# Formulas and properties111The study was carried out within the framework of
the state contract of the Sobolev Institute of Mathematics (project No.
0314-2019-0002), and the Committee of Science in Education and the Science
Ministry of the Republic of Kazakhstan (Grant No. AP08855544).

S.V. Sudoplatov

###### Abstract

We study links between first-order formulas and arbitrary properties for
families of theories, classes of structures and their isomorphism types.
Possibilities for ranks and degrees for formulas and theories with respect to
given properties are described. Characteristics for generic sentences and
generic theories with respect to properties are described and characterized.

Key words: formula, property, rank, degree, generic sentence, generic theory.

First-order formulas are used to express semantic and syntactic definable
properties. Since in general there are more natural properties than definable
ones, these formulas can express them in a partial way. In the present paper
we study links between formulas and arbitrary properties, consider
characteristics reflecting measures of their correspondence.

The paper is organized as follows. Preliminary notions, notations and results
are represented in Section 1. In Section 2 we consider links between formulas
and properties for semantic and syntactic families. We study and characterize
these links with respect to constructions of formulas, set-theoretic
operations and closures. In Section 3 we study, characterize and describe rank
values and degree values for formulas with respect to given properties. In
Section 4, we describe spectra for cardinalities of definable properties. In
Section 5, generic sentences and theories with respect to properties are
introduced, and their links and ranks and described.

Throughout we use the standard terminology in Mathematical Logic [1], notions,
notations and results of [2, 3, 4].

##  1 Preliminaries

Let $\Sigma$ be a language. If $\Sigma$ is relational we denote by
$\mathcal{T}_{\Sigma}$ the family of all theories of the language $\Sigma$. If
$\Sigma$ contains functional symbols $f$ then $\mathcal{T}_{\Sigma}$ is the
family of all theories of the language $\Sigma^{\prime}$, which is obtained by
replacements of all $n$-ary symbols $f$ with $(n+1)$-ary predicate symbols
$R_{f}$ interpreted by $R_{f}=\\{(\bar{a},b)\mid f(\bar{a})=b\\}$.

Following [2] we define the rank ${\rm RS}(\cdot)$ for families
$\mathcal{T}\subseteq\mathcal{T}_{\Sigma}$, similar to Morley rank for a fixed
theory, and a hierarchy with respect to these ranks in the following way.

By $F(\Sigma)$ we denote the set of all formulas in the language $\Sigma$ and
by ${\rm Sent}(\Sigma)$ the set of all sentences in $F(\Sigma)$.

For a sentence $\varphi\in{\rm Sent}(\Sigma)$ we denote by
$\mathcal{T}_{\varphi}$ the set of all theories $T\in\mathcal{T}$ with
$\varphi\in T$.

Any set $\mathcal{T}_{\varphi}$ is called the $\varphi$-neighbourhood, or
simply a neighbourhood, for $\mathcal{T}$, or the ($\varphi$-)definable subset
of $\mathcal{T}$. The set $\mathcal{T}_{\varphi}$ is also called (formula- or
sentence-)definable (by the sentence $\varphi$) with respect to $\mathcal{T}$,
or (sentence-)$\mathcal{T}$-definable, or simply $s$-definable.

Definition [2]. For the empty family $\mathcal{T}$ we put the rank ${\rm
RS}(\mathcal{T})=-1$, for finite nonempty families $\mathcal{T}$ we put ${\rm
RS}(\mathcal{T})=0$, and for infinite families $\mathcal{T}$ — ${\rm
RS}(\mathcal{T})\geq 1$.

For a family $\mathcal{T}$ and an ordinal $\alpha=\beta+1$ we put ${\rm
RS}(\mathcal{T})\geq\alpha$ if there are pairwise inconsistent
$\Sigma(\mathcal{T})$-sentences $\varphi_{n}$, $n\in\omega$, such that ${\rm
RS}(\mathcal{T}_{\varphi_{n}})\geq\beta$, $n\in\omega$.

If $\alpha$ is a limit ordinal then ${\rm RS}(\mathcal{T})\geq\alpha$ if ${\rm
RS}(\mathcal{T})\geq\beta$ for any $\beta<\alpha$.

We set ${\rm RS}(\mathcal{T})=\alpha$ if ${\rm RS}(\mathcal{T})\geq\alpha$ and
${\rm RS}(\mathcal{T})\not\geq\alpha+1$.

If ${\rm RS}(\mathcal{T})\geq\alpha$ for any $\alpha$, we put ${\rm
RS}(\mathcal{T})=\infty$.

A family $\mathcal{T}$ is called $e$-totally transcendental, or totally
transcendental, if ${\rm RS}(\mathcal{T})$ is an ordinal.

If $\mathcal{T}$ is $e$-totally transcendental, with ${\rm
RS}(\mathcal{T})=\alpha\geq 0$, we define the degree ${\rm ds}(\mathcal{T})$
of $\mathcal{T}$ as the maximal number of pairwise inconsistent sentences
$\varphi_{i}$ such that ${\rm RS}(\mathcal{T}_{\varphi_{i}})=\alpha$.

Definition [5]. An infinite family $\mathcal{T}$ is called $e$-minimal if for
any sentence $\varphi\in\Sigma(\mathcal{T})$, $\mathcal{T}_{\varphi}$ is
finite or $\mathcal{T}_{\neg\varphi}$ is finite.

By the definition a family $\mathcal{T}$ is $e$-minimal iff ${\rm
RS}(\mathcal{T})=1$ and ${\rm ds}(\mathcal{T})=1$ [2], and iff $\mathcal{T}$
has unique accumulation point [5].

In the paper [6] the notion of $E$-closure was introduced and characterized as
follows:

Proposition 1.1. If $\mathcal{T}\subseteq\mathcal{T}_{\Sigma}$ is an infinite
set and $T\in\mathcal{T}_{\Sigma}\setminus\mathcal{T}$ then $T\in{\rm
Cl}_{E}(\mathcal{T})$ (i.e., $T$ is an accumulation point for $\mathcal{T}$
with respect to $E$-closure ${\rm Cl}_{E}$) if and only if for any sentence
$\varphi\in T$ the set $\mathcal{T}_{\varphi}$ is infinite.

The following theorem characterizes the property of $e$-total transcendency
for countable languages.

Theorem 1.2 [2]. For any family $\mathcal{T}$ with
$|\Sigma(\mathcal{T})|\leq\omega$ the following conditions are equivalent:

$(1)$ $|{\rm Cl}_{E}(\mathcal{T})|=2^{\omega}$;

$(2)$ $e$-${\rm Sp}(\mathcal{T})=2^{\omega}$;

$(3)$ ${\rm RS}(\mathcal{T})=\infty$.

Theorem 1.3 [3]. For any language $\Sigma$ either ${\rm
RS}(\mathcal{T}_{\Sigma})$ is finite, if $\Sigma$ consists of finitely many
$0$-ary and unary predicates, and finitely many constant symbols, or ${\rm
RS}(\mathcal{T}_{\Sigma})=\infty$, otherwise.

For a language $\Sigma$ we denote by $\mathcal{T}_{\Sigma,n}$ the family of
all theories in $\mathcal{T}_{\Sigma}$ having $n$-element models,
$n\in\omega$, as well as by $\mathcal{T}_{\Sigma,\infty}$ the family of all
theories in $\mathcal{T}_{\Sigma}$ having infinite models.

Theorem 1.4 [3]. For any language $\Sigma$ either ${\rm
RS}(\mathcal{T}_{\Sigma,n})=0$, if $\Sigma$ is finite or $n=1$ and $\Sigma$
has finitely many predicate symbols, or ${\rm
RS}(\mathcal{T}_{\Sigma,n})=\infty$, otherwise.

Theorem 1.5 [3]. For any language $\Sigma$ either ${\rm
RS}(\mathcal{T}_{\Sigma,\infty})$ is finite, if $\Sigma$ is finite and without
predicate symbols of arities $m\geq 2$ as well as without functional symbols
of arities $n\geq 1$, or ${\rm RS}(\mathcal{T}_{\Sigma,\infty})=\infty$,
otherwise.

By the definition the families $\mathcal{T}_{\Sigma}$,
$\mathcal{T}_{\Sigma,n}$, $\mathcal{T}_{\Sigma,\infty}$ are $E$-closed. Thus,
combining Theorem 1.2 with Theorems 1.3–1.5 we obtain the following
possibilities of cardinalities for the families $\mathcal{T}_{\Sigma}$,
$\mathcal{T}_{\Sigma,n}$, $\mathcal{T}_{\Sigma,\infty}$ depending on $\Sigma$
and $n\in\omega$:

Proposition 1.6. For any language $\Sigma$ either either
$\mathcal{T}_{\Sigma}$ is countable, if if $\Sigma$ consists of finitely many
$0$-ary and unary predicates, and finitely many constant symbols, or
$|\mathcal{T}_{\Sigma}|\geq 2^{\omega}$, otherwise.

Proposition 1.7. For any language $\Sigma$ either $\mathcal{T}_{\Sigma,n}$ is
finite, if $\Sigma$ is finite or $n=1$ and $\Sigma$ has finitely many
predicate symbols, or $|\mathcal{T}_{\Sigma,n}|\geq 2^{\omega}$, otherwise.

Proposition 1.8. For any language $\Sigma$ either
$\mathcal{T}_{\Sigma,\infty}$ is at most countable, if $\Sigma$ is finite and
without predicate symbols of arities $m\geq 2$ as well as without functional
symbols of arities $n\geq 1$, or $|\mathcal{T}_{\Sigma,\infty}|\geq
2^{\omega}$, otherwise.

Definition [4]. If $\mathcal{T}$ is a family of theories and $\Phi$ is a set
of sentences, then we put
$\mathcal{T}_{\Phi}=\bigcap\limits_{\varphi\in\Phi}\mathcal{T}_{\varphi}$ and
the set $\mathcal{T}_{\Phi}$ is called (type-) or (diagram-)definable (by the
set $\Phi$) with respect to $\mathcal{T}$, or
(diagram-)$\mathcal{T}$-definable, or simply $d$-definable.

Clearly, finite unions of $d$-definable sets are again $d$-definable.
Considering infinite unions $\mathcal{T}^{\prime}$ of $d$-definable sets
$\mathcal{T}_{\Phi_{i}}$, $i\in I$, one can represent them by sets of
sentences with infinite disjunctions $\bigvee\limits_{i\in I}\varphi_{i}$,
$\varphi_{i}\in\Phi_{i}$. We call these unions $\mathcal{T}^{\prime}$ are
called $d_{\infty}$-definable sets.

Definition [4]. Let $\mathcal{T}$ be a family of theories, $\Phi$ be a set of
sentences, $\alpha$ be an ordinal $\leq{\rm RS}(\mathcal{T})$ or $-1$. The set
$\Phi$ is called $\alpha$-ranking for $\mathcal{T}$ if ${\rm
RS}(\mathcal{T}_{\Phi})=\alpha$. A sentence $\varphi$ is called
$\alpha$-ranking for $\mathcal{T}$ if $\\{\varphi\\}$ is $\alpha$-ranking for
$\mathcal{T}$.

The set $\Phi$ (the sentence $\varphi$) is called ranking for $\mathcal{T}$ if
it is $\alpha$-ranking for $\mathcal{T}$ with some $\alpha$.

Proposition 1.9 [4]. For any ordinals $\alpha\leq\beta$, if ${\rm
RS}(\mathcal{T})=\beta$ then ${\rm RS}(\mathcal{T}_{\varphi})=\alpha$ for some
($\alpha$-ranking) sentence $\varphi$. Moreover, there are ${\rm
ds}(\mathcal{T})$ pairwise $\mathcal{T}$-inconsistent $\beta$-ranking
sentences for $\mathcal{T}$, and if $\alpha<\beta$ then there are infinitely
many pairwise $\mathcal{T}$-inconsistent $\alpha$-ranking sentences for
$\mathcal{T}$.

Theorem 1.10 [4]. Let $\mathcal{T}$ be a family of a countable language
$\Sigma$ and with ${\rm RS}(\mathcal{T})=\infty$, $\alpha$ be a countable
ordinal, $n\in\omega\setminus\\{0\\}$. Then there is a $d_{\infty}$-definable
subfamily $\mathcal{T}^{\ast}\subset\mathcal{T}$ such that ${\rm
RS}(\mathcal{T}^{\ast})=\alpha$ and ${\rm ds}(\mathcal{T}^{\ast})=n$.

Theorem 1.11 [7]. For any two disjoint subfamilies $\mathcal{T}_{1}$ and
$\mathcal{T}_{2}$ of an $E$-closed family $\mathcal{T}$ the following
conditions are equivalent:

$(1)$ $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$ are separated by some sentence
$\varphi$: $\mathcal{T}_{1}\subseteq\mathcal{T}_{\varphi}$ and
$\mathcal{T}_{2}\subseteq\mathcal{T}_{\neg\varphi}$;

$(2)$ $E$-closures of $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$ are disjoint in
$\mathcal{T}$: ${\rm Cl}_{E}(\mathcal{T}_{1})\cap{\rm
Cl}_{E}(\mathcal{T}_{2})\cap\mathcal{T}=\emptyset$;

$(3)$ $E$-closures of $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$ are disjoint:
${\rm Cl}_{E}(\mathcal{T}_{1})\cap{\rm Cl}_{E}(\mathcal{T}_{2})=\emptyset$.

Definition [6]. Let $\mathcal{T}_{0}$ be a family of theories. A subset
$\mathcal{T}^{\prime}_{0}\subseteq\mathcal{T}_{0}$ is said to be generating if
$\mathcal{T}_{0}={\rm Cl}_{E}(\mathcal{T}^{\prime}_{0})$. The generating set
$\mathcal{T}^{\prime}_{0}$ (for $\mathcal{T}_{0}$) is minimal if
$\mathcal{T}^{\prime}_{0}$ does not contain proper generating subsets. A
minimal generating set $\mathcal{T}^{\prime}_{0}$ is least if
$\mathcal{T}^{\prime}_{0}$ is contained in each generating set for
$\mathcal{T}_{0}$.

Theorem 1.12 [6]. If $\mathcal{T}^{\prime}_{0}$ is a generating set for a
$E$-closed set $\mathcal{T}_{0}$ then the following conditions are equivalent:

$(1)$ $\mathcal{T}^{\prime}_{0}$ is the least generating set for
$\mathcal{T}_{0}$;

$(2)$ $\mathcal{T}^{\prime}_{0}$ is a minimal generating set for
$\mathcal{T}_{0}$;

$(3)$ any theory in $\mathcal{T}^{\prime}_{0}$ is isolated by some set
$(\mathcal{T}^{\prime}_{0})_{\varphi}$, i.e., for any
$T\in\mathcal{T}^{\prime}_{0}$ there is $\varphi\in T$ such that
$(\mathcal{T}^{\prime}_{0})_{\varphi}=\\{T\\}$;

$(4)$ any theory in $\mathcal{T}^{\prime}_{0}$ is isolated by some set
$(\mathcal{T}_{0})_{\varphi}$, i.e., for any $T\in\mathcal{T}^{\prime}_{0}$
there is $\varphi\in T$ such that $(\mathcal{T}_{0})_{\varphi}=\\{T\\}$.

##  2 Relations between formulas and properties

Definition. Let $\Sigma$ be a language,
$\varphi\rightleftharpoons\varphi(\overline{x})$ be a formula in $F(\Sigma)$,
$P_{s}$ be a subclass of the class $K(\Sigma)$ of all structures $\mathcal{A}$
in the language $\Sigma$. We say that $\varphi(\overline{x})$ partially
(respectively, totally) satisfies $P_{s}$, denoted by $\varphi\rhd_{\rm
ps}P_{s}$ or $\varphi\rhd^{\exists}_{s}P_{s}$ ($\varphi\rhd_{\rm ts}P_{s}$ or
$\varphi\rhd^{\forall}_{s}P_{s}$), if there are $\mathcal{A}\in P_{s}$ and
$\overline{a}\in A$ (for any $\mathcal{A}\in P_{s}$ there is $\overline{a}\in
A$) such that $\mathcal{A}\models\varphi(\overline{a})$.

If $P_{\rm is}$ is a subclass of the class ${\rm ITK}(\Sigma)$ of isomorphism
types for the class $K(\Sigma)$ then we say that $\varphi(\overline{x})$
partially (respectively, totally) satisfies $P_{\rm its}$, denoted by
$\varphi\rhd_{\rm pits}P_{\rm its}$ or $\varphi\rhd^{\exists}_{\rm its}P_{\rm
its}$ ($\varphi\rhd_{\rm tits}P_{\rm its}$ or $\varphi\rhd^{\forall}_{\rm
its}P_{\rm its}$) if $\varphi\rhd_{\rm ps}P_{s}$ ($\varphi\rhd_{\rm ts}P_{s}$,
where $P_{s}$ consists of all structures whose isomorphism types belong to
$P_{\rm its}$.

If $P_{t}$ is a subset of the set $\mathcal{T}_{\Sigma}$ of all complete
theories in the language $\Sigma$ then we say that $\varphi(\overline{x})$
partially (respectively, totally) satisfies $P_{t}$, denoted by
$\varphi\rhd_{\rm pt}P_{t}$ or $\varphi\rhd^{\exists}_{t}P_{t}$
($\varphi\rhd_{\rm tt}P_{t}$ or $\varphi\rhd^{\forall}_{t}P_{t}$), if there
are $T\in P_{t}$, $\mathcal{M}\models T$, and $\overline{a}\in M$ (for any
$T\in P_{t}$ there are $\mathcal{M}\models T$ and $\overline{a}\in M$) such
that $\mathcal{M}\models\varphi(\overline{a})$.

We write $\not\\!\rhd_{\xi}$ if a $\rhd_{\xi}$-relation does not hold.

Remark 2.1. By the definition we have the following obvious properties.

1\. If $P_{s}\neq\emptyset$ and $\varphi\rhd_{\rm ts}P_{s}$ then
$\varphi\rhd_{\rm ps}P_{s}$. Similarly, if $\varphi\rhd_{\rm tits}P_{\rm its}$
for nonempty $P_{\rm its}$ then $\varphi\rhd_{\rm pits}P_{\rm its}$, and if
$\varphi\rhd_{\rm tt}P_{t}$ for nonempty $P$ then $\varphi\rhd_{\rm pt}P_{t}$.

2\. For any singleton $P_{s}$, $\varphi\rhd_{\rm ps}P_{s}$ implies
$\varphi\rhd_{\rm ts}P_{s}$. Similarly, $\varphi\rhd_{\rm pits}P_{\rm its}$
implies $\varphi\rhd_{\rm tits}P_{\rm its}$ for any singleton $P_{\rm its}$,
and $\varphi\rhd_{\rm pt}P_{t}$ implies $\varphi\rhd_{\rm tt}P$ for any
singleton $P_{t}$.

3\. If $\varphi\rhd_{\rm ps}\\{\mathcal{A}\\}$ and
$\mathcal{A}\equiv\mathcal{B}$ then $\varphi\rhd_{\rm ps}\\{\mathcal{B}\\}$.
It implies that the relations $\varphi\rhd_{\rm pt}P_{t}$ and
$\varphi\rhd_{\rm tt}P_{t}$ do not depend on the choice of models
$\mathcal{M}\models T$ for $T\in P_{t}$.

4\. (Reflexivity) For any sentence $\varphi$ and a (nonempty) family
$\mathcal{T}_{\varphi}\subseteq\mathcal{T}_{\Sigma}$ we have $\varphi\rhd_{\rm
tt}\mathcal{T}_{\varphi}$ (and $\varphi\rhd_{\rm pt}\mathcal{T}_{\varphi}$).

5\. (Monotony) If $\varphi\rhd_{\rm ps}P_{s}$, $\varphi\vdash\psi$ and
$P_{s}\subseteq P^{\prime}_{s}\subseteq K(\Sigma)$ then $\psi\rhd_{\rm
ps}P^{\prime}_{s}$. If $\varphi\rhd_{\rm ts}P_{s}$, $\varphi\vdash\psi$ and
$P_{s}\supseteq P^{\prime}_{s}$ then $\psi\rhd_{\rm ps}P^{\prime}_{s}$. If
$\varphi\rhd_{\rm pits}P_{\rm its}$, $\varphi\vdash\psi$ and $P_{\rm
its}\subseteq P^{\prime}_{\rm its}\subseteq{\rm ITK}(\Sigma)$ then
$\psi\rhd_{\rm pits}P^{\prime}_{\rm its}$. If $\varphi\rhd_{\rm tits}P_{\rm
its}$, $\varphi\vdash\psi$ and $P_{\rm its}\supseteq P^{\prime}_{\rm its}$
then $\psi\rhd_{\rm ps}P^{\prime}_{\rm its}$. If $\varphi\rhd_{\rm pt}P_{t}$,
$\varphi\vdash\psi$ and $P_{t}\subseteq
P^{\prime}_{t}\subseteq\mathcal{T}_{\Sigma}$ then $\psi\rhd_{\rm
ps}P^{\prime}_{t}$. If $\varphi\rhd_{\rm tt}P_{t}$, $\varphi\vdash\psi$ and
$P_{t}\supseteq P^{\prime}_{t}$ then $\psi\rhd_{\rm tt}P^{\prime}_{t}$.

For a property $P_{s}$ we denote by ${\rm ITK}(P_{s})$ the class of
isomorphism types for structures in $P_{s}$, and by ${\rm Th}(P_{s})$ the set
$\\{T\in\mathcal{T}_{\Sigma}\mid\mathcal{A}\models T$ for some $\mathcal{A}\in
P_{s}\\}$.

For a property $P_{\rm its}$ we denote by $K(P_{\rm its})$ the class of all
structures whose isomorphism types are represented in $P_{\rm its}$, and by
${\rm Th}(P_{\rm its})$ the set ${\rm Th}(K(P_{\rm its}))$.

For a property $P_{t}$ we denote by $K(P_{t})$ the class of all models of
theories in $P_{t}$, and by ${\rm ITK}(P_{t})$ the class ${\rm
ITK}(K(P_{t}))$.

In terms of these notations by the definition we have the following natural
links between semantic properties $P_{s}$ and $P_{\rm its}$ and syntactic
properties $P_{t}$:

Proposition 2.2. For any formula $\varphi\in F(\Sigma)$ and properties
$P_{s}$, $P_{\rm its}$, $P_{t}$ the following conditions hold:

$(1)$ $\varphi\rhd_{\rm ps}P_{s}$ iff $\varphi\rhd_{\rm pits}{\rm
ITK}(P_{s})$, and iff $\varphi\rhd_{\rm pt}{\rm Th}(P_{s})$;

$(2)$ $\varphi\rhd_{\rm ts}P_{s}$ iff $\varphi\rhd_{\rm tits}{\rm
ITK}(P_{s})$, and iff $\varphi\rhd_{\rm tt}{\rm Th}(P_{s})$;

$(3)$ $\varphi\rhd_{\rm pits}P_{\rm its}$ iff $\varphi\rhd_{\rm ps}K(P_{\rm
its})$, and iff $\varphi\rhd_{\rm pt}{\rm Th}(P_{\rm its})$;

$(4)$ $\varphi\rhd_{\rm tits}P_{\rm its}$ iff $\varphi\rhd_{\rm ts}K(P_{\rm
its})$, and iff $\varphi\rhd_{\rm tt}{\rm Th}(P_{\rm its})$;

$(5)$ $\varphi\rhd_{\rm pt}P_{t}$ iff $\varphi\rhd_{\rm ps}K(P_{t})$, and iff
$\varphi\rhd_{\rm pits}{\rm ITK}(P_{t})$;

$(6)$ $\varphi\rhd_{\rm tt}P_{t}$ iff $\varphi\rhd_{\rm ts}K(P_{t})$, and iff
$\varphi\rhd_{\rm tits}{\rm ITK}(P_{t})$.

In the items $(3)$ and $(4)$ the class $K(P_{\rm its})$ can be replaced by a
subclass $K^{\prime}$ such that ${\rm ITK}(K^{\prime})=P_{\rm its}$.
Similarly, in the items $(5)$ and $(6)$ the class $K(P_{t})$ can be replaced
by a subclass $K^{\prime}$ such that ${\rm Th}(K^{\prime})=P_{t}$, and
independently ${\rm ITK}(P_{t})$ can be replaced by a subclass
$K^{\prime\prime}$ such that ${\rm Th}(K^{\prime\prime})=P_{t}$.

By Proposition 2.2 semantic properties $P_{s}$ and $P_{\rm its}$ can be
naturally transformed into syntactic ones $P_{t}$, and vice versa. It means
that natural model-theoretic properties such as $\omega$-categoricity,
stability, simplicity etc. can be formulated both for theories, for structures
and for their isomorphism types.

The links between $\rhd$-relations which pointed out in Proposition 2.2 allow
to reduce our consideration to the relations $\rhd_{\rm pt}$ and $\rhd_{\rm
tt}$. Besides, for the simplicity we will principally consider sentences
$\varphi$ instead of formulas in general. Reductions of formulas
$\psi(\overline{x})$ to sentences use the operators
$\psi(\overline{x})\mapsto\forall\overline{x}\,\psi(\overline{x})$ and
$\psi(\overline{x})\mapsto\exists\overline{x}\,\psi(\overline{x})$.

Proposition 2.3. For any sentences $\varphi,\psi\in{\rm Sent}(\Sigma)$ and
properties $P_{t},P^{\prime}_{t}\subseteq\mathcal{T}_{\Sigma}$ the following
conditions hold:

$(1)$ if $(\varphi\wedge\psi)\rhd_{\rm pt}(P_{t}\cap P^{\prime}_{t})$ then
$\varphi\rhd_{\rm pt}P_{t}$ and $\psi\rhd_{\rm pt}P^{\prime}_{t}$; the
converse implication does not hold: there are
$\varphi^{\prime},\psi^{\prime}\in{\rm Sent}(\Sigma)$ and
$P^{\prime\prime}_{t}\in\mathcal{T}_{\Sigma}$ such that
$\varphi^{\prime}\rhd_{\rm pt}P^{\prime\prime}_{t}$, $\psi^{\prime}\rhd_{\rm
pt}P^{\prime\prime}_{t}$, and
$(\varphi^{\prime}\wedge\psi^{\prime})\not\\!\rhd_{\rm
pt}P^{\prime\prime}_{t}$;

$(2)$ $(\varphi\wedge\psi)\rhd_{\rm tt}P_{t}$ iff $\varphi\rhd_{\rm tt}P_{t}$
and $\psi\rhd_{\rm tt}P_{t}$;

$(3)$ if $\varphi\rhd_{\rm tt}P_{t}$ and $\psi\rhd_{\rm tt}P^{\prime}_{t}$
then $(\varphi\wedge\psi)\rhd_{\rm tt}(P_{t}\cap P^{\prime}_{t})$; the
converse implication does not hold: there are
$\varphi^{\prime},\psi^{\prime}\in{\rm Sent}(\Sigma)$ and
$P^{\prime\prime}_{t},P^{\prime\prime\prime}_{t}\in\mathcal{T}_{\Sigma}$ such
that $(\varphi\wedge\psi)\rhd_{\rm tt}(P^{\prime\prime}_{t}\cap
P^{\prime\prime\prime}_{t})$ whereas $\varphi^{\prime}\not\\!\rhd_{\rm
tt}P^{\prime\prime}_{t}$ and $\psi^{\prime}\not\\!\rhd_{\rm
tt}P^{\prime\prime\prime}_{t}$.

Proof. $(1)$ If $(\varphi\wedge\psi)\rhd_{\rm pt}(P_{t}\cap P^{\prime}_{t})$
then there is $T\in P_{t}\cap P^{\prime}_{t}$ with $(\varphi\wedge\psi)\in T$.
Since $\varphi,\psi\in T$, $T\in P_{t}$ and $T\in P^{\prime}_{t}$, we obtain
$\varphi\rhd_{\rm pt}P_{t}$ and $\psi\rhd_{\rm pt}P^{\prime}_{t}$. Therefore
it suffices to notice for $(1)$ that sentences
$\varphi^{\prime},\psi^{\prime}$ asserting distinct finite cardinalities $m$
and $n$ for universes partially satisfy a property $P^{\prime\prime}_{t}$
containing a theory $T_{1}$ with an $m$-element model and a theory $T_{2}$
with an $n$-element model. At the same time
$(\varphi^{\prime}\wedge\psi^{\prime})\not\\!\rhd_{\rm
pt}P^{\prime\prime}_{t}$ since $(\varphi^{\prime}\wedge\psi^{\prime})$ is
inconsistent.

$(2)$ If $(\varphi\wedge\psi)\rhd_{\rm tt}P_{t}$ then $(\varphi\wedge\psi)$,
and so, $\varphi$ and $\psi$ belong to all theories in $P_{t}$ implying
$\varphi\rhd_{\rm tt}P_{t}$ and $\psi\rhd_{\rm tt}P_{t}$. Conversely, if
$\varphi\rhd_{\rm tt}P_{t}$ and $\psi\rhd_{\rm tt}P_{t}$ then $\varphi$,
$\psi$, and so $(\varphi\wedge\psi)$ belong to all theories in $P_{t}$
implying $(\varphi\wedge\psi)\rhd_{\rm tt}P_{t}$.

$(3)$ If $\varphi\rhd_{\rm tt}P_{t}$ and $\psi\rhd_{\rm tt}P^{\prime}_{t}$
then $\varphi\in\bigcap P_{t}$ and $\psi\in\bigcap P^{\prime}_{t}$, implying
$(\varphi\wedge\psi)\in\bigcap P_{t}\cap\bigcap P^{\prime}_{t}$, i.e.,
$(\varphi\wedge\psi)\rhd_{\rm tt}(P_{t}\cap P^{\prime}_{t})$. Finally, if
$P^{\prime\prime}_{t}$ and $P^{\prime\prime\prime}_{t}$ are nonempty with
$P^{\prime\prime}_{t}\cap P^{\prime\prime\prime}_{t}=\emptyset$, and
$\varphi^{\prime}$, $\psi^{\prime}$ are inconsistent sentences then
$(\varphi\wedge\psi)\rhd_{\rm tt}(P^{\prime\prime}_{t}\cap
P^{\prime\prime\prime}_{t})$ and $\varphi^{\prime}\not\\!\rhd_{\rm
pt}P^{\prime\prime}_{t}$ and $\psi^{\prime}\not\\!\rhd_{\rm
pt}P^{\prime\prime\prime}_{t}$ implying $\varphi^{\prime}\not\\!\rhd_{\rm
tt}P^{\prime\prime}_{t}$ and $\psi^{\prime}\not\\!\rhd_{\rm
tt}P^{\prime\prime\prime}_{t}$. $\Box$

Proposition 2.4. For any sentences $\varphi,\psi\in{\rm Sent}(\Sigma)$ and
properties $P_{t},P^{\prime}_{t}\subseteq\mathcal{T}_{\Sigma}$ the following
conditions hold:

$(1)$ if $\varphi\rhd_{\rm pt}P_{t}$ or $\psi\rhd_{\rm pt}P^{\prime}_{t}$ then
$(\varphi\vee\psi)\rhd_{\rm pt}(P_{t}\cup P^{\prime}_{t})$; the converse
implication does not hold: there are $\varphi^{\prime},\psi^{\prime}\in{\rm
Sent}(\Sigma)$ and $P^{\prime\prime}_{t}\in\mathcal{T}_{\Sigma}$ such that
$(\varphi^{\prime}\vee\psi^{\prime})\rhd_{\rm pt}P^{\prime\prime}_{t}$, and
$\varphi^{\prime}\not\\!\rhd_{\rm pt}P^{\prime\prime}_{t}$ or
$\psi^{\prime}\not\\!\rhd_{\rm pt}P^{\prime\prime}_{t}$;

$(2)$ $(\varphi\vee\psi)\rhd_{\rm pt}P_{t}$ iff $\varphi\rhd_{\rm pt}P_{t}$ or
$\psi\rhd_{\rm pt}P_{t}$;

$(3)$ if $\varphi\rhd_{\rm tt}P_{t}$ and $\psi\rhd_{\rm tt}P^{\prime}_{t}$
then $(\varphi\vee\psi)\rhd_{\rm tt}(P_{t}\cup P^{\prime}_{t})$; the converse
implication does not hold: there are $\varphi^{\prime},\psi^{\prime}\in{\rm
Sent}(\Sigma)$ and $P^{\prime\prime}_{t}\in\mathcal{T}_{\Sigma}$ such that
$(\varphi^{\prime}\vee\psi^{\prime})\rhd_{\rm tt}(P^{\prime\prime}_{t})$
whereas $\varphi^{\prime}\not\\!\rhd_{\rm tt}P^{\prime\prime}_{t}$ and
$\psi^{\prime}\not\\!\rhd_{\rm tt}P^{\prime\prime}_{t}$.

Proof. $(1)$ If $\varphi\rhd_{\rm pt}P_{t}$ or $\psi\rhd_{\rm
pt}P^{\prime}_{t}$ then $\varphi\in T$ for some $T\in P_{t}$ or $\psi\in
T^{\prime}$ for some $T^{\prime}\in P^{\prime}_{t}$. Therefore $T$ or
$T^{\prime}$ witness that $(\varphi\vee\psi)\rhd_{\rm pt}(P_{t}\cup
P^{\prime}_{t})$. If $\varphi^{\prime}$ is a tautology and $\psi^{\prime}$ is
an inconsistent sentence, then for any nonempty
$P^{\prime\prime}_{t}\subseteq\mathcal{T}_{\Sigma}$ we have
$(\varphi^{\prime}\vee\psi^{\prime})\rhd_{\rm pt}P^{\prime\prime}_{t}$,
$\varphi^{\prime}\rhd_{\rm pt}P^{\prime\prime}_{t}$, and
$\psi^{\prime}\not\\!\rhd_{\rm pt}P^{\prime\prime}_{t}$.

$(2)$ It holds since a sentence $(\varphi\vee\psi)$ belongs to a complete
theory $T$ if and only if $\varphi\in T$ or $\psi\in T$.

$(3)$ If $\varphi\rhd_{\rm tt}P_{t}$ and $\psi\rhd_{\rm tt}P^{\prime}_{t}$
then $\varphi\in\bigcap P_{t}$ and $\psi\in\bigcap P^{\prime}_{t}$. Therefore
$(\varphi\vee\psi)\in\bigcap P_{t}$ and $(\varphi\vee\psi)\in\bigcap
P^{\prime}_{t}$ implying $(\varphi\vee\psi)\in\bigcap P_{t}\cap\bigcap
P^{\prime}_{t}$, hence $(\varphi\vee\psi)\rhd_{\rm tt}(P_{t}\cup
P^{\prime}_{t})$.

Now let $P^{\prime\prime}_{t}=\mathcal{T}_{\Sigma}$, $\varphi^{\prime}$ be a
sentence belonging to some but not all theories in $P^{\prime\prime}_{t}$. For
the sentence $\psi^{\prime}=\neg\varphi^{\prime}$ we have
$(\varphi^{\prime}\vee\psi^{\prime})\rhd_{\rm tt}(P^{\prime\prime}_{t})$ since
$(\varphi\vee\psi)$ is a tautology, $\varphi^{\prime}\not\\!\rhd_{\rm
tt}P^{\prime\prime}_{t}$ and $\psi^{\prime}\not\\!\rhd_{\rm
tt}P^{\prime\prime}_{t}$ by the choice of $\varphi^{\prime}$. $\Box$

Proposition 2.5. For any sentence $\varphi\in{\rm Sent}(\Sigma)$ and a
property $P_{t}\subseteq\mathcal{T}_{\Sigma}$ the following conditions hold:

$(1)$ $\varphi\rhd_{\rm pt}P_{t}$ iff $\neg\varphi\not\\!\rhd_{\rm tt}P_{t}$;

$(2)$ $\varphi\rhd_{\rm tt}P_{t}$ iff $\neg\varphi\not\\!\rhd_{\rm pt}P_{t}$.

Proof. $(1)$ If $\varphi\rhd_{\rm pt}P_{t}$ then there is $T\in P_{t}$ such
that $\varphi\in T$. Since $T$ is complete then $\neg\varphi\notin T$ implying
$\neg\varphi\not\\!\rhd_{\rm tt}P_{t}$. Conversely if
$\neg\varphi\not\\!\rhd_{\rm tt}P_{t}$ then $\neg\varphi$ does not belong to
some theory $T\in P_{t}$. Since $T$ is complete then $\varphi\in T$ implying
$\varphi\rhd_{\rm pt}P_{t}$.

$(2)$ is immediately follows from $(1)$. $\Box$

Proposition 2.6. For any formula $\varphi\in F(\Sigma)$ and a property
$P_{t}\subseteq\mathcal{T}_{\Sigma}$ the following conditions hold:

$(1)$ if $\varphi=\forall x\psi$ and $\varphi\rhd_{\rm pt}P_{t}$ then
$\psi\rhd_{\rm pt}P_{t}$;

$(2)$ if $\varphi=\forall x\psi$ and $\varphi\rhd_{\rm tt}P_{t}$ then
$\psi\rhd_{\rm tt}P_{t}$;

$(3)$ if $\varphi=\exists x\psi$ and $\psi\rhd_{\rm pt}P_{t}$ then
$\psi\rhd_{\rm pt}P_{t}$;

$(3)$ if $\varphi=\exists x\psi$ and $\psi\rhd_{\rm tt}P_{t}$ then
$\psi\rhd_{\rm tt}P_{t}$.

Proof. $(1)$ Let $\varphi=\forall x\psi$ and $\varphi\rhd_{\rm pt}P_{t}$. Then
there are $T\in P_{t}$, $\mathcal{M}\models T$, $\overline{a}\in M$ such that
$\mathcal{M}\models\varphi(\overline{a})$. It implies
$\mathcal{M}\models\forall x\psi(x,\overline{a})$, therefore there is $b\in M$
with $\mathcal{M}\models x\psi(b,\overline{a})$, i.e., $\psi\rhd_{\rm
pt}\\{T\\}$, hence $\psi\rhd_{\rm pt}P_{t}$.

$(2)$ We repeat arguments for $(1)$ replacing some $T\in P_{t}$ by an
arbitrary one.

$(3)$ Let $\varphi=\exists x\psi$ and $\psi\rhd_{\rm pt}P_{t}$. Then there are
$T\in P_{t}$, $\mathcal{M}\models T$, $\overline{a},b\in M$ such that
$\mathcal{M}\models\psi(b,\overline{a})$. It implies
$\mathcal{M}\models\exists x\psi(x,\overline{a})$, i.e., $\varphi\rhd_{\rm
pt}\\{T\\}$, hence $\psi\rhd_{\rm pt}P_{t}$.

$(4)$ As for $(2)$ we repeat arguments for $(3)$ replacing some $T\in P_{t}$
by an arbitrary one. $\Box$

The following two theorems assert that the relations $\rhd_{\rm pt}$ and
$\rhd_{\rm tt}$ are preserved under $E$-closures.

Theorem 2.7. For any sentence $\varphi\in{\rm Sent}(\Sigma)$ and a property
$P_{t}\subseteq\mathcal{T}_{\Sigma}$ the following conditions are equivalent:

$(1)$ $\varphi\rhd_{\rm pt}P_{t}$,

$(2)$ $\varphi\rhd_{\rm pt}{\rm Cl}_{E}(P_{t})$,

$(3)$ $\varphi\rhd_{\rm pt}P^{\prime}_{t}$ for any/some $P^{\prime}_{t}$ with
${\rm Cl}_{E}(P^{\prime}_{t})={\rm Cl}_{E}(P_{t})$.

Proof. $(1)\Rightarrow(2)$ holds in view of $P_{t}\subseteq{\rm
Cl}_{E}(P_{t})$ and the monotony of the relation $\rhd_{\rm pt}$.

$(2)\Rightarrow(3)$. It suffices to show that $\varphi\rhd_{\rm
pt}P^{\prime}_{t}$ for any $P^{\prime}_{t}$ with ${\rm
Cl}_{E}(P^{\prime}_{t})={\rm Cl}_{E}(P_{t})$. Since $\varphi\rhd_{\rm pt}{\rm
Cl}_{E}(P_{t})$ there is a theory $T\in{\rm Cl}_{E}(P_{t})$ with $\varphi\in
T$. If $T\in P^{\prime}t$ we have $\varphi\rhd_{\rm pt}P^{\prime}_{t}$.
Otherwise $T$ is an accumulation point of $P^{\prime}_{t}$ implying, in view
of Proposition 1.1, that $varphi$ belongs to infinitely many theories in
$P^{\prime}_{t}$. Therefore $\varphi\rhd_{\rm pt}P^{\prime}_{t}$.

$(3)\Rightarrow(1)$ is obvious. $\Box$

Theorem 2.8. For any sentence $\varphi\in{\rm Sent}(\Sigma)$ and a property
$P_{t}\subseteq\mathcal{T}_{\Sigma}$ the following conditions are equivalent:

$(1)$ $\varphi\rhd_{\rm tt}P_{t}$,

$(2)$ $\varphi\rhd_{\rm tt}{\rm Cl}_{E}(P_{t})$,

$(3)$ $\varphi\rhd_{\rm tt}P^{\prime}_{t}$ for any/some $P^{\prime}_{t}$ with
${\rm Cl}_{E}(P^{\prime}_{t})={\rm Cl}_{E}(P_{t})$.

Proof. $(1)\Rightarrow(2)$. Let $\varphi\rhd_{\rm tt}P_{t}$. If $P_{t}$ is
finite then ${\rm Cl}_{E}(P_{t})=P_{t}$ and we have $\varphi\rhd_{\rm tt}{\rm
Cl}_{E}(P_{t})$. If $P_{t}$ is finite then by Proposition 1.1, ${\rm
Cl}_{E}(P_{t})$ consists of theories in $P_{t}$ and of theories
$T\in\mathcal{T}_{\Sigma}\setminus P_{t}$ such that for any sentence $\psi\in
T$ the set $(P_{t})_{\psi}$ is infinite. Since $\varphi\rhd_{\rm tt}P_{t}$,
$\varphi$ belongs to each such theory $T$. Thus, $\varphi\rhd_{\rm tt}{\rm
Cl}_{E}(P_{t})$.

$(2)\Rightarrow(1)$ and $(2)\Rightarrow(3)$ are obvious.

$(3)\Rightarrow(2)$ follows assuming $\varphi\rhd_{\rm tt}P^{\prime}_{t}$ for
any/some $P^{\prime}_{t}$ with ${\rm Cl}_{E}(P^{\prime}_{t})={\rm
Cl}_{E}(P_{t})$ repeating the arguments for $(1)\Rightarrow(2)$. $\Box$

For a property $P_{t}\subseteq\mathcal{T}_{\Sigma}$ we denote by
$\nabla(P_{t})$ the set of all sentences $\varphi\in{\rm Sent}(\Sigma)$ with
$\varphi\rhd_{\rm pt}P_{t}$, and by $\triangle(P_{t})$ the set of all
sentences $\varphi\in{\rm Sent}(\Sigma)$ with $\varphi\rhd_{\rm tt}P_{t}$.

By the definition $\nabla(\emptyset)=\emptyset$, $\triangle(\emptyset)={\rm
Sent}(\Sigma)$, $\nabla(\mathcal{T}_{\Sigma})$ consists of all consistent
sentences $\varphi\in{\rm Sent}(\Sigma)$, and
$\triangle(\mathcal{T}_{\Sigma})$ consists of all tautologies $\varphi\in{\rm
Sent}(\Sigma)$.

Proposition 2.9. For any property $P_{t}\subseteq\mathcal{T}_{\Sigma}$ the
following conditions hold:

$(1)$ $\nabla(P_{t})=\bigcup P_{t}$;

$(2)$ $\nabla(P_{t})$ is consistent iff $|P_{t}|\leq 1$, and $\nabla(P_{t})$
is a complete theory iff $P_{t}$ is a singleton;

$(3)$ $\triangle(P_{t})=\bigcap P_{t}$;

$(4)$ $\triangle(P_{t})$ is a consistent theory iff $P_{t}\neq\emptyset$, and
$\triangle(P_{t})$ is complete iff $P_{t}$ is a singleton;

$(5)$ for any $P_{t}\neq\emptyset$, $\nabla(P_{t})\supseteq\triangle(P_{t})$,
and $\nabla(P_{t})=\triangle(P_{t})$ iff $P_{t}$ is a singleton.

Proof. $(1)$ If $\varphi\in\nabla(P_{t})$ then $\varphi\rhd_{\rm pt}P_{t}$ and
$\varphi\in T$ for some $T\in P_{t}$ implying $\nabla(P_{t})\subseteq\bigcup
P_{t}$. Conversely, if $\varphi\in\bigcup P_{t}$ then $\varphi\in T$ for some
$T\in P_{t}$ implying $\varphi\rhd_{\rm pt}P_{t}$ and therefore
$\varphi\in\nabla(P_{t})$.

$(2)$ Since $\nabla(\emptyset)=\emptyset$ it is consistent. If $P_{t}=\\{T\\}$
then $\nabla(P_{t})=T$, i.e., $\nabla(P_{t})$ is consistent and complete. If
$P_{t}$ contains two distinct theories $T_{1}$ and $T_{2}$ then $T_{1}\cup
T_{2}\subseteq\nabla(P_{t})$ implying that $\nabla(P_{t})$ is inconsistent
since there are sentences $\psi$ such that $\psi\in T_{1}$ and $\neg\psi\in
T_{2}$.

$(3)$ If $\varphi\in\triangle(P_{t})$ then $\varphi\rhd_{\rm tt}P_{t}$ and
$\varphi\in T$ for any $T\in P_{t}$ implying $\triangle(P_{t})\subseteq\bigcap
P_{t}$. Conversely, if $\varphi\in\bigcap P_{t}$ then $\varphi\in T$ for any
$T\in P_{t}$ implying $\varphi\rhd_{\rm tt}P_{t}$ and therefore
$\varphi\in\triangle(P_{t})$.

$(4)$ Since $\triangle(\emptyset)={\rm Sent}(\Sigma)$ it is inconsistent. If
$P_{t}\neq\emptyset$ then $\triangle(P_{t})=\bigcap P_{t}$ by $(3)$ implying
that $\triangle(P_{t})$ is a consistent theory as an intersection of complete
theories. If $\triangle(P_{t})$ is complete then both $P_{t}$ is nonempty and
does not contain two distinct theories, i.e., $P_{t}$ is a singleton.
Conversely, if $P_{t}=\\{T\\}$ then $\triangle(P_{t})=T$ which is a complete
theory.

$(5)$ If $P_{t}\neq\emptyset$ then by $(1)$ and $(3)$ we have
$\nabla(P_{t})=\bigcup P_{t}\supseteq\bigcap P_{t}=\triangle(P_{t})$. If
$P_{t}=\\{T\\}$ then $\triangle(P_{t})=T=\nabla(P_{t})$. If $T_{1},T_{2}\in
P_{t}$ for some $T_{1}\neq T_{2}$ then $\nabla(P_{t})=\bigcup P_{t}\supseteq
T_{1}\cup T_{2}\supsetneq T_{1}\cap T_{2}\supseteq\bigcap
P_{t}=\triangle(P_{t})$. $\Box$

Theorems 1.11, 2.7, 2.8 and Proposition 2.9 immediately imply the following
corollary on the separability of properties with respect to the relations
$\rhd_{\rm pt}$ and $\rhd_{\rm tt}$.

Corollary 2.10. $(1)$ For any properties
$P_{1},P_{2}\subseteq\mathcal{T}_{\Sigma}$ the following conditions hold:

$(1)$ there exists $\varphi\in{\rm Sent}(\Sigma)$ such that $\varphi\rhd_{\rm
pt}P_{1}$ and $\neg\varphi\rhd_{\rm pt}P_{2}$ iff $P_{1}$ and $P_{2}$ are
nonempty and $|P_{1}\cup P_{2}|\geq 2$; in particular, there exists
$\varphi\in{\rm Sent}(\Sigma)$ such that $\varphi\rhd_{\rm pt}P_{1}$ and
$\neg\varphi\rhd_{\rm pt}P_{1}$ iff $|P_{1}|\geq 2$;

$(2)$ there exists $\varphi\in{\rm Sent}(\Sigma)$ such that $\varphi\rhd_{\rm
tt}P_{1}$ and $\neg\varphi\rhd_{\rm tt}P_{2}$ iff ${\rm Cl}_{E}(P_{1})\cap{\rm
Cl}_{E}(P_{2})=\emptyset$.

Corollary 2.11. For any nonempty property $P_{t}\subseteq\mathcal{T}_{\Sigma}$
the following conditions hold:

$(1)$ the set $\triangle(P_{t})$ forms a filter
$\triangle(P_{t})/\\!\\!\equiv$ on $\\{\equiv\\!\\!(\varphi)\mid\varphi\in{\rm
Sent}(\Sigma)\\}$ with respect to $\vdash$;

$(2)$ the filter $\triangle(P_{t})/\\!\\!\equiv$ is principal iff $\bigcap
P_{t}$ is forced by some its sentence, i.e., $\bigcap P_{t}$ is a finitely
axiomatizable theory, which is incomplete for $|P_{t}|\geq 2$;

$(3)$ the filter $\triangle(P_{t})/\\!\\!\equiv$ is an ultrafilter iff $P_{t}$
is a singleton.

Proof. $(1)$ holds by Monotony and Proposition 2.3, (2).

$(2)$ immediately follows by Proposition 2.9, (3).

$(3)$ is satisfied in view of Proposition 2.9, (4). $\Box$

##  3 Ranks of sentences and spectra with respect to properties

In this section we introduce a measure of complexity for sentences satisfying
a property using the ${\rm RS}$-rank for families of theories [2, 3, 4].

Definition. For a sentence $\varphi\in{\rm Sent}(\Sigma)$ and a property
$P=P_{t}\subseteq\mathcal{T}_{\Sigma}$ we put ${\rm RS}_{P}(\varphi)={\rm
RS}(P_{\varphi})$, and ${\rm ds}_{P}(\varphi)={\rm ds}(P_{\varphi})$ if ${\rm
ds}(P_{\varphi})$ is defined.

If $P=\mathcal{T}_{\Sigma}$ then we omit $P$ and write ${\rm RS}(\varphi)$,
${\rm ds}(\varphi)$ instead of ${\rm RS}_{P}(\varphi)$ and ${\rm
ds}_{P}(\varphi)$, respectively.

Clearly, if $P\subseteq P^{\prime}\subseteq\mathcal{T}_{\Sigma}$ and
$\varphi\in{\rm Sent}(\Sigma)$ then ${\rm RS}_{P}(\varphi)\leq{\rm
RS}_{P^{\prime}}(\varphi)$, and if ${\rm RS}_{P}(\varphi)={\rm
RS}_{P^{\prime}}(\varphi)\in{\rm Ord}$ then ${\rm ds}_{P}(\varphi)\leq{\rm
ds}_{P^{\prime}}(\varphi)$. Besides, we have:

Proposition 3.1. $(1)$ $\varphi\rhd_{\rm tt}P$ iff ${\rm
RS}_{P}(\neg\varphi)=-1$.

$(2)$ $\varphi\rhd_{\rm pt}P$ iff ${\rm RS}_{P}(\varphi)\geq 0$.

Definition. For a sentence $\varphi\in{\rm Sent}(\Sigma)$ and a property
$P\subseteq\mathcal{T}_{\Sigma}$ we say that $\varphi$ is $P$-totally
transcendental if ${\rm RS}_{P}(\varphi)$ is an ordinal. If
$P=\mathcal{T}_{\Sigma}$ then a $P$-totally transcendental sentence is said to
be totally transcendental.

A sentence $\varphi$ is co-($P$)-totally transcendental if $\neg\varphi$ is
$P$-totally transcendental.

We omit $P$ and say about totally transcendental and co-totally transcendental
sentences if $P=\mathcal{T}_{\Sigma}$.

By the definition each sentence $\varphi\in{\rm Sent}(\Sigma)$ obtains the
characteristics ${\rm RS}_{P}(\varphi)$ and ${\rm RS}_{P}(\neg\varphi)$ saying
that $\varphi$ is (co)-rich enough with respect to the property $P$, The
characteristics ${\rm ds}_{P}(\varphi)$ and ${\rm ds}_{P}(\neg\varphi)$, if
they are defined, give an additional information on “$P$-richness” of
$\varphi$.

For instance, if ${\rm RS}_{P}(\varphi)=0$ and ${\rm ds}_{P}(\varphi)=n$ then
$\varphi$ is $P$-finite satisfying exactly $n$ theories in $P$. Respectively,
if ${\rm RS}_{P}(\neg\varphi)=0$ and ${\rm ds}_{P}(\neg\varphi)=n$ then
$\varphi$ is $P$-cofinite, i.e., it does not satisfy exactly $n$ theories in
$P$.

Clearly, $\varphi$ is both $P$-finite and $P$-cofinite iff $P$ is nonempty and
is finite.

Theorem 3.2. For a language $\Sigma$ there is a totally transcendental
sentence $\varphi\in{\rm Sent}(\Sigma)$ iff $\Sigma$ has finitely many
predicate symbols.

Proof. If $\Sigma$ has finitely many predicate symbols we choose a sentence
$\varphi$ assigning that a universe is a singleton. Since functional symbols
have unique interpretations and there are finitely many possibilities for
(non)empty language predicate we obtain ${\rm RS}(\varphi)=0$ that is
$\varphi$ is totally transcendental.

Conversely, if $\Sigma$ has infinitely many predicate symbols then each
consistent sentence $\varphi$ obtains a $2$-tree with respect to (non)empty
predicates $Q\in\Sigma\setminus\Sigma(\varphi)$. This $2$-tree witnesses that
${\rm RS}(\varphi)=\infty$, i.e., $\varphi$ is not totally transcendental.
$\Box$.

Remark 3.3. If $\Sigma$ is finite then for the proof of Theorem 3.2 it
suffices to choose a sentence $\varphi$ assigning that a universe is finite,
since there are finitely many many possibilities, up to isomorphism, for
interpretations of language symbols implying ${\rm RS}(\varphi)=0$. $\Box$

The following definition introduces values for richness of a sentence with
respect to a property.

Definition. For a language $\Sigma$, a property
$P\subseteq\mathcal{T}_{\Sigma}$, an ordinal $\alpha$ and a natural number
$n\geq 1$, a sentence $\varphi\in{\rm Sent}(\Sigma)$ is called
$(P,\alpha,n)$-(co-)rich if ${\rm RS}_{P}(\varphi)=\alpha$ and ${\rm
ds}_{P}(\varphi)=n$ (respectively, ${\rm RS}_{P}(\neg\varphi)=\alpha$ and
${\rm ds}_{P}(\neg\varphi)=n$).

A sentence $\varphi\in{\rm Sent}(\Sigma)$ is called $(P,\infty)$-(co-)rich if
${\rm RS}_{P}(\varphi)=\infty$ (respectively, ${\rm
RS}_{P}(\neg\varphi)=\infty$).

If $P=\mathcal{T}_{\Sigma}$ we write that $\varphi$ is $(\alpha,n)$-(co-)rich
instead of $(P,\alpha,n)$-(co-)rich, and $\infty$-(co-)rich instead of
$(P,\infty)$-(co-)rich.

If for a property $P$ there is a $(P,\ast)$-(co-)rich sentence $\varphi$, we
say that $P$ has a $(P,\ast)$-(co-)rich sentence, where $\ast=\alpha,n$ or
$\alpha=\infty$.

By the definition if a sentence $\varphi$ is $(P,\alpha,n)$-rich then ${\rm
RS}(P_{\varphi})=\alpha$, ${\rm ds}(P_{\varphi})=n$.

Theorem 3.4. $(1)$ If a property $P\subseteq\mathcal{T}_{\Sigma}$ has a
$(P,\alpha,m)$-rich sentence $\varphi$ which is $(P,\beta,n)$-co-rich then
${\rm RS}(P)={\rm max}\\{\alpha,\beta\\}$, ${\rm ds}(P)=m$ for $\alpha>\beta$,
${\rm ds}(P)=n$ for $\alpha<\beta$, and ${\rm ds}(P)=m+n$ for $\alpha=\beta$.

$(2)$ If for a property $P\subseteq\mathcal{T}_{\Sigma}$, ${\rm RS}(P)=\alpha$
and ${\rm ds}(P)=n$, then for each sentence $\varphi\in{\rm Sent}(\Sigma)$ the
following assertions hold:

(i) ${\rm RS}_{P}(\varphi)\leq\alpha$,

(ii) if ${\rm RS}_{P}(\varphi)=\alpha$ then $\varphi$ is $(P,\alpha,m)$-rich
for some $m\leq n$, and for $m=n$ either $\varphi\rhd_{\rm tt}P$ or $\varphi$
is $(P,\beta,k)$-co-rich for some $\beta<\alpha$ and $k\in\omega$, and if
$m<n$ then $\varphi$ is $(P,\alpha,n-m)$-co-rich.

Proof. (1) For the sentence $\varphi$ we have ${\rm RS}_{P}(\varphi)=\alpha$,
${\rm ds}_{P}(\varphi)=m$, ${\rm RS}_{P}(\neg\varphi)=\beta$, ${\rm
ds}_{P}(\neg\varphi)=n$. It means that $P$ is divided into two disjoint parts
$P_{\varphi}$ and $P_{\neg\varphi}$ with given characteristics ${\rm
RS}(P_{\varphi})=\alpha$, ${\rm ds}(P_{\varphi})=m$, ${\rm
RS}(P_{\neg\varphi})=\beta$, ${\rm ds}(P_{\neg\varphi})=n$.

If ${\rm RS}(P)=0$ then $|P|={\rm ds}(P)$, ${\rm RS}_{P}(\varphi)={\rm
RS}_{P}(\neg\varphi)=0$, ${\rm ds}(P)=|P_{\varphi}|+|P_{\neg\varphi}|=m+n$.

If ${\rm RS}(P)>0$ then a tree witnessing the value ${\rm RS}(P)=\gamma$ can
be transformed step-by-step using theories either in $P_{\varphi}$ or in
$P_{\neg\varphi}$: in each step witnessing ${\rm RS}(P)=\gamma$ there are
infinitely many branches of previous values related to $P_{\varphi}$ or to
$P_{\neg\varphi}$.

In the first case, related to $P_{\varphi}$, we have $\gamma=\alpha$ and in
the second case, related to $P_{\neg\varphi}$, $\gamma=\beta$. If
$\alpha>\beta$ a tree for $P_{\varphi}$ witnesses $l=m$. If $\alpha<\beta$ a
tree for $P_{\neg\varphi}$ witnesses $l=n$. If $\alpha=\beta$ then both trees
for $P_{\varphi}$ and for $P_{\neg\varphi}$ witness $\gamma=\alpha$ and
$l=m+n$, since there are exactly $l+m$ $s$-definable subsets of $P$ having the
rank $\gamma$ and the degree $1$.

$(2)$ We have (i) by the monotony of rank (if $P_{1}\subseteq P_{2}$ then
${\rm RS}(P_{1})\leq{\rm RS}(P_{2})$) and the inclusion $P_{\varphi}\subseteq
P$. (ii) holds by the monotony of degree for a fixed rank (if $P_{1}\subseteq
P_{2}$ and ${\rm RS}(P_{1})={\rm RS}(P_{2})\in{\rm Ord}$ then ${\rm
ds}(P_{1})\leq{\rm ds}(P_{2})$) and the inclusion $P_{\varphi}\subseteq P$.
Besides if ${\rm ds}(P_{\varphi})=m=n={\rm ds}(P)$ then $P$ can not have a
tree in $P_{\neg\varphi}=P\setminus P_{\varphi}$ witnessing ${\rm
RS}(P_{\neg\varphi})=\alpha$ since otherwise ${\rm ds}(P)$ should be more than
$n$. Therefore either $\neg\varphi$ is $P$-inconsistent, i.e.,
$\varphi\rhd_{\rm tt}P$, or $\varphi$ is $(P,\beta,k)$-co-rich for some
$\beta<\alpha$ and $k\in\omega$. If $m<n$ then $\varphi$ is
$(P,\alpha,n-m)$-co-rich in view of (1). $\Box$

By Theorem 3.4 for any $e$-totally transcendental property $P$ and any
$\alpha\leq{\rm RS}(P)$ there are $s$-definable subfamilies $P_{\varphi}$ with
${\rm RS}(P_{\varphi})=\alpha$. Similarly all values $m\leq{\rm ds}(P)$ are
also realized by appropriate $s$-definable subfamilies.

Thus the spectrum ${\rm Sp}_{\rm Rd}(P)$ for the pairs $({\rm
RS}_{P}(\varphi),{\rm ds}_{P}(\varphi))$ with nonempty $P_{\varphi}$ forms the
set

$\\{({\rm RS}(P),m)\mid 1\leq m\leq{\rm
ds}(P)\\}\cup\\{(\alpha,m)\mid\alpha<{\rm
RS}(P),m\in\omega\setminus\\{0\\}\\},$ (1)

which is an initial segment $O[(\beta,n)]$ consisting of all pairs
$(\alpha,m)\in{\rm Ord}\times(\omega\setminus{0})$ with $\alpha\leq\beta$ and
$m\leq n$ for $\alpha=\beta$, ${\rm RS}(P)=\beta$, ${\rm ds}(P)=n$.

Remark 3.5. If ${\rm RS}(P)=\infty$ then $s$-definable subfamilies
$P_{\varphi}$ can have only values ${\rm RS}(P_{\varphi})=\infty$ or both the
value ${\rm RS}(P)=\infty$ and pairs forming some initial segment
$O[(\beta,n)]$.

Indeed, let $P$ be a family of theories in a language $\Sigma$ of independent
$0$-ary predicates $Q_{l}$, $l\in\lambda$, $\lambda\geq\omega$, such that each
sentence $Q^{\delta_{1}}_{i_{1}}\wedge\ldots\wedge Q^{\delta_{k}}_{i_{k}}$,
$i_{1}<\ldots<i_{k}<\lambda$, $\delta_{1},\ldots,\delta_{k}\in\\{0,1\\}$,
$k\in\omega$, is $P$-consistent. Each $P$-consistent sentence $\varphi\in{\rm
Sent}(\Sigma)$ is divided into $2$-tree witnessing that ${\rm
RS}_{P}(\varphi)=\infty$. In such a case we say that the spectrum ${\rm
Sp}_{\rm Rd}(P)$ equals $\\{\infty\\}$.

The family $P$ above can be extended by a family $P^{\prime}$ with dependent
predicates $Q_{l}$ producing given ${\rm RS}$-rank and ${\rm ds}$-degree for a
subfamily with, say, $Q_{0}\leftrightarrow Q_{1}$. Therefore the arguments for
Theorem 3.4 produce an initial segment $O[(\beta,n)]$ for the spectrum ${\rm
Sp}_{\rm Rd}(P^{\prime})$ of $s$-definable family $P^{\prime}$. Thus, ${\rm
Sp}_{\rm Rd}(P\cup P^{\prime})=O[(\beta,n)]\cup\\{\infty\\}$.

Since each nonempty $s$-definable subfamily has a spectrum of the form
$O[(\beta,n)]$, or $\\{\infty\\}$, or $O[(\beta,n)]\cup\\{\infty\\}$, initial
segments are well-ordered with respect to the relation $\subseteq$, and the
ordinal ${\rm RS}(P)$-ranks are bounded by $|\mathcal{T}_{\Sigma}|\leq 2^{{\rm
max}\\{|\Sigma|,\omega\\}}$, all values ${\rm Sp}_{\rm Rd}(P)$, for nonempty
properties $P\subseteq\mathcal{T}_{\Sigma}$, are exhausted by these three
possibilities, and we obtain the following:

Theorem 3.6. For any nonempty property $P\subseteq\mathcal{T}_{\Sigma}$ one of
the following possibilities holds for some $\beta\in{\rm Ord}$ and
$n\in\omega\setminus\\{0\\}$:

$(1)$ ${\rm Sp}_{\rm Rd}(P)=O[(\beta,n)]$,

$(2)$ ${\rm Sp}_{\rm Rd}(P)=\\{\infty\\}$,

$(3)$ ${\rm Sp}_{\rm Rd}(P)=O[(\beta,n)]\cup\\{\infty\\}$.

All possibilities above are realized by appropriate languages $\Sigma$ and
properties $P\subseteq\mathcal{T}_{\Sigma}$.

Remark 3.7. Any value ${\rm Sp}_{\rm Rd}(P)$ can be naturally extended till
$\overline{\rm Sp}_{\rm Rd}(P)={\rm Sp}_{\rm Rd}(P)\cup\\{-1\\}$ corresponding
to the value ${\rm RS}(\emptyset)=-1$ of the empty subfamily of
$\mathcal{T}_{\Sigma}$. It is also natural to put $\overline{\rm Sp}_{\rm
Rd}(P)=\\{-1\\}$ for empty $P\subset\mathcal{T}_{\Sigma}$. In view of Theorem
3.6 we have the following description of extended spectra $\overline{\rm
Sp}_{\rm Rd}(P)$:

i) $\overline{\rm Sp}_{\rm Rd}(P)=\\{-1\\}$,

ii) $\overline{\rm Sp}_{\rm Rd}(P)=O[(\beta,n)]\cup\\{-1\\}$,

iii) $\overline{\rm Sp}_{\rm Rd}(P)=\\{-1,\infty\\}$,

iv) $\overline{\rm Sp}_{\rm Rd}(P)=O[(\beta,n)]\cup\\{-1,\infty\\}$.

Theorem 3.8 [4]. Let $\mathcal{T}$ be a family of a countable language
$\Sigma$ and with ${\rm RS}(\mathcal{T})=\infty$, $\alpha$ be a countable
ordinal, $n\in\omega\setminus\\{0\\}$. Then there is a $d_{\infty}$-definable
subfamily $\mathcal{T}^{\ast}\subset\mathcal{T}$ such that ${\rm
RS}(\mathcal{T}^{\ast})=\alpha$ and ${\rm ds}(\mathcal{T}^{\ast})=n$.

Theorems 3.6 and 3.8 immediately imply:

Corollary 3.9. Let $\mathcal{T}$ be a family of a countable language $\Sigma$
and with ${\rm RS}(\mathcal{T})=\infty$, $\alpha$ be a countable ordinal,
$n\in\omega\setminus\\{0\\}$. Then there is a $d_{\infty}$-definable property
$P\subset\mathcal{T}$ such that ${\rm Sp}_{\rm Rd}(P)=O[(\alpha,n)]$.

##  4 Spectra for cardinalities of definable subproperties

In this section we study some refinements of the relation $\rhd_{\rm pt}$.

For a cardinality $\lambda\geq 1$, a sentence $\varphi\in{\rm Sent}(\Sigma)$
and a property $P\subseteq\mathcal{T}_{\Sigma}$ we write
$\varphi\rhd^{\lambda}_{\rm pt}P$ if $\varphi$ satisfies exactly $\lambda$
theories in $P$, i.e., $|P_{\varphi}|=\lambda$.

By the definition if $P\neq\emptyset$ and $\varphi\rhd_{\rm tt}P$ then
$\varphi\rhd^{|P|}_{\rm pt}P$, and conversely $\varphi\rhd^{|P|}_{\rm pt}P$
implies $\varphi\rhd_{\rm tt}P$ for finite $P$. For infinite $P$ the converse
implication can fail. Moreover, since infinite sets can be divided into two
parts of same cardinality, one can easily introduce an expansion $P^{\prime}$
of $P$ by a $0$-ary predicate $Q$ such that $Q\rhd^{|P^{\prime}|}_{\rm
pt}P^{\prime}$ and $\neg Q\rhd^{|P^{\prime}|}_{\rm pt}P^{\prime}$, implying
that $Q\not\rhd_{\rm tt}P^{\prime}$.

For a property $P$ we denote by ${\rm Sp}_{\rm pt}(P)$ the set
$\\{\lambda\mid\varphi\rhd^{\lambda}_{\rm pt}P$ for some sentence
$\varphi\\}$. This set is called the ${\rm pt}$-spectrum of $P$.

By the definition $|P|\in{\rm Sp}_{\rm pt}(P)$ for any nonempty $P$, and
$\lambda\leq|P|$ for any $\lambda\in{\rm Sp}_{\rm pt}(P)$, i.e., ${\rm
sup}{\rm Sp}_{\rm pt}(P)=||P|+1|$.

A natural question arises on a description of ${\rm pt}$-spectra.

This question is easily answered for finite $P$, since in such a case all
subsets of theories are separated as $s$-definable singletons from their
complements, and we obtain the following:

Proposition 4.1. For any finite property $P\subseteq\mathcal{T}_{\Sigma}$,
${\rm Sp}_{\rm pt}(P)=(|P|+1)\setminus\\{0\\}$.

The following assertion generalizes Proposition 4.1 in terms of isolated
points due Theorem 1.12:

Proposition 4.2. If $P$ has exactly $n\in\omega\setminus\\{0\\}$ isolated
points then ${\rm Sp}_{\rm pt}(P)\cap\omega=(n+1)\setminus\\{0\\}$.

Proof. Let $T_{1},\ldots,T_{n}$ are isolated points in $P$. If $k\in{\rm
Sp}_{\rm pt}(P)\cap\omega$, i.e., there is a sentence $\varphi$ with
$\varphi\rhd^{k}_{\rm pt}P$ then $P_{\varphi}$ consists of isolated points
$T_{i_{1}},\ldots,T_{i_{k}}$ since elements of the finite set $P_{\varphi}$
are separated as $s$-definable singletons from their complements in
$P_{\varphi}$. Then $k\in(n+1)\setminus\\{0\\}$. Conversely, if
$k\in\omega\setminus{\rm Sp}_{\rm pt}(P)$ then $k>n$ since each $k\leq n$
equals $|P_{\varphi}|$ for some sentence $\varphi$, implying $k\in{\rm
Sp}_{\rm pt}(P)$. $\Box$

Proposition 4.3. For any nonempty property $P\subseteq\mathcal{T}_{\Sigma}$
either ${\rm Sp}_{\rm pt}(P)\cap\omega$ equals an initial segment
$(n+1)\setminus\\{0\\}$ for some $n\in\omega\setminus\\{0\\}$ or ${\rm
Sp}_{\rm pt}(P)\cap\omega=\omega\setminus\\{0\\}$.

Proof. If there are sentences $\varphi$ with finite $P_{\varphi}$ then either
there is $P_{\varphi}$ with the greatest finite cardinality, implying ${\rm
Sp}_{\rm pt}(P)\cap\omega=(n+1)\setminus\\{0\\}$ by the arguments of
Proposition 4.2, or the finite cardinalities $|P_{\varphi}|$ are unbounded
that means ${\rm Sp}_{\rm pt}(P)\cap\omega=\omega\setminus\\{0\\}$. $\Box$

Proposition 4.4. For any infinite property $P\subseteq\mathcal{T}_{\Sigma}$
there is a nonempty set $Y\subseteq|P|$ of infinite cardinalities such that
either there is $n\in\omega\setminus\\{0\\}$ with ${\rm Sp}_{\rm
pt}(P)=Y\cup(n+1)\setminus\\{0\\}$, or ${\rm Sp}_{\rm
pt}(P)=Y\cup\omega\setminus\\{0\\}$, or ${\rm Sp}_{\rm pt}(P)=Y$. All values
$Y\cup(n+1)\setminus\\{0\\}$, $Y\cup\omega\setminus\\{0\\}$ and $Y$, for a
nonempty set $Y$ of infinite cardinalities and $n\in\omega\setminus\\{0\\}$,
are realized as ${\rm Sp}_{\rm pt}(P)$ for appropriate properties $P$.

Proof. Since ${\rm sup}{\rm Sp}_{\rm pt}(P)\leq|P|$ and Proposition 4.3
describes all possibilities for ${\rm Sp}_{\rm pt}(P)\cap\omega$, it suffices,
for a nonempty set $Y$ of infinite cardinalities and
$n\in\omega\setminus\\{0\\}$, to find a property $P_{1}$ with ${\rm Sp}_{\rm
pt}(P_{1})=Y\cup(n+1)\setminus\\{0\\}$, a property $P_{2}$ with ${\rm Sp}_{\rm
pt}(P_{2})=Y\cup\omega\setminus\\{0\\}$, and a property $P_{3}$ with ${\rm
Sp}_{\rm pt}(P_{2})=Y$. For the property $P_{1}$ one can take a finite
$n$-element family $Y_{n}$, expanded by a $0$-ary predicate $Q_{0}$ marking
all theories in $Y_{n}$, and extend $Y_{n}$ by families $Y_{\lambda}$, for
each $\lambda\in Y$, of $\lambda$ theories of $\lambda$ independent $0$-ary
predicates $Q^{\lambda}_{i}$ expanded by a $0$-ary predicate $Q_{\lambda}$
marking all theories in $Y_{\lambda}$. Any $P_{1}$-consistent sentence
$\varphi$ satisfies either $k\leq n$ theories in $Y_{n}$ or $\lambda$ many
theories in $Z_{\lambda}$ and possibly $\mu$ many theories in $Y_{\mu}$ for
finitely many $\mu<\lambda$. It means that the cardinalities
$|(P_{1})_{\varphi}|$ witness the equality ${\rm Sp}_{\rm
pt}(P_{1})=Y\cup(n+1)\setminus\\{0\\}$.

For the property $P_{2}$ we repeat the process for $P_{1}$ replacing the part
$Y_{n}$ by an $e$-minimal family $Y^{\lambda}_{0}$ consisting of some
$\lambda\in Y$ theories all of which are marked by new $0$-ary predicate
$Q_{0}$. Realizing this process we obtain that $s$-definable sets are finite,
cofinite, or consists of $\lambda$ theories for $\lambda\in Y$. Thus, ${\rm
Sp}_{\rm pt}(P_{1})=Y\cup\omega\setminus\\{0\\}$.

For the property $P_{3}$ we repeat the process for $P_{1}$ without the part
$Y_{n}$ obtaining ${\rm Sp}_{\rm pt}(P_{1})=Y$. $\Box$

Summarizing Propositions 4.1–4.4 we obtain the following theorem describing
${\rm pt}$-spectra of properties.

Theorem 4.5. For any nonempty property $P\subseteq\mathcal{T}_{\Sigma}$ one of
the following conditions holds:

$(1)$ ${\rm Sp}_{\rm pt}(P)=(n+1)\setminus\\{0\\}$ for some
$n\in\omega\setminus\\{0\\}$; it is satisfied iff $P$ is finite with $|P|=n$;

$(2)$ ${\rm Sp}_{\rm pt}(P)=Y\cup(n+1)\setminus\\{0\\}$ for some nonempty set
$Y\subseteq|P|$ of infinite cardinalities and $n\in\omega\setminus\\{0\\}$;

$(3)$ ${\rm Sp}_{\rm pt}(P)=Y\cup\omega\setminus\\{0\\}$ for some nonempty set
$Y\subseteq|P|$ of infinite cardinalities;

$(4)$ ${\rm Sp}_{\rm pt}(P)=Y$ for some nonempty set $Y\subseteq|P|$ of
infinite cardinalities.

All values $(n+1)\setminus\\{0\\}$, $Y\cup(n+1)\setminus\\{0\\}$,
$Y\cup\omega\setminus\\{0\\}$, and $Y$, for a nonempty set $Y$ of infinite
cardinalities and $n\in\omega\setminus\\{0\\}$, are realized as ${\rm Sp}_{\rm
pt}(P)$ for an appropriate property $P$.

The following assertion shows that $Y$ in Theorem 4.5 is finite for a property
$P$ with ${\rm RS}(P)=1$.

Proposition 4.6. If ${\rm RS}(P)=1$ then ${\rm Sp}_{\rm
pt}(P)=Y\cup\omega\setminus\\{0\\}$ for some finite nonempty set
$Y\subseteq|P|$ of infinite cardinalities with $|Y|\leq{\rm ds}(P)$.

Proof. Since ${\rm RS}(P)=1$, $P$ is divided into $k={\rm ds}(P)$ infinite
$s$-definable $e$-minimal parts $P_{1},\ldots,P_{k}$. Each part $P_{i}$ has
only finite and cofinite $s$-definable subset producing ${\rm Sp}_{\rm
pt}(P_{i})=\\{|P_{i}|\\}\cup\omega\setminus\\{0\\}$. Since each $s$-definable
subset of $P$ is a Boolean combination of $s$-definable subsets of $P_{i}$ and
$|Z_{i}\cup Z_{j}|={\rm max}\\{|Z_{i}|,|Z_{j}|\\}$ for infinite $s$-definable
$Z_{i}\subseteq P_{i}$, $Z_{j}\subseteq P_{J}$, $i,j\leq k$, we obtain ${\rm
Sp}_{\rm pt}(P)=\\{|P_{1}|,\ldots,|P_{k}|\\}\cup\omega\setminus\\{0\\}$.
$\Box$

Remark 4.7. Describing possibilities for ${\rm pt}$-spectra ${\rm Sp}_{\rm
pt}(P)$ we admit that properties $P$ may be not $E$-closed. If we assume that
$P$ is infinite and $E$-closed then we have two cases: either $P$ is
$e$-totally transcendental with the least generating set of a cardinality
$\mu_{1}\leq{\rm max}\\{|\Sigma|,\omega\\}$ and with $\mu_{2}\leq{\rm
max}\\{|\Sigma|,\omega\\}$ accumulation points, or ${\rm RS}(P)=\infty$ with
$|P|\geq 2^{\omega}$ by Theorem 1.2. In the first case values for
$\lambda\in{\rm Sp}_{\rm pt}(P)$ are exhausted by all cardinalities in
$\omega\setminus\\{0\\}$ and by some infinite cardinalities $\leq{\rm
max}\\{|\Sigma|,\omega\\}$. In particular, for countable $\Sigma$, since ${\rm
RS}(P)$ is a countable ordinal, we have ${\rm Sp}_{\rm
pt}(P)=(\omega+1)\setminus\\{0\\}$. In the second case values for
$\lambda\in{\rm Sp}_{\rm pt}(P)$ are exhausted by cardinalities in
$\omega\setminus\\{0\\}$ or by cardinalities of some its initial segment,
depending on existence of the least generating set for $P$, and by some
infinite cardinalities $\leq{\rm max}\\{|\Sigma|,\omega\\}$ and cardinalities
$\geq 2^{\omega}$. In particular, for countable $\Sigma$, ${\rm Sp}_{\rm
pt}(P)$ includes $2^{\omega}$, and possibly $\omega$ depending on existence of
infinite totally transcendental definable part. Thus, in Theorem 4.5, for
$E$-closed $P$ some cases are not realized:
$\\{\omega\\}\cup(n+1)\setminus\\{0\\}$, $\\{\omega\\}$, $\\{\omega_{1}\\}$,
$\\{\omega,\omega_{1}\\}$, etc.

By Remark 4.7 using Theorem 1.2 we have the following:

Theorem 4.8. For any nonempty $E$-closed property
$P\subseteq\mathcal{T}_{\Sigma}$ with at most countable language $\Sigma$ one
of the following possibilities holds:

$(1)$ ${\rm Sp}_{\rm pt}(P)=(n+1)\setminus\\{0\\}$ for some
$n\in\omega\setminus\\{0\\}$, if $P$ is finite with $|P|=n$;

$(2)$ ${\rm Sp}_{\rm pt}(P)=\\{2^{\omega}\\}\cup(n+1)\setminus\\{0\\}$ for
some $n\in\omega$, if $P$ is infinite and has $n$ isolated points;

$(3)$ ${\rm Sp}_{\rm pt}(P)=(\omega+1)\setminus\\{0\\}$, if $P$ is infinite
and totally transcendental;

$(4)$ ${\rm Sp}_{\rm
pt}(P)=\\{\omega,2^{\omega}\\}\cup\omega\setminus\\{0\\}$, if $P$ has an
infinite totally transcendental definable subfamily but $P$ itself is not
totally transcendental;

$(5)$ ${\rm Sp}_{\rm pt}(P)=\\{2^{\omega}\\}\cup\omega\setminus\\{0\\}$, if
$P$ has infinitely many isolated points but does not have infinite totally
transcendental definable subfamilies.

Remark 4.9. Possibilities in Theorem 4.7 give low bounds for correspondent
cases in uncountable languages.

##  5 $P$-generic sentences and $P$-generic theories

Definition. (Cf. [8, 9, 10]) For a property $P\subseteq\mathcal{T}_{\Sigma}$ a
sentence $\varphi\in{\rm Sent}(\Sigma)$ is called $P$-generic if ${\rm
RS}_{P}(\varphi)={\rm RS}(P)$, and ${\rm ds}_{P}(\varphi)={\rm ds}(P)$ if
${\rm ds}(P)$ is defined.

If $P=\mathcal{T}_{\Sigma}$ then we omit $P$ and a $P$-generic sentence is
called generic.

By the definition we have the following:

Proposition 5.1. Any $P$-generic sentence $\varphi$ is $(P,{\rm RS}(P),{\rm
ds}(P))$-rich if ${\rm RS}(P)$ is an ordinal, and $(P,\infty)$-rich if ${\rm
RS}(P)=\infty$. And vice versa, each $(P,{\rm RS}(P),{\rm ds}(P))$-rich
sentence, for an ordinal ${\rm RS}(P)$, is $P$-generic, and each
$(P,\infty)$-rich sentence, for ${\rm RS}(P)=\infty$, is $P$-generic.

Proposition 5.2. If $\varphi\rhd_{\rm tt}P$ then $\varphi$ is $P$-generic.

In view of Proposition 5.2 any property $P\subseteq\mathcal{T}_{\Sigma}$ has
$P$-generic sentences.

Corollary 5.3. If a property $P\subseteq\mathcal{T}_{\Sigma}$ is finite and
$\varphi\in{\rm Sent}(\Sigma)$ then $\varphi\rhd_{\rm tt}P$ iff $\varphi$ is
$P$-generic.

Proof. By Proposition 5.2 it suffices to show that if $\varphi$ is $P$-generic
then $\varphi\rhd_{\rm tt}P$. If $P=\emptyset$ then both ${\rm RS}(P)={\rm
RS}_{P}(\varphi)=-1$ and $\varphi\rhd_{\rm tt}P$. If $P$ consists of $n\geq 1$
theories, then ${\rm RS}(P)=0$, ${\rm ds}(P)=n$. Assuming that $\varphi$ is
$P$-generic we have ${\rm RS}(P_{\varphi})=0$, ${\rm ds}(P_{\varphi})=n$
implying that $\varphi$ belongs to all $n$ theories in $P$ and
$\varphi\rhd_{\rm tt}P$. $\Box$

Remark 5.4. In view of Corollary 5.3 the converse implication for Proposition
5.2 holds iff $P$ is finite. Indeed, if $P=\mathcal{T}_{\Sigma}$ for a
countable language $\Sigma$ with ${\rm RS}(P)=\infty$, which is characterized
in Theorem 1.2, then we can construct a $2$-tree of sentences
$\varphi_{\delta}$, $\delta\in{\,}<\omega 2$ witnessing this value of ${\rm
RS}(P)$. It means that $P$ is divided into two disjoint definable parts
$P_{\varphi_{0}}$ and $P_{\varphi_{1}}$ with ${\rm RS}(P_{\varphi_{0}})={\rm
RS}(P_{\varphi_{1}})=\infty$. Thus, $\varphi_{0}$ and $\varphi_{1}$ are
generic whereas $\varphi_{0}\not\\!\\!\rhd_{\rm tt}P$ and
$\varphi_{1}\not\\!\\!\rhd_{\rm tt}P$. Moreover, this effect works both for an
arbitrary property $P$ with ${\rm RS}(P)=\infty$ and for an arbitrary property
$P$ with ${\rm RS}(P)\in{\rm Ord}$ and ${\rm RS}(P)\geq 1$. In the latter case
we can remove a branch in the tree witnessing the values ${\rm RS}(P)$ and
${\rm ds}(P)$ just considering a sentence $\varphi_{1}\wedge\neg\psi$, where
$\varphi_{1}$ is a tautology and $P_{\psi}$ is nonempty with ${\rm
RS}(P_{\psi})<{\rm RS}(P)$. In such a case $\varphi_{1}\wedge\neg\psi$ is
$P$-generic and $(\varphi_{1}\wedge\neg\psi)\not\\!\\!\rhd_{\rm tt}P$. $\Box$

In view of Remark 5.4 we have the following:

Proposition 5.5. For a property $P\subseteq\mathcal{T}_{\Sigma}$ there is a
$P$-generic sentence $\varphi\in{\rm Sent}(\Sigma)$ with minimal/least
$P_{\varphi}$ iff $P$ is finite. If that $\varphi$ exists then
$P_{\varphi}=P$.

By Proposition 5.5 for a property $P\subseteq\mathcal{T}_{P}$ with ${\rm
RS}(P)\geq 1$, $P$-generic sentences $\varphi$ produce infinite descending
chains of $s$-definable subfamilies $P_{\varphi}$.

Proposition 5.6. If for a property $P\subseteq\mathcal{T}_{\Sigma}$, ${\rm
RS}(P)=\alpha\in{\rm Ord}$, ${\rm ds}(P)=1$, then for any sentence
$\varphi\in{\rm Sent}(\Sigma)$ either ${\rm RS}_{P}(\varphi)=\alpha$ and ${\rm
ds}_{P}(\varphi)=1$, or ${\rm RS}_{P}(\neg\varphi)=\alpha$ and ${\rm
ds}_{P}(\neg\varphi)=1$.

Proof. By the conjecture for $P$ and the monotony for pairs of values of ${\rm
RS}$ and ${\rm ds}$ we have ${\rm RS}_{P}(\varphi)\leq\alpha$ and ${\rm
RS}_{P}(\neg\varphi)\leq\alpha$ for any $\varphi\in{\rm Sent}(\Sigma)$, and if
the ${\rm RS}$-rank equals $\alpha$ then the ${\rm ds}$-degree equals $1$. We
can not have ${\rm RS}_{P}(\varphi)<\alpha$ and ${\rm
RS}_{P}(\neg\varphi)<\alpha$ for $\varphi\in{\rm Sent}(\Sigma)$ by Theorem
3.4. Thus ${\rm RS}_{P}(\varphi)=\alpha$ and ${\rm ds}_{P}(\varphi)=1$, or
${\rm RS}_{P}(\neg\varphi)=\alpha$ and ${\rm ds}_{P}(\neg\varphi)=1$. The
latter conditions can not be satisfied simultaneously since otherwise ${\rm
ds}(P)\geq 2$. $\Box$

Since any property $P\subseteq\mathcal{T}_{\Sigma}$ with ${\rm
RS}(P)=\alpha\in{\rm Ord}$ is represented as a disjoint union of ${\rm ds}(P)$
$s$-definable subfamilies $P_{1},\ldots,P_{{\rm ds}(P)}$ with ${\rm
RS}=\alpha$ and ${\rm ds}=1$, Proposition 5.6 immediately implies:

Corollary 5.7. For any property $P\subseteq\mathcal{T}_{\Sigma}$ with ${\rm
RS}(P)=\alpha\in{\rm Ord}$ and any sentence $\varphi\in{\rm Sent}(\Sigma)$
either $\varphi$ is $P$-generic or $\neg\varphi$ is $P$-generic, or, for ${\rm
ds}(P)>1$ with non-$P$-generic $\varphi$ and $\neg\varphi$, $\varphi$ is
represented as a disjunction of $k$ $(P,\alpha,1)$-rich sentences and
$\neg\varphi$ is represented as a disjunction of $m$ $(P,\alpha,1)$-rich
sentences such that $k+m={\rm ds}(P)$, $k>0$, $m>0$.

Remark 5.8. By Proposition 5.6 for any property
$P\subseteq\mathcal{T}_{\Sigma}$ with ${\rm RS}(P)=\alpha\in{\rm Ord}$ and
${\rm ds}(P)=1$ there is unique ultrafilter $U_{P}$ consisting of $P$-generic
sentences. By Proposition 5.5 this ultrafilter is principal if and only if $P$
is finite, i.e., in such a case it is a singleton. Anyway $U_{P}$ produces a
theory $T\in\mathcal{T}_{\Sigma}$ consisting of $P$-generic sentences only.
This theory $T$ is called $P$-generic.

If $P$ is infinite then $T$ belongs to the $E$-closure ${\rm Cl}_{E}(P)$ [2,
6] of $P$ as unique element of $\alpha$-th Cantor–Bendixson derivative of
${\rm Cl}_{E}(P)$, i.e., an element of ${\rm Cl}_{E}(P)$ having
Cantor–Bendixson rank ${\rm CB}({\rm Cl}_{E}(P))=\alpha$ [2].

If ${\rm ds}(P)>1$ we can divide $P$ into ${\rm ds}(P)$ $s$-definable parts
$P_{i}$ with ${\rm RS}(P_{i})={\rm RS}(P)$ and ${\rm ds}(P_{i})=1$, each of
which has unique $P_{i}$-generic theory $T_{i}$. The set
$\\{T_{1},\ldots,T_{{\rm ds}(P)}\\}$ is called the set of $P$-generic
theories.

Thus we have the following:

Proposition 5.9. Each $e$-totally transcendental property $P$ has finitely
many, exactly ${\rm ds}(P)\geq 1$, $P$-generic theories. These theories have
Cantor–Bendixson rank ${\rm CB}({\rm Cl}_{E}(P))=\alpha={\rm RS}(P)$.

Now we extend the results above on generic sentences and theories for
properties $P$ with ${\rm RS}(P)=\infty$.

Proposition 5.10. If for a property $P\subseteq\mathcal{T}_{\Sigma}$, ${\rm
RS}(P)=\infty$, then for any sentence $\varphi\in{\rm Sent}(\Sigma)$ either
${\rm RS}_{P}(\varphi)=\infty$ or ${\rm RS}_{P}(\neg\varphi)=\infty$.

Proof. Assume that for a sentence $\varphi\in{\rm Sent}(\Sigma)$ we have ${\rm
RS}_{P}(\varphi)<\infty$ and ${\rm RS}_{P}(\neg\varphi)<\infty$. We can
suppose that $\varphi$ and $\neg\varphi$ are both $P$-consistent. Then
$\varphi$ is $(P,\alpha,m)$-rich and $(P,\beta,n)$-co-rich for some
$\alpha,\beta\in{\rm Ord}$, $m,n\in\omega$. Applying Theorem 3.4 we obtain
${\rm RS}(P)\in{\rm Ord}$ contradicting ${\rm RS}(P)=\infty$. $\Box$

Proposition 5.10 immediately implies:

Corollary 5.11. For any property $P\subseteq\mathcal{T}_{\Sigma}$ with ${\rm
RS}(P)=\infty$ and any sentence $\varphi\in{\rm Sent}(\Sigma)$ either
$\varphi$ is $P$-generic or $\neg\varphi$ is $P$-generic.

Remark 5.12. By Proposition 5.10 for any property
$P\subseteq\mathcal{T}_{\Sigma}$ with ${\rm RS}(P)=\infty$ there is
ultrafilters $U_{P}$ consisting of $P$-generic sentences. Since the condition
${\rm RS}(P)=\infty$ implies the existence of $2$-tree of $\Sigma$-sentences
there are at least continuum many these ultrafilters, and by Theorem 1.2 there
are exactly continuum many ones for the language $\Sigma$ with
$|\Sigma|\leq\omega$. Each $U_{P}$ produces a theory
$T\in\mathcal{T}_{\Sigma}$ consisting of $P$-generic sentences only. This
theory $T$ is called $P$-generic.

The $P$-generic theories form the perfect kernel with respect to
Cantor–Bendixson derivatives of ${\rm Cl}_{E}(P)$, i.e., the set of element of
${\rm Cl}_{E}(P)$ having Cantor–Bendixson rank ${\rm CB}({\rm
Cl}_{E}(P))=\infty$.

Applying Theorem 3.6 and summarizing Remarks 5.8 and 5.12 we obtain the
following:

Theorem 5.13. $(1)$ For any nonempty property $P\subseteq\mathcal{T}_{\Sigma}$
there are ${\rm ds}(P)$ $P$-generic theories if $P$ is totally transcendental,
and at least continuum many if $P$ is not totally transcendental. In the
latter case either all theories in $P$ are $P$-generic if ${\rm Sp}_{\rm
Rd}(P)=\\{\infty\\}$, or $P$ has at least $\beta\cdot\omega+n$ non-$P$-generic
theories if ${\rm Sp}_{\rm Rd}(P)=O[(\beta,n)]\cup\\{\infty\\}$.

$(2)$ The ${\rm CB}$-rank of each $P$-generic theory equals ${\rm RS}(P)$.

Definition [5]. For a property $P\subseteq\mathcal{T}_{\Sigma}$ a sentence
$\varphi\in{\rm Sent}(\Sigma)$ is called $P$-complete if $\varphi$ isolates a
unique theory $T$ in $P$, i.e., $P_{\varphi}$ is a singleton. In such a case
the theory $T\in P_{\varphi}$ is called $P$-finitely axiomatizable (by the
sentence $\varphi$).

Since $P$-finitely axiomatizable theories are isolated points we have the
following:

Proposition 5.14. For any nonempty property $P\subseteq\mathcal{T}_{\Sigma}$ a
$P$-finitely axiomatizable theory $T$ is $P$-generic iff $P$ is finite.

##  6 Conclusion

We studied links between formulas and properties, considered and described
characteristics for properties with respect to satisfying formulas, their
ranks and degrees. Possibilities of spectra for ranks, degrees, and
cardinalities of definable properties are shown. Generic formulas and theories
are introduced and characterized.

Possibilities for ranks and degrees for formulas and theories with respect to
given properties are described. Their top values form generic sentences and
theories which are also described and characterized.

There are many natural model-theoretic and other properties that can be
studied and described in this context. In this case, the relations $\rhd_{\rm
pt}$ and $\rhd_{\rm tt}$ are preserved under natural closures. At the same
time the operator of $E$-closure does not preserve a series of natural model-
theoretic properties. For instance, there are families of strongly minimal
whose accumulation points have the strict order properties. It implies that
families of $\omega$-stabile, superstable and stable theories can be not
$E$-closed in these classes. Natural questions arise on characteristics and
characterizations of families in these classes $\mathcal{T}$ whose
$E$-closures are contained in $\mathcal{T}$.

## References

  * [1] Ershov Yu. L. Mathematical logic / Yu. L. Ershov, E. A. Palyutin. — Moscow : FIZMATLIT, 2011\. — 356 p. [in Russian]
  * [2] Sudoplatov S. V. Ranks for families of theories and their spectra / S. V. Sudoplatov // arXiv:1901.08464v1 [math.LO]. — 2019. — 17 p.
  * [3] Markhabatov N. D. Ranks for families of all theories of given languages / N. D. Markhabatov, S. V. Sudoplatov // Eurasian Mathematical Journal. — 2021 (to appear). arXiv:1901.09903v1 [math.LO]. — 2019. — 9 p.
  * [4] Markhabatov N. D. Definable subfamilies of theories, related calculi and ranks / N. D. Markhabatov, S. V. Sudoplatov // Siberian Electronic Mathematical Reports. — 2020. — Vol. 17. — P. 700–714.
  * [5] Sudoplatov S. V. Approximations of theories / S. V. Sudoplatov // Siberian Electronic Mathematical Reports. — 2020. — Vol. 17. — P. 715–725.
  * [6] Sudoplatov S. V. Closures and generating sets related to combinations of structures / S. V. Sudoplatov // Bulletin of Irkutsk State University. Series “Mathematics”. — 2016. — Vol. 16. — P. 131–144.
  * [7] Sudoplatov S. V. Hierarchy of families of theories and their rank characteristics / S. V. Sudoplatov // Bulletin of Irkutsk State University. Series “Mathematics”. — 2020. — Vol. 33. — P. 80–95.
  * [8] Poizat B. Groupes Stables / B. Poizat. — Villeurbanne : Nur Al-Mantiq Wal-Ma’rifah, 1987.
  * [9] Truss J. K. Generic Automorphisms of Homogeneous Structures / J. K. Truss // Proceedings of the London Mathematical Society. — 1992. — Vol. 65, No. 3. — P. 121–141.
  * [10] Tent K. A Course in Model Theory (Lecture Notes in Logic. No. 40) / K. Tent, M. Ziegler. — Cambridge : Cambridge University Press, 2012. — 258 p.

Sergey Vladimirovich Sudoplatov, Doctor of Sciences (Physics and Mathematics),
docent; Leading researcher, Sobolev Institute of Mathematics, Novosibirsk;
Head of Chair, Novosibirsk State Technical University; Professor, Novosibirsk
State University.
e-mail<EMAIL_ADDRESS>