically
true sentence χ, and φ⊢T is
equivalent to φ⊢Tθ for any
identically false sentence θ. So below we consider only
relations of form φ⊢Tψ and their
natural modifications.
Ordinary axioms and rules for calculi of sentences can be
naturally transformed for the relations
φ⊢Tψ obtaining T-calculi, i.e., calculi with respect to families
T.
Clearly, φ⊢∅ψ for any sentences φ
and ψ. Therefore there are sentences φ and ψ
such that φ⊢Tψ but
φ⊢ψ. Indeed, if φ and ψ are
sentences in a language Σ satisfying ⊢φ and
⊢ψ then we have φ⊢ψ whereas
φ⊢∅ψ. Besides, for the set
TΣ of all theories in the language Σ and
for T=(TΣ)ψ we have
φ⊢Tψ. Additionally, for any sentence
φ which does not belong to theories in a family
T, i.e., Tφ=∅, and for any
sentence ψ we have φ⊢Tψ.
The following obvious proposition asserts that the relation
φ⊢Tψ is monotone under ⊢ and
inclusion:
Proposition 2.1. For any sentences
φ,φ′,ψ,ψ′ and families
T,T′, if φ′⊢φ,
ψ⊢ψ′, and T′⊆T then
φ⊢Tψ implies
φ′⊢T′ψ′.
The following proposition asserts the finite character for the
relations φ⊢Tψ.
Proposition 2.2. For any sentences φ, ψ
and a family T of theories the following conditions
are equivalent:
(1)* φ⊢Tψ;*
(2)* φ⊢T0ψ for any finite
T0⊆T;*
(3)* φ⊢{T}ψ for any singleton
{T}⊆T.*
Proof. The implications (1)⇒(2) and
(2)⇒(3) hold by Proposition 2.1.
(3)⇒(1). In view of φ⊢∅ψ it
suffices to show φ⊢Tψ for nonempty
T having φ⊢{T}ψ for any singleton
{T}⊆T. But if T∈Tφ then
T∈{T}φ and using φ⊢{T}ψ we
obtain T∈{T}ψ implying T∈Tψ. Thus,
φ⊢Tψ. □
Proposition 2.3. For any sentences φ and ψ
in a language Σ the following conditions are equivalent:
(1)* φ⊢ψ;*
(2)* φ⊢TΣψ;*
(3)* φ⊢Tψ for any (finite) family (singleton) T⊆TΣ;*
(4)* φ⊢Tψ for any (finite) family (singleton) T;*
(5)* T∪{φ}⊢ψ for any
T∈TΣ.*
Proof. (4)⇒(3) and (3)⇒(2) are obvious
using Proposition 2.2.
(2)⇒(1). Assume on contrary that
φ⊢TΣψ and
φ⊢ψ. Then φ∧¬ψ is
consistent. Extending {φ∧¬ψ} till a complete
theory T in the language Σ we obtain
T∈(TΣ)φ and
T∈/(TΣ)ψ contradicting
φ⊢TΣψ.
(1)⇒(4). If φ⊢ψ then for any theory
T with φ∈T we have ψ∈T, hence
Tφ⊆Tψ for any family
T, i.e., φ⊢Tψ.
(3)⇔(5). φ⊢{T}ψ means that
φ∈T implies ψ∈T. So if φ∈T then
T⊢ψ implying T∪{φ}⊢ψ. Otherwise if
φ∈/T then ¬φ∈T. Therefore we have
{φ,¬φ}⊢ψ implying
T∪{φ}⊢ψ. Conversely, assuming on contrary
φ⊢{T}ψ we have φ∈T and
ψ∈/T, so ¬ψ∈T. Hence
T∪{φ}⊢ψ since T is complete theory and
containing ¬ψ it can not force ψ, i.e., T can not
contain ψ. □
Definition. If T is a family of theories and
Φ is a set of sentences, then we put
TΦ=φ∈Φ⋂Tφ
and the set TΦ is called (type- or diagram-)definable (by the set Φ) with respect to
T, or (diagram-)T-definable,
or simply d-definable.
By the definition we have the following properties:
-
Any d-definable subfamily of E-closed family T
is again E-closed.
-
T{φ}=Tφ.
-
TΦ={T∈T∣Φ⊆T}.
-
TΦ=T if and only if
Φ⊆∩T. In particular,
T∅=T.
TΦ∪Ψ=TΦ∩TΨ.
-
TΦ=(TΦ)Ψ for any Ψ
consisting of sentences ψ with Φ⊢ψ. In
particular, the operation (⋅)Φ is idempotent:
(TΦ)Φ=TΦ.
T{φ1,…,φn}=Tφ1∧…∧φn,
i.e., definable sets TΦ by finite Φ are
sentence-definable.
- TΦ=TΨ, where Ψ is the
closure of Φ under conjunctions.
By the latter property, studying d-definable sets, we will
usually consider sets Φ closed under conjunctions. Moreover,
by Property 5, considering d-definable families we can
additionally assume that any Φ is closed under logical
conclusions with respect to ⊢. It means that it suffices to
assume that Φ corresponds a filter with respect to the family
of d-definable subsets of T.
- For any sets Φ and Ψ containing all their logical
conclusions,
TΦ∩Ψ=TΦ∪TΨ.
Indeed, if T∈TΦ∩Ψ then
Φ∩Ψ⊆T. Assuming
T∈/TΦ∪TΨ we have
Φ⊆T and Ψ⊆T. So there are
sentences φ∈Φ∖T and ψ∈Ψ∖T. Then φ∨ψ∈/T. But by conjecture,
φ∨ψ∈Φ∩Ψ contradicting
Φ∩Ψ⊆T. Conversely, if
T∈TΦ∪TΨ then Φ⊆T
or Ψ⊆T implying Φ∩Ψ⊆T and
T∈TΦ∩Ψ.
- For any T∈T and Φ⊆T with
Φ⊢φ for all φ∈T,
TΦ={T}. So any set of axioms for T isolates
T in T. In particular, since T is an ultrafilter
and axiomatized by itself, TT={T}.
The following proposition gives obvious criteria for d-definable
sets to be s-definable.
Proposition 2.4. For any d-definable set
T=TΦ, where Φ is closed under
conjunctions, and a sentence φ∈Φ the following
conditions are equivalent:
(1)* T is s-definable by φ:
T=Tφ;*
(2)* φ⊢Tψ for any ψ∈Φ;*
(3)* each ψ∈Φ with ψ⊢φ satisfies
Tφ=Tψ;*
(3)* there are no T∈T containing
φ∧¬ψ for any ψ∈Φ.*
The sentence φ with Tφ=∅ and
satisfying the conditions in Proposition 2.4 is called T-isolating, T-principal or T-complete for Φ, and Φ is called T-isolated or T-principal.
By Proposition 2.3, TΣ-isolating sentences are
isolating for Φ, in the ordinary sense. Besides, if Φ is
forced by some φ∈Φ then for any family T,
Φ is T-isolated, but not vice versa.
Clearly, each d-definable set TΦ equals the set
Tθ, where θ=⋀Φ with possibly
infinite conjunction and Tθ is the set of all
theories T∈T containing conjunctive members of
θ.
By Property 8 finite unions of d-definable sets are again
d-definable. Considering infinite unions T′ of
d-definable sets TΦi, i∈I, we can
represent them by sets of formulas with infinite disjunctions
i∈I⋁φi, φi∈Φi. We call
these unions T′ as d∞-definable sets.
Now the definability for subfamilies of T can be
extended for infinite unions, intersections and their complements.
Notice that since all singletons {T}⊆T are
d-definable, each subfamily T′⊆T
is d∞-definable.
The relations φ⊢Tψ can be naturally
spread to sets Φ and Ψ of sentences producing relations
Φ⊢TΨ meaning
TΦ⊆TΨ.
By Proposition 2.1 the relations Φ⊢TΨ are
again monotone:
Proposition 2.5. For any sets Φ,Φ′,Ψ,Ψ′
of sentences and families T,T′, if
Φ′⊢Φ, Ψ⊢Ψ′, and
T′⊆T then
Φ⊢TΨ implies
Φ′⊢T′Ψ′.
Proposition 2.2 implies the following:
Proposition 2.6. For any sets Φ and Ψ of
sentences and a family T of theories the following
conditions are equivalent:
(1)* Φ⊢TΨ;*
(2)* Φ⊢T0Ψ for any finite
T0⊆T;*
(3)* Φ⊢{T}Ψ for any singleton
{T}⊆T.*
Proposition 2.3 immediately implies
Proposition 2.7. For any sets Φ and Ψ of
sentences in a language Σ the following conditions are
equivalent:
(1)* Φ⊢Ψ, i.e., each sentence in Ψ is forced by
some conjunction of sentences in Φ;*
(2)* Φ⊢TΣΨ;*
(3)* Φ⊢TΨ for any (finite)
family (singleton) T⊆TΣ;*
(4)* Φ⊢TΨ for any (finite)
family (singleton) T.*
Extending the list for criteria of Φ⊢TΨ we
have the following:
Theorem 2.8. For any sets Φ and Ψ of
sentences and a family T of theories the following
conditions are equivalent:
(1)* Φ⊢TΨ;*
(2)* Φ⊢ClE(T)Ψ.*
Proof. Since T⊆ClE(T) we have
(2)⇒(1) by Proposition 2.5.
(1)⇒(2). Assume that Φ⊢TΨ. It
suffices to show that if φ∈Φ, ψ∈Ψ with
Tφ⊆Tψ then (ClE(T))φ⊆(ClE(T))ψ. Let T∈(ClE(T))φ By the hypothesis we can assume that
T∈ClE(T)∖T and using
Proposition 1.1 we have infinite Tχ for any
χ∈T. Since φ∈T,
(Tφ)χ=Tφ∧χ are
also infinite for any χ∈T and therefore
Tφ⊆Tψ implies that all
(Tψ)χ are infinite. Thus again by Proposition
1.1, T∈ClE(Tψ)=(ClE(T))ψ. □
Theorem 2.8 immediately implies the following:
Corollary 2.9. For any sets Φ and Ψ of
sentences, and families T, T′,
T′′ of theories such that T′ generates
ClE(T) and
T′⊆T′′⊆ClE(T), the following conditions are equivalent:
(1)* Φ⊢TΨ;*
(2)* Φ⊢T′Ψ;*
(3)* Φ⊢T′′Ψ.*
Remark 2.10. Notice that in general case Corollary 2.9 can
not be extended to families T′′⊆ClE(T). Indeed, taking any theory T∈/ClE(T) we have, by Proposition 1.1, a sentence
χ∈T such that (ClE(T))χ is finite.
Since T∈/(ClE(T))χ and (ClE(T))χ is finite, there is a sentence
θ∈T such that (ClE(T))θ=∅. Thus for any inconsistence
sentence φ we have θ⊢ClE(T)φ whereas θ⊢ClE(T)∪{T}φ. □
The assertions above show that for any family T there
are calculi, connected with ordinary calculi for first-order
sentences [13], both for the relations
φ⊢Tψ and Φ⊢TΨ,
which satisfy monotone properties, are reflexive
(Φ⊢TΦ) and transitive (if
Φ⊢TΨ and Ψ⊢TX
then Φ⊢TX).
Definition. Sets Φ and Ψ of sentences are called
T-equivalent, written
Φ≡TΨ, if Φ⊢TΨ
and Ψ⊢TΦ, i.e.,
TΦ=TΨ.
Sentences φ and ψ are called T-equivalent, written
φ≡Tψ, if
{φ}≡T{ψ}.
Clearly, the relations ≡T are equivalent
relations both for sentences and for sets of sentences.
Proposition 2.7 and Theorem 2.8 immediately implies the following:
Proposition 2.11. For any sets Φ and Ψ of
sentences in a language Σ the following conditions are
equivalent:
(1)* Φ⊢Ψ and Ψ⊢Φ, i.e., Φ and
Ψ force each other;*
(2)* Φ≡TΣΨ;*
(3)* Φ≡TΨ for any (finite)
family (singleton) T⊆TΣ;*
(4)* Φ≡TΨ for any (finite)
family (singleton) T.*
Corollary 2.12. For any sentences φ and ψ
in a language Σ the following conditions are equivalent:
(1)* φ⊢ψ and ψ⊢φ;*
(2)* φ≡TΣψ;*
(3)* φ≡Tψ for any (finite) family (singleton) T⊆TΣ;*
(4)* φ≡Tψ for any (finite) family (singleton) T.*
Theorem 2.8 implies
Corollary 2.13. For any sets Φ and Ψ of
sentences, and families T, T′,
T′′ of theories such that T′ generates
ClE(T) and
T′⊆T′′⊆ClE(T), the following conditions are equivalent:
(1)* Φ≡TΨ;*
(2)* Φ≡T′Ψ;*
(3)* Φ≡T′′Ψ.*
3 Compactness and E-closed families
Definition. A d-definable set TΦ is called
T-consistent if TΦ=∅,
and TΦ is called locally
T-consistent if for any finite Φ0⊆Φ,
TΦ0 is T-consistent.
Clearly, locally T-consistent T-principal
sets TΦ are T-consistent.
Notice also that there are locally T-consistent
d-definable sets TΦ which are not
T-consistent. Indeed, let, for instance, T
be an e-minimal family which does not contain its unique
accumulation point T. Then by the definition of accumulation
point, TT is locally T-consistent whereas
TT=∅.
The following Compactness Theorem shows that this effect
does not occur for E-closed families.
Theorem 3.1. For any nonempty E-closed family
T, every locally T-consistent
d-definable set TΦ is T-consistent.
Proof. If all neighbourhoods Tφ,
φ∈Φ, contain same theory T∈T then
TΦ is T-consistent. So we can assume
that Φ is infinite, closed under conjunctions,
non-T-principal, and for any φ∈Φ,
Tφ contains infinitely many theories in
T. Now we extend step-by-step the set Φ till a
non-principal ultrafilter T of sentences of the language
Σ(T) such that each ψ∈T satisfies
∣Tψ∣≥ω. Applying Proposition 1.1 we
obtain T∈ClE(T)=T, and by
T⊃Φ we have T∈TΦ, i.e.,
TΦ is T-consistent. □
Theories T∈T belonging to locally
T-consistent d-definable sets TΦ are
called their realizations.
The following proposition, along Proposition 1.1 and compactness
above, clarifies the mechanism of construction of ClE(T) via realizations of d-definable subfamilies
of T.
Proposition 3.2. For any family T, ClE(T) consists of elements of T and of
accumulation points realizing locally T-consistent
d-definable sets TΦ.
Proof. By monotonicity property in Proposition 2.5, implying
TΦ⊇TΨ for
Φ⊆Ψ, it suffices to note that for any theory T,
T∈ClE(T) if and only if T is a (unique)
realization of locally T-consistent d-definable
subfamily TT. □
The following theorem gives a criterion of existence of
d-definable family which is not s-definable.
Theorem 3.3. For any E-closed family T,
there is a d-definable family TΦ which is not
s-definable if and only if T is infinite.
Proof. If T is finite then each theory
T∈T is isolated by some sentence φ. So each
nonempty subfamily of T is s-definable by some
disjunction of the sentences φ. Thus, since the empty
subfamily of T is s-definable, by an inconsistent
sentence, then each d-definable family TΦ is
s-definable.
Now we assume that T is infinite. By compactness,
since T is E-closed and infinite, the set Φ of
all sentences φ such that ∣T¬φ∣=1
is T-consistent. Taking an arbitrary theory
T∈TΦ we obtain a d-definable singleton
TT={T} which can not be s-definable by choice of
Φ. □
Remark 3.4. Theorem 3.3 does not hold for families
T which are not E-closed. Indeed, take an arbitrary
e-minimal family T, which does not contain its
(unique) accumulation point T. Repeating arguments for the proof
of Theorem 3.3 we find the set Φ which is locally
T-consistent but TΦ=∅ in view
of T∈/T. Since all s-definable subfamilies of
T are either finite or cofinite, the only possibility
for new d-definable subfamily of T is
TΦ. Since TΦ is empty,
T does not have d-definable subfamilies which are
not s-definable. □
4 Dynamics of ranks with respect to definable subfamilies of theories
Let T be a family of theories, Φ be a set of
sentences, α be an ordinal ≤RS(T) or
−1. The set Φ is called α-ranking for
T if RS(TΦ)=α. A sentence
φ is called α-ranking for T if
RS(T{φ})=α.
The set Φ (the sentence φ) is called ranking
for T if it is α-ranking for T with
some α.
Definition [9]. For a family T, a theory
T is T-finitely axiomatizable, or finitely axiomatizable with respect to T, or T-relatively finitely axiomatizable, if
Tφ={T} for some
Σ(T)-sentence φ.
For a family T of a language Σ, a sentence
φ of the language Σ is called T-complete if φ isolates a unique theory in
T, i.e., Tφ is a singleton.
Proposition 4.1. (1) A set Φ (a sentence
φ) is (−1)-ranking for T if and only
if T=∅ or Φ (respectively
φ) is inconsistent with theories in T.
(2)* A set Φ (a sentence φ) is
[math]-ranking for T, with ds(TΦ)=n, if and only if Φ (respectively φ) is consistent exactly with some
n∈ω∖{0} theories in T.*
(3)* Any [math]-ranking sentence φ for T, with
ds(Tφ)=n, is T-equivalent to
a disjunction of n (pairwise inconsistent)
T-complete sentences.*
Proof. (1) and (2) immediately follow from the definition.
(3) In view of RS(Tφ)=0 and ds(Tφ)=n we have
Tφ={T1,…,Tn} for some distinct
theories T1,…,Tn∈T. Since the theories Ti
are distinct, there are sentences ψi∈Ti such that
¬ψi∈Tj for j=i. Thus the formulas
[TABLE]
[TABLE]
[TABLE]
[TABLE]
are
T-complete, pairwise inconsistent and such that their
disjunction is T-equivalent to φ. □
Remark 4.2. By Proposition 4.1, if T∈T then
Φ=T is [math]-ranking, with TT={T}. More
generally, for any distinct T1,…,Tn∈T the set
T1∨…∨Tn={φ1∨…∨φn∣φi∈Ti} is
[math]-ranking, with ds(TT1∨…∨Tn)=n.
As shown in Remark 4.2 each finite subset
T0⊆T is d-definable, and
Proposition 4.1 gives a characterization for T0 to be
s-definable.
The following theorem produces a characterization for a subfamily
T′⊆T to be d-definable.
Theorem 4.3. A subfamily
T′⊆T is d-definable in
T if and only if T′ is E-closed in
T, i.e., T′=ClE(T′)∩T.
Proof. In view of Remark 4.2 we can assume that T′ is
infinite. Let T′ be d-definable, i.e.,
T′=TΦ for some set Φ. By
Proposition 1.1, all theories in ClE(T′)
contain the set Φ, i.e., ClE(T′)∩T⊆TΦ. Indeed, if a theory
T∈ClE(T′) does not contain a sentence
φ∈Φ then ¬φ∈T and
(T′)¬φ=∅ contradicting T∈ClE(T′). Since T′⊆ClE(T′)∩T, we have T′=ClE(T′)∩T, i.e., the subfamily
T′ is E-closed in T.
Now let the subfamily T′ be E-closed in
T. Denote by Φ the set ⋂T′,
i.e., the set of all Σ(T)-sentences belonging to
all theories in T′. Clearly,
T′⊆TΦ. If
T′⊂TΦ, i.e., there is
T∈TΦ∖T′ then T∈/ClE(T′). Applying Proposition 1.1 we find a sentence
φ∈T such that (T′)φ is finite, say,
(T′)φ={T1,…,Tn}. Since Ti=T
there are sentences ψi∈T∖Ti, i=1,…,n.
For the sentence
χ=φ∧ψ1∧…∧ψn we have
χ∈T and (T′)χ=∅. It implies
¬χ∈Φ, contradicting T∈TΦ. □
The following proposition shows that s-definable subsets of a
family T witnessing RS(T)=β
produce a hierarchy of α-ranking sentences for all ordinals
α≤β.
Proposition 4.4. For any ordinals α≤β, if
RS(T)=β then RS(Tφ)=α for some (α-ranking)* sentence φ. Moreover, there are
ds(T) pairwise T-inconsistent
β-ranking sentences for T, and if α<β
then there are infinitely many pairwise T-inconsistent
α-ranking sentences for T.*
Proof. Since the Boolean algebra F(T) is superatomic
by Theorem 1.1, each Tφ belongs to a hierarchy
with respect to the rank RS(⋅) starting with
singletons, e-minimal subfamilies, etc. Thus, each
Tφ obtains a value RS(Tφ)=α in this hierarchy such that all
α≤β are witnessed by some Tφ. By
the definition of RS(⋅), T can be divided
onto ds(T) disjoint parts Tφ
having the rank β. Again by the definition, if
α<β then there are infinitely many pairwise
T-inconsistent α-ranking sentences for
T. □
By Proposition 4.4, for every family T with RS(T)=β≥0 the possibilities for RS(T′) with T′⊆T are
realized by s-definable subsets Tφ with RS(Tφ)=α for all α≤β. Thus
the following natural question arises for families T
which are not e-totally transcendental.
Question. Let T be a family with RS(T)=∞. What are the RS-possibilities
for s-definable / d-definable subfamilies of T?
As shown in Remark 4.2 every finite subset of T is
d-definable and [math]-ranking. So in fact the question arises for
α≥0 with s-definable subfamilies, and for α>0
with d-definable subfamilies.
Partially answering the question we notice that obtaining an
s-definable / d-definable subfamily Tβ of
T with RS(Tβ)=β≥0 we
have, by Proposition 4.4, s-definable / d-definable
subfamilies Tα of T with RS(Tα)=α, for all ordinals
α≤β. Thus, the required ordinals α form an
initial segment.
Illustrating the question we notice that, in some more or less
general cases, the possibility for α=0 with s-definable
subfamilies can be realized:
Remark 4.5. If a family T has an
α-ranking sentence, for α≥0, it does not imply
that T is e-totally transcendental. Indeed, any
family T, for instance, of functional language,
e-totally transcendental or not, and with a theory T of an
one-element algebra has a [math]-ranking sentence φ saying
that the universe is a singleton. Clearly,
Tφ={T}.
At the same time there are many examples of families of theories
without nonempty s-definable e-totally transcendental
subfamilies. Indeed, taking, for instance, a family
TΣ of all theories in a language Σ
containing infinitely many predicate symbols, we can not control,
by a sentence, links between all predicates. In particular, there
are at least continuum many possibilities arbitrarily varying
empty/nonempty predicates. These variations produce unbounded
ranks for any nonempty s-definable subfamilies
Tφ implying RS(Tφ)=∞.
The following theorem gives an answer to the question for
d-definable subfamilies of theories in countable languages. The
arguments for this answer can be naturally spread for arbitrary
languages.
Theorem 4.6. Let T be a family of a
countable language Σ and with RS(T)=∞, α∈{0,1},
n∈ω∖{0}. Then there is a d-definable
subfamily TΦ such that RS(TΦ)=α and ds(TΦ)=n.
Proof. We fix a family T of countable language
Σ, with RS(T)=∞, a countable ordinal
α, and n∈ω∖{0}. By Theorem 4.3 it
suffices to find an E-closed subfamily T′ in
T with RS(T′)=1 and ds(T′)=n.
If α=0 then T′ exists by Remark 4.2.
If α=1 we take n pairwise inconsistent sentences
φi, i=1,…,n, such that RS(Tφi)=∞ and for each
Tφi find E-closed, in
Tφi (and so in T), e-minimal
subfamily Ti′ in the following way. We enumerate the
set of all Σ-sentences which force φi:
ψik, k∈ω, and form Ti′ step-by-step
with respect to that enumeration using the following subfamilies
Tik of Tφi with
Tik⊇Ti,k+1.
At the initial step if Tψi0 is cofinite in
Tφi, we set
Ti0=Tψi0, if
Tφi=Tψi0, and
Ti0=Tψi0∪{T0}, with an
arbitrary theory
T0i∈Ti∖Tψi0, if
Ti=Tψi0. If
Tψi0 is co-infinite we repeat the process
replacing ψi0 by φi∧¬ψi0: for
infinite Tφi∧¬ψi0 instead of
Tψi0.
Let at the step k a family Tik is already formed
with some theories T0i,…,Tri added to families
Tψis or
Tφi∧¬ψis. Now we consider the
sentence ψi,k+1. If (Tik)ψi,k+1 is
cofinite in Tik, we set
Ti,k+1=Tik, if
Tik=(Tik)ψi,k+1 modulo
{T0i,…,Tri}, and
Ti,k+1=(Tik)ψi,k+1∪{Tr+1i},
with an arbitrary theory
Tr+1i∈Ti,k+1∖((Tik)ψi,k+1∪{T0i,…,Tri}),
if Tik=(Tik)ψi,k+1 modulo
{T0i,…,Tri}. If (Tik)ψi,k+1
is co-infinite we repeat the process for infinite
(Tik)φi∧¬ψi,k+1.
By the construction the subfamilies Ti′ consisting of
the theories T0i,…, Tri, …, are infinite and
can not be divided into two infinite parts by Σ-sentences.
Indeed, Ti′ is infinite because each set
{T0i,…,Tri} is extended in some step by some new
theory since
Tik=(Tik)ψi,k+1 modulo
{T0i,…,Tri} for some ψi,k+1 negating all
theories in {T0i,…,Tri} and some theory in
Tik. The subfamilies Ti′ are
e-minimal since each Σ-sentence is equivalent to some
ψik modulo φi and each ψik can divide
only Ti0,…,Ti,k−1 modulo
{T0i,…,Tri}.
Thus, the subfamilies Ti′ of T are
e-minimal, i=1,…,n. By Proposition 1.6 we have RS(Ti′)=1 and ds(Ti′)=1, and by
Proposition 1.7 we can assume that the families Ti′
are E-closed in T. Hence, for
T′=T1′∪…∪Tn′, which
is d-definable by Theorem 4.3, we have RS(T′)=1 and ds(T′)=n. □
Remark 4.7. Notice that the arguments in the proof of
Theorem 4.6 do not work for α≥2 since taking infinitely
many disjoint s-definable infinite subfamilies
Tφi we can not guarantee that
φi⊢Tψj for infinitely many
T-disjoint sentences ψj. Thus constructing
d-definable e-minimal subfamilies Ti of
Tφi it is possible to obtain RS(i⋃Ti)≥3, not RS(i⋃Ti)=2.
At the same time, constructing countably many d-definable
subfamilies Ti of Tφi,
i∈ω, with pairwise inconsistent φi, we can
choose some infinite I⊆ω, such that accumulation
points Ti for Ti, i∈I, form an e-minimal
family. Thus, possibly loosing the d-definability we obtain a
d∞-definable subfamily T′=i∈I⋃Ti with RS(T′)=2 and ds(T′)=1. Taking some n disjoint T′ we
obtain a subfamily T′′, being the union of
T′, with RS(T′)=2 and ds(T′)=n.
Now we can continue the process for greater countable ordinals
α obtaining a d∞-definable subfamily
T∗⊂T with RS(T∗)=α and ds(T∗)=n
for given n∈ω∖{0}. □
Theorem 4.6 and Remark 4.7 imply the following:
Theorem 4.8. Let T be a family of a
countable language Σ and with RS(T)=∞, α be a countable ordinal,
n∈ω∖{0}. Then there is a d∞-definable
subfamily T∗⊂T such that RS(T∗)=α and ds(T∗)=n.
Example 4.9. Let TΣ be a family of all
theories of a countable language Σ with RS(T∗)=∞ [2], say of unary
predicates Qn, n∈ω. Taking a countable d-definable
subfamily T⊂TΣ with either empty
or complete predicates Qn such that complete predicates in
T are linearly ordered and indexes for complete
predicates form an infinite set I⊂ω with infinite
ω∖I we can assume that T is
e-minimal has unique accumulation point witnessing RS(T)=1 and ds(T)=1. Taking indexes
in ω∖I we can define countably many disjoint
e-minimal d-definable subfamilies Tk with unique
accumulation point for the set of all accumulation points of
Tk witnessing RS=2 and ds=1. Now
applying Theorem 4.8 we can continue the process obtaining RS=α and ds=n for arbitrary countable ordinal
α and n∈ω∖{0}. □
Definition. An α-ranking set Φ for
T, and TΦ are called T-irreducible if for any T-inconsistent
Ψ,X⊇Φ, i.e., with
TΨ∩TX⊇TΦ, RS(TΨ)<α
or RS(TX)<α. An α-ranking
sentence φ for T, and Tφ
are called T-irreducible if the singleton
{φ} is T-irreducible.
If T is fixed, T-irreducible sets are
called simply irreducible.
By the definition each T-inconsistent set Φ, with
TΦ=∅, is irreducible, as well as
singletons TΦ.
Moreover, nonempty E-closed families TΦ of rank
α are irreducible if and only if ds(TΦ)=1.
Indeed, if TΦ is irreducible it can not be divided
by a sentence into two parts of rank α implying ds(TΦ)=1. Conversely, having
T-inconsistent Ψ,X⊇Φ, with
TΨ∩TX⊇TΦ, RS(TΨ)=α
and RS(TX)=α, we obtain, by
compactness, some T-inconsistent ψ∈Ψ and
χ∈X such that RS((TΦ)ψ)=α and RS((TΦ)χ)=α contradicting ds(TΦ)=1.
Since each family T with RS(T)=α≥0 has a finite degree ds(T)=n, there are pairwise inconsistent sentences
φ1,…,φn such that
T=Tφ1∪˙…
∪˙Tφn, RS(Tφi)=α and ds(Tφi)=1, i=1,…,n.
Thus, all e-totally transcendental E-closed families and, in
particular, d-definable α-ranking E-closed families are
reduced to irreducible ones:
Proposition 4.10. Any e-totally transcendental
E-closed family T is represented as a finite
disjoint union of s-definable irreducible subfamilies of rank
α=RS(T).