The subset relation and $2$-stratified sentences in set theory and class theory

Recent work by Hamkins and Kikuchi [HK16, HK17] initiates the study of the structure $\langle M,\subseteq^{\mathcal{M}}\rangle$ where $\subseteq^{\mathcal{M}}$ is the definable subset relation in a model of set theory or class theory, $\mathcal{M}$, with domain $M$. [HK16] shows that in models of set theory the definable subset relation is an atomic unbounded relatively complemented distributive lattice, which is a complete and decidable theory. In [HK17], it is shown that if $\mathcal{M}=\langle M,\in^{\mathcal{M}}\rangle$ is a model of set theory that satisfies some very mild set-theoretic conditions (conditions that are satisfied in all models of $\mathrm{ZFC}$ and even in all nonstandard models of finite set theory), then the structure $\langle M,\subseteq^{\mathcal{M}}\rangle$ is $\omega$-saturated. Thus, if $\mathcal{M}$ is countable, then the structure $\langle M,\subseteq^{\mathcal{M}}\rangle$ is unique up to isomorphism. In addition, [HK17] also shows that, in any model of von Neumann-Bernays-Gödel class theory plus the class formulation of the axiom choice the definable subset relation is an infinite atomic Boolean algebra that is $\omega$-saturated. Thereby showing that, as is the case with set theory, the theory of the subset relation in class theory is a complete decidable theory whose structure, for countable models of class theory, is unique. These results are presented as evidence that the subset relation alone is insufficient to serve as a foundation for mathematics. Indeed, the fact that the theory of the definable subset relation in set theory and class theory is a complete theory shows that, after some point, varying the axioms of set theory or class theory in ways that we know fundamentally change the mathematics that can be done (for example, adding or removing the Axiom of Infinity or the Axiom of Choice, or varying the amount of comprehension or collection that is available) does not alter the first order theory of the subset ordering of the universe. Similarly, it also shows that whatever fragment of set theory that is expressible in the language that only consists of the subset relation must be too weak to be able to express any assertion that is not decided by the weak fragment of set theory or class theory that fixes the complete theory of the definable subset ordering. This paper makes these observations precise by identifying the minimal subsystems of set theory and class theory that fix the complete theories of the subset ordering identified in [HK16, HK17]. We also identify the precise fragment of set theory and class theory that is expressible using only the subset relation, thereby showing that weak subsystems of set theory and class theory decide every sentence in this fragment.

The weak system Adjunctive Set Theory with Boolean operations ($\mathrm{BAS}$) is axiomatised by extensionality, emptyset, axioms asserting that for all sets $x$ and $y$, the sets $x\cap y$, $x\cup y$, $x-y$ and $\{x\}$ exist, and an axiom asserting that there is no universal set. In section 3, it is shown that if $\mathcal{M}=\langle M,\in^{\mathcal{M}}\rangle$ is a model of $\mathrm{BAS}$ then $\langle M,\subseteq^{\mathcal{M}}\rangle$ is an atomic unbounded relatively complemented distributive lattice with the same number of elements as atoms. Conversely, we show that every atomic unbounded relatively complemented distributive lattice with the same number of atoms as elements can be realised as the definable subset relation of a model of $\mathrm{BAS}$. It follows that if $\mathcal{M}=\langle M,\in^{\mathcal{M}}\rangle$ is a model of Tarski’s Adjunctive Set Theory, then $\langle M,\subseteq^{\mathcal{M}}\rangle$ is an atomic unbounded relatively complemented distributive lattice if and only if $\mathcal{M}$ satisfies $\mathrm{BAS}$.

In section 4, we identify the $2$-stratified sentences as the exact fragment of the language of set theory that is expressible in the structure $\langle M,\subseteq^{\mathcal{M}}\rangle$, where $\mathcal{M}$ is a model of set theory with domain $M$ and $\subseteq^{\mathcal{M}}$ is the definable subset relation. We present a translation, $\tau$, of $2$-stratified formulae of set theory into formulae in language of orderings such that if $\mathcal{M}$ is a model of set theory with domain $M$ and $\phi$ is a $2$-stratified sentence in the language of set theory, then $\mathcal{M}\models\phi$ if and only if $\langle M,\subseteq^{\mathcal{M}}\rangle\models\phi^{\tau}$. Since the theory of the definable subset relation in the theory $\mathrm{BAS}$ is complete, this shows that any extension of $\mathrm{BAS}$ decides every $2$-stratified sentence in the language of set theory. We show that this result is optimal by expressing a version of the Axiom of Choice using a $3$-stratified sentence of set theory.

Section 5 turns to investigating the definable subset relation in models of class theory. The system Adjunctive Class Theory with Boolean operations ($\mathrm{BAC}$) consists of extensionality for classes, axioms asserting that the sets are closed downwards under the subset relation and satisfy the theory $\mathrm{BAS}$, and axioms asserting that for all classes $X$ and $Y$, the classes $X\cap Y$, $X\cup Y$ and the complement of $X$ exist. We show that if $\mathcal{M}$ is a model of $\mathrm{BAC}$ with domain $M$ and sets $\mathcal{S}^{\mathcal{M}}$, then $\langle M,\subseteq^{\mathcal{M}}\rangle$ is an infinite atomic Boolean algebra and $\mathcal{S}^{\mathcal{M}}$ is a proper ideal of $\langle M,\subseteq^{\mathcal{M}}\rangle$ that is the same size as, and contains all of, the atoms of $\langle M,\subseteq^{\mathcal{M}}\rangle$. Conversely, every infinite atomic Boolean algebra can be realised as the subset ordering of a model $\mathcal{M}$ of $\mathrm{BAC}$. While the translation, $\tau$, introduced in section 4 shows that $\mathrm{BAC}$ decides every $2$-stratified sentence in the language of set theory (the language that only includes $\in$), we note that there is a $2$-stratified sentence of the language of class theory including a unary predicate distinguishing the sets that is independent of $\mathrm{BAC}$. Using a complete decidable theory studied in [DM], which in this paper we call $\mathrm{IABA}_{\mathrm{Ideal}}$, we identify an extension $\mathrm{BAC}^{+}$ of $\mathrm{BAC}$ such that for models $\mathcal{M}$ of this theory, the structure $\langle M,\mathcal{S}^{\mathcal{M}},\subseteq^{\mathcal{M}}\rangle$ satisfies $\mathrm{IABA}_{\mathrm{Ideal}}$, where $M$ is the underlying set of $\mathcal{M}$, $\subseteq^{\mathcal{M}}$ is the definable subset relation and $\mathcal{S}^{\mathcal{M}}$ distinguishes the sets of $\mathcal{M}$. This shows that the theory $\mathrm{BAC}^{+}$, a subsystem of von Neumann-Bernays-Gödel class theory, decides every $2$-stratified sentence in the language of class theory including a unary predicate that distinguishes the sets.

The results in this paper are influenced by the work of Gris̆hin [Gri] on subsystems of Quine’s ‘New Foundations’ Set Theory ($\mathrm{NF}$). Gris̆hin shows that the Simple Theory of Types restricted to only two types ($\mathrm{TST}_{2}$) is a complete and decidable theory with essentially the same expressive power as the theory of infinite atomic Boolean algebras. The results of section 4 of this paper are an analogue of this result of Gris̆hin’s for set theories that refute the existence of a universal set. Gris̆hin also shows that the fragment of Quine’s ‘New Foundations’ Set Theory axiomatised by extensionality and all $2$-stratifiable instances of the comprehension scheme ($\mathrm{NF}_{2}$) is finitely axiomatised by extensionality and axioms asserting that there exists a universal set ($V$) and for all sets $x$ and $y$, $x\cap y$, $x\cup y$, $\{x\}$ and $V-x$. Moreover, models of $\mathrm{NF}_{2}$ can be obtained from any infinite atomic Boolean algebra with the same number of atoms as elements. The set theory $\mathrm{BAS}$ introduced in this paper is the theory $\mathrm{NF}_{2}$ with the axiom asserting the existence of a universal set replaced with its negation.

Let $\mathcal{L}_{\mathrm{po}}$ be the language of partial orders– first-order logic endowed with a binary (ordering) relation $\sqsubseteq$. We will make reference to the following $\mathcal{L}_{\mathrm{po}}$ theories:

• •

  The theory of partial orders ($\mathrm{PO}$) is the $\mathcal{L}_{\mathrm{po}}$-theory with axioms asserting that $\sqsubseteq$ is reflexive, antisymmetric and transitive.

• •

  The theory of lattices ($\mathrm{Lat}$) is the $\mathcal{L}_{\mathrm{po}}$-theory extending $\mathrm{PO}$ with axioms asserting that there exists a least element ($0$), and that every pair of elements $x$ and $y$ have both a least upper bound ($x\dot{\lor}y$) and greatest lower bound ($x\dot{\land}y$)¹¹1Dots will be used to distinguish the algebraic lattice operations from logical connectives and set-theoretic operations..

We use $\mathbf{Atm}(x)$ to abbreviate the $\mathcal{L}_{\mathrm{po}}$-formula that asserts, in the theory $\mathrm{Lat}$, that $x$ is an atom. I.e. $(x\neq 0)\land\forall y(y\sqsubseteq x\Rightarrow y=0\lor y=x)$.

• •

  The theory of set-theoretic mereology ($\mathrm{Mer}$) is the $\mathcal{L}_{\mathrm{po}}$-theory extending $\mathrm{Lat}$ with the axioms:

  • (Atomic) for all $x$, there exists $y$ such that $\mathbf{Atm}(y)$ and $y\sqsubseteq x$;

  • (Unbounded) for all $x$, there exists $y$ such that $x\sqsubseteq y$ and $x\neq y$;

  • (Relatively Complemented) for all $x$ and $y$, there exists an element $x\begin{picture}(12.0,6.0)(-4.0,-2.0)\put(-1.0,0.0){ \makebox(0.0,0.0)[]{$-$} } \put(-1.0,2.0){ \makebox(0.0,0.0)[]{$\cdot$} } \end{picture}y$ that satisfies the equations

    $$y\dot{\land}(x\begin{picture}(12.0,6.0)(-4.0,-2.0)\put(-1.0,0.0){ \makebox(0.0,0.0)[]{$-$} } \put(-1.0,2.0){ \makebox(0.0,0.0)[]{$\cdot$} } \end{picture}y)=0\textrm{ and }x=(x\dot{\land}y)\dot{\lor}(x\begin{picture}(12.0,6.0)(-4.0,-2.0)\put(-1.0,0.0){ \makebox(0.0,0.0)[]{$-$} } \put(-1.0,2.0){ \makebox(0.0,0.0)[]{$\cdot$} } \end{picture}y);$$

  • (Distributive) for all $x$, $y$ and $z$,

    $$x\dot{\land}(y\dot{\lor}z)=(x\dot{\land}y)\dot{\lor}(x\dot{\land}z)\textrm{ and }x\dot{\lor}(y\dot{\land}z)=(x\dot{\lor}y)\dot{\land}(x\dot{\lor}z).$$

  I.e. $\mathrm{Mer}$ is the theory of atomic unbounded relatively complemented distributive lattices.

• •

  The theory of infinite atomic Boolean algebras ($\mathrm{IABA}$) is the $\mathcal{L}_{\mathrm{po}}$-theory extending $\mathrm{Lat}$ with the axioms:

  • (Atomic) for all $x$, there exists $y$ such that $\mathbf{Atm}(y)$ and $y\sqsubseteq x$;

  • (Top) there exists a greatest element ($1$);

  • (Complemented) for all $x$, there exists an element $\begin{picture}(12.0,6.0)(-4.0,-2.0)\put(-1.0,0.0){ \makebox(0.0,0.0)[]{$-$} } \put(-1.0,2.0){ \makebox(0.0,0.0)[]{$\cdot$} } \end{picture}x$ that satisfies the equations

    $$x\dot{\lor}\begin{picture}(12.0,6.0)(-4.0,-2.0)\put(-1.0,0.0){ \makebox(0.0,0.0)[]{$-$} } \put(-1.0,2.0){ \makebox(0.0,0.0)[]{$\cdot$} } \end{picture}x=1\textrm{ and }x\dot{\land}\begin{picture}(12.0,6.0)(-4.0,-2.0)\put(-1.0,0.0){ \makebox(0.0,0.0)[]{$-$} } \put(-1.0,2.0){ \makebox(0.0,0.0)[]{$\cdot$} } \end{picture}x=0;$$

  • (Distributive) for all $x$, $y$ and $z$,

    $$x\dot{\land}(y\dot{\lor}z)=(x\dot{\land}y)\dot{\lor}(x\dot{\land}z)\textrm{ and }x\dot{\lor}(y\dot{\land}z)=(x\dot{\lor}y)\dot{\land}(x\dot{\lor}z);$$

  • (Infinity Scheme) for all $n\in\mathbb{N}$ with $n>0$, the axiom that asserts that there are at least $n$ distinct atoms.

Let $\mathcal{L}$ denote the language of set theory– first-order logic endowed with a binary (membership) relation $\in$. We will have cause to consider the following subsystems of $\mathrm{ZF}$:

• •

  Adjunctive Set Theory ($\mathrm{AS}$) is the $\mathcal{L}$-theory with axioms:

  • ($\mathrm{Emp}$) $\exists x\forall y(y\notin x)$;

  • ($\mathrm{Adj}$) $\forall x\forall y\exists z\forall u(u\in z\iff(u\in x\lor u=y))$.

  The axiom $\mathrm{Adj}$ guarantees that for all sets $x$ and $y$, the set $x\cup\{y\}$ exists.

• •

  Adjunctive Set Theory with Boolean operations ($\mathrm{BAS}$) is the $\mathcal{L}$-theory extending $\mathrm{AS}$ with the axioms:

  • ($\mathrm{Ext}$) $\forall x\forall y(x=y\iff\forall z(z\in x\iff z\in y))$;

  • ($\mathrm{Union}$) $\forall x\forall y\exists z\forall u(u\in z\iff u\in x\lor u\in y)$;

  • ($\mathrm{Intersection}$) $\forall x\forall y\exists z\forall u(u\in z\iff u\in x\land u\in y)$;

  • ($\mathrm{RelComp}$) $\forall x\forall y\exists z\forall u(u\in z\iff u\in x\land u\notin y)$;

  • ($\mathrm{UB}$) $\forall x\exists y(y\notin x)$.

Adjunctive Set Theory was first introduced by Tarski, and is known to interpret Robinson’s Arithmetic. It follows that both $\mathrm{AS}$ and $\mathrm{BAS}$ are essentially undecidable theories. We refer the reader to [Che] for a survey of essentially undecidable theories. As usual, $x\subseteq y$ if $\forall z(z\in x\Rightarrow z\in y)$. If $\phi$ is a formula, then $\mathbf{Var}(\phi)$ will be used to denote the set of variables, both free and bound, that appear in $\phi$.

The notion of stratification was first introduced by Quine in [Qui] where it is used to define the set theory $\mathrm{NF}$. We refer the reader to [For] for a survey of $\mathrm{NF}$ and related systems of set theory that avoid the set-theoretic paradoxes by restricting comprehension using the notion of stratified formula. Note that the formula $x\subseteq y$ and all of the axioms of $\mathrm{BAS}$ are $2$-stratified $\mathcal{L}$-formulae.

Let $\mathcal{L}_{\mathrm{cl}}$ be the single-sorted language of class theory– first-order logic endowed with a binary membership relation ($\in$) and a unary predicate ($\mathcal{S}$) that distinguishes sets from classes. Therefore an $\mathcal{L}_{\mathrm{cl}}$-structure is a triple $\mathcal{M}=\langle M,\mathcal{S}^{\mathcal{M}},\in^{\mathcal{M}}\rangle$ with $\mathcal{S}^{\mathcal{M}}\subseteq M$. In order make the presentation of $\mathcal{L}_{\mathrm{cl}}$-formulae more readable, we will pretend that $\mathcal{L}_{\mathrm{cl}}$ is a two-sorted language with sorts sets (referred to using lower case Roman letters $x,y,z,u,v,\ldots$) that are elements of the domain that satisfy $\mathcal{S}$ and classes (referred to using upper case Roman letters $X,Y,Z,U,V,\ldots$) that are any element of the domain. Therefore, in the axiomatisations presented below, $\exists x(\cdots)$ is an abbreviation for $\exists x(\mathcal{S}(x)\land\cdots)$, $\forall x(\cdots)$ is an abbreviation for $\forall x(\mathcal{S}(x)\Rightarrow\cdots)$ and $\exists x(x=X)$ is an abbreviation for $\mathcal{S}(X)$, etc. We use $\mathrm{NBG}$ to denote the $\mathcal{L}_{\mathrm{cl}}$-axiomatisation of the version of von Neumann-Gödel-Bernays Class Theory without the axiom of choice that is presented in [Men, Chapter 4]. We will study the following subsystems of $\mathrm{NBG}$:

• •

  Adjunctive Set Theory with Classes ($\mathrm{CAS}$ ²²2Note that this theory is stronger than the adjunctive class theory (ac) described in [Vis, p6-7]) is the $\mathcal{L}_{\mathrm{cl}}$-theory with axioms:

  • ($\mathrm{Mem}$) $\forall X\forall Y(X\in Y\Rightarrow\exists x(x=X)$;

  • ($\mathrm{Subset}$) $\forall x\forall X(\forall y(y\in X\Rightarrow y\in x)\Rightarrow\exists z(z=X))$;

  • ($\mathrm{Emp}$) $\exists x\forall y(y\notin x)$;

  • ($\mathrm{Adj}$) $\forall x\forall y\exists z\forall u(u\in z\iff(u\in x\lor u=y))$.

• •

  Adjunctive Class Theory with Boolean operations ($\mathrm{BAC}$) is the $\mathcal{L}_{\mathrm{cl}}$-theory extending $\mathrm{CAS}$ with the axioms:

  • ($\mathrm{CExt}$) $\forall X\forall Y(X=Y\iff\forall x(x\in X\iff x\in Y))$;

  • ($\mathrm{Union}$) $\forall x\forall y\exists z\forall u(u\in z\iff u\in x\lor u\in y)$;

  • ($\mathrm{UB}$) $\forall x\exists y(y\notin x)$;

  • ($\mathrm{CUnion}$) $\forall X\forall Y\exists Z\forall x(x\in Z\iff x\in X\lor x\in Y)$;

  • ($\mathrm{CIntersection}$) $\forall X\forall Y\exists Z\forall x(x\in Z\iff x\in X\land x\in Y)$;

  • ($\mathrm{CComp}$) $\forall X\exists Y\forall x(x\in Y\iff x\notin X)$.

Hamkins and Kikuchi [HK16, Theorem 9 and Corollary 10] observe that the subset relation in any model of $\mathrm{ZF}$ is an atomic unbounded relatively complemented distributive lattice ordering of the universe and, extending [Erš], show that this theory is complete and decidable.

[HK16] note that this result also holds for models of certain subsystems of $\mathrm{ZF}$ such as finite set theory and Kripke-Platek Set Theory. In the next section we will see that Theorem 2.1 holds when $\mathcal{M}$ satisfies $\mathrm{BAS}$.

In [HK17, Section 5], Hamkins and Kikuchi extend their analysis of the theory and structure of the subset ordering to models of class theory. They show that if $\mathcal{M}$ is a model of Von Neumann-Bernays-Gödel set theory with a class version of the axiom of choice, then the structure $\langle M,\subseteq^{\mathcal{M}}\rangle$ is an $\omega$-saturated model of $\mathrm{IABA}$. Even if $\mathcal{M}$ does not satisfy the axiom of choice, $\langle M,\subseteq^{\mathcal{M}}\rangle$ satisfies $\mathrm{IABA}$. It is a well-known result due to Tarski [Tar] that the theory $\mathrm{IABA}$ complete and decidable.

This section establishes that $\mathrm{ZF}$ can be replaced by $\mathrm{BAS}$ in Theorem 2.1 and, conversely, that every model of $\mathrm{Mer}$ with the same number of atoms as elements can be realised as the subset relation of a model of $\mathrm{BAS}$.

• Proof

  Work inside $\mathcal{M}$. Now, $\langle M,\subseteq^{\mathcal{M}}\rangle$ is clearly reflexive and transitive, and $\mathrm{Ext}$ ensures that $\langle M,\subseteq^{\mathcal{M}}\rangle$ is antisymmetric. Now, the axiom $\mathrm{Emp}$ ensures that $\emptyset$ exists and this set is a $\subseteq^{\mathcal{M}}$-least element. The axioms $\mathrm{Union}$ and $\mathrm{Intersection}$ ensure that for all sets $x$ and $y$, $x\cap y$ and $x\cup y$ exist. In the order $\langle M,\subseteq^{\mathcal{M}}\rangle$, $x\cap y$ is the greatest lower bound of $x$ and $y$, and $x\cup y$ is the least upper bound of $x$ and $y$. Since for all $x$, $y$ and $z$,

  $$x\cap(y\cup z)=(x\cap y)\cup(x\cap z)\textrm{ and }x\cup(y\cap z)=(x\cup y)\cap(x\cup z),$$

  $\langle M,\subseteq^{\mathcal{M}}\rangle$ is a distributive lattice. The existence of $\emptyset$ and the axiom $\mathrm{Adj}$ ensures that for all $x$, $\{x\}$ exists. Note that for all $x$, $\{x\}$ is an atom of $\langle M,\subseteq^{\mathcal{M}}\rangle$. Now, if $y\neq\emptyset$, then there exists $x\in y$ and $\{x\}$ is an atom below $y$ in $\langle M,\subseteq^{\mathcal{M}}\rangle$. Therefore, $\langle M,\subseteq^{\mathcal{M}}\rangle$ is atomic. The axiom $\mathrm{RelComp}$ ensures that for all sets $x$ and $y$, $x-y$ exists. Since for all $x$ and $y$,

  $$y\cap(x-y)=\emptyset\textrm{ and }x=(x\cap y)\cup(x-y),$$

  $\langle M,\subseteq^{\mathcal{M}}\rangle$ is relatively complemented. Finally, for all $x$, the axiom $\mathrm{UB}$ ensures that there exists $y$ such that $y\notin x$. The axiom $\mathrm{Adj}$ ensures that $x\cup\{y\}$ exists. Now, $x\subseteq x\cup\{y\}\neq x$ and so $\langle M,\subseteq^{\mathcal{M}}\rangle$ is unbounded. To see that $\langle M,\subseteq^{\mathcal{M}}\rangle$ has the same number of atoms as elements, observe that $f=\{\langle x,\{x\}\rangle\mid x\in M\}$ is a bijection between the elements of $\langle M,\subseteq^{\mathcal{M}}\rangle$ and the atoms of $\langle M,\subseteq^{\mathcal{M}}\rangle$. □

We now turn to showing that if $\mathcal{M}=\langle M,\sqsubseteq^{\mathcal{M}}\rangle$ is a model of $\mathrm{Mer}$ with the same number of atoms as elements, then there exists a membership relation on $M$ that makes $M$ a model of $\mathrm{BAS}$ whose subset relation is exactly $\sqsubseteq^{\mathcal{M}}$. The intuition behind this result is that a model of set theory can be obtained from a subset relation and a map that sends a set to its own singleton (see [HK16, Theorem 13]). The set theory $\mathrm{BAS}$ is so weak that this singleton map can be chosen to be any bijection between elements and atoms of a structure satisfying $\mathrm{Mer}$.

• Proof

  Let $x,y\in M$. Work inside $\mathcal{M}$. Note that if $x\sqsubseteq y$, then for all atoms $z$, if $z\sqsubseteq x$, then $z\sqsubseteq y$. Conversely, suppose that for all atoms $z$, if $z\sqsubseteq x$, then $z\sqsubseteq y$. Now, $x\begin{picture}(12.0,6.0)(-4.0,-2.0)\put(-1.0,0.0){ \makebox(0.0,0.0)[]{$-$} } \put(-1.0,2.0){ \makebox(0.0,0.0)[]{$\cdot$} } \end{picture}y=0$, because otherwise there would be an atom that sits below both $x\begin{picture}(12.0,6.0)(-4.0,-2.0)\put(-1.0,0.0){ \makebox(0.0,0.0)[]{$-$} } \put(-1.0,2.0){ \makebox(0.0,0.0)[]{$\cdot$} } \end{picture}y$ and $y$ contradicting the fact that $y\dot{\land}(x\begin{picture}(12.0,6.0)(-4.0,-2.0)\put(-1.0,0.0){ \makebox(0.0,0.0)[]{$-$} } \put(-1.0,2.0){ \makebox(0.0,0.0)[]{$\cdot$} } \end{picture}y)=0$. Therefore, $x=x\dot{\land}y$ and so $x\sqsubseteq y$. □

• Proof

  It follows immediately from Lemma 3.2 that $\langle M,\in^{*}\rangle\models\mathrm{Ext}$ and $\subseteq^{\langle M,\in^{*}\rangle}=\sqsubseteq^{\mathcal{M}}$. Note that the $\sqsubseteq^{\mathcal{M}}$-least element $0$ is such that $\langle M,\in^{*}\rangle\models\forall y(y\notin 0)$. Therefore $\langle M,\in^{*}\rangle\models\mathrm{Emp}$. Now, let $x,y\in M$. Working in $\mathcal{M}$, let $z=x\dot{\lor}f(y)$. For all $u\in M$,

  $$f(u)\sqsubseteq^{\mathcal{M}}z\textrm{ if and only if }f(u)\sqsubseteq^{\mathcal{M}}x\textrm{ or }f(u)=f(y)$$

  $$\textrm{if and only if }\langle M,\in^{*}\rangle\models(u\in x\lor u=y).$$

  Therefore $\langle M,\in^{*}\rangle\models\mathrm{Adj}$. To show that Union holds in $\langle M,\in^{*}\rangle$, let $x,y\in M$. Working in $\mathcal{M}$, let $z=x\dot{\lor}y$. Let $u\in M$. It is clear that if $f(u)\sqsubseteq^{\mathcal{M}}x$ or $f(u)\sqsubseteq^{\mathcal{M}}y$, then $f(u)\sqsubseteq^{\mathcal{M}}z$. Conversely, suppose that $f(u)\not\sqsubseteq^{\mathcal{M}}x$ and $f(u)\not\sqsubseteq^{\mathcal{M}}y$. Then, working inside $\mathcal{M}$, $x\dot{\land}f(u)=y\dot{\land}f(u)=0$ and

  $$f(u)\dot{\land}z=f(u)\dot{\land}(x\dot{\lor}y)=(f(u)\dot{\land}x)\dot{\lor}(f(u)\dot{\land}y)=0.$$

  Therefore $f(u)\sqsubseteq^{\mathcal{M}}z$. We have shown that

  $$\langle M,\in^{*}\rangle\models\forall u(u\in z\iff u\in x\lor u\in y).$$

  So, $\langle M,\in^{*}\rangle\models\mathrm{Union}$. To see that $\langle M,\in^{*}\rangle$ satisfies $\mathrm{Intersection}$, let $x,y\in M$. Working inside $\mathcal{M}$, let $z=x\dot{\land}y$. Let $u\in M$. If $f(u)\sqsubseteq^{\mathcal{M}}z$, then $f(u)\sqsubseteq^{\mathcal{M}}x$ and $f(u)\sqsubseteq^{\mathcal{M}}y$. Conversely, suppose $f(u)\sqsubseteq^{\mathcal{M}}x$ and $f(u)\sqsubseteq^{\mathcal{M}}y$. Therefore, inside $\mathcal{M}$,

  $$f(u)\dot{\lor}z=f(u)\dot{\lor}(x\dot{\land}y)=(f(u)\dot{\lor}x)\dot{\land}(f(u)\dot{\lor}y)=x\dot{\land}y=z.$$

  Therefore $f(u)\sqsubseteq^{\mathcal{M}}z$. This shows that

  $$\langle M,\in^{*}\rangle\models\forall u(u\in z\iff u\in x\land u\in y).$$

  So, $\langle M,\in^{*}\rangle\models\mathrm{Intersection}$. To show that $\langle M,\in^{*}\rangle$ satisfies $\mathrm{RelComp}$, let $x,y\in M$. Inside $\mathcal{M}$, let $z=x\begin{picture}(12.0,6.0)(-4.0,-2.0)\put(-1.0,0.0){ \makebox(0.0,0.0)[]{$-$} } \put(-1.0,2.0){ \makebox(0.0,0.0)[]{$\cdot$} } \end{picture}y$. Let $u\in M$. Note that, inside $\mathcal{M}$, $x\begin{picture}(12.0,6.0)(-4.0,-2.0)\put(-1.0,0.0){ \makebox(0.0,0.0)[]{$-$} } \put(-1.0,2.0){ \makebox(0.0,0.0)[]{$\cdot$} } \end{picture}y\sqsubseteq x$ and, since $y\dot{\land}(x\begin{picture}(12.0,6.0)(-4.0,-2.0)\put(-1.0,0.0){ \makebox(0.0,0.0)[]{$-$} } \put(-1.0,2.0){ \makebox(0.0,0.0)[]{$\cdot$} } \end{picture}y)=0$, if $f(u)\sqsubseteq x\begin{picture}(12.0,6.0)(-4.0,-2.0)\put(-1.0,0.0){ \makebox(0.0,0.0)[]{$-$} } \put(-1.0,2.0){ \makebox(0.0,0.0)[]{$\cdot$} } \end{picture}y$, then $f(u)\not\sqsubseteq y$. Therefore, if $f(u)\sqsubseteq^{\mathcal{M}}z$, then $f(u)\sqsubseteq^{\mathcal{M}}x$ and $f(u)\not\sqsubseteq^{\mathcal{M}}y$. Conversely, suppose that $f(u)\sqsubseteq^{\mathcal{M}}x$ and $f(u)\not\sqsubseteq^{\mathcal{M}}y$. Therefore, inside $\mathcal{M}$,

  $$f(u)=f(u)\dot{\land}x=f(u)\dot{\land}((x\dot{\land}y)\dot{\lor}z)=(f(u)\dot{\land}(x\dot{\land}y))\dot{\lor}(f(u)\dot{\land}z)=0\dot{\lor}(f(u)\dot{\land}z)=f(u)\dot{\land}z.$$

  Therefore, $f(u)\sqsubseteq^{\mathcal{M}}z$. This shows that

  $$\langle M,\in^{*}\rangle\models\forall u(u\in z\iff u\in x\land u\notin y).$$

  So, $\langle M,\in^{*}\rangle\models\mathrm{RelComp}$. To see that $\langle M,\in^{*}\rangle$ satisfies $\mathrm{UB}$, let $x\in M$. Since $\mathcal{M}$ is unbounded, there exists $z\in M$ such that $x\sqsubseteq^{\mathcal{M}}z$ and $x\neq z$. Since $x\neq z$, by Lemma 3.2, there is an atom $u$ such $u\sqsubseteq^{\mathcal{M}}z$ and $u\not\sqsubseteq^{\mathcal{M}}x$. Now, letting $y=f^{-1}(u)$, we see that $\langle M,\in^{*}\rangle\models(y\notin x)$. Therefore $\langle M,\in^{*}\rangle\models\mathrm{UB}$. □

• Proof

  Let $A=\{x\in M\mid\mathcal{M}\models\mathbf{Atm}(x)\}$. Note that $A$ is infinite. Since $\mathcal{M}$ is countable, $|M|=|A|$ and the result now follows from Theorem 3.3. □

In contrast with Corollary 3.4, there are uncountable models of $\mathrm{Mer}$ with fewer atoms than elements and thus models of $\mathrm{Mer}$ that are not isomorphic to the subset relation of any model of $\mathrm{BAS}$.

Theroem 3.3 also shows that $\mathrm{BAS}$ is the minimal set theory whose subset relation is an atomic unbounded relatively complemented lattice order.

• Proof

  The fact that $(I)\Rightarrow(II)$ follows from 3.1. To see that $(II)\Rightarrow(I)$, assume that $\langle M,\subseteq^{\mathcal{M}}\rangle\models\mathrm{Mer}$. The fact that $\langle M,\subseteq^{\mathcal{M}}\rangle$ is antisymmetric immediately implies that $\mathcal{M}\models\mathrm{Ext}$. Inside $\mathcal{M}$, the axioms $\mathrm{Emp}$ and $\mathrm{Adj}$ ensure that for all sets $x$, $\{x\}$ exists. Consider $f=\{\langle x,\{x\}\rangle\mid x\in M\}$. The existence of singletons coupled with extensionality in $\mathcal{M}$ ensure that $f$ is a bijection between $M$ and the atoms of $\langle M,\subseteq^{\mathcal{M}}\rangle$. As in Theorem 3.3, define $\in^{*}\subseteq M\times M$ by: for all $x,y\in M$,

  $$x\in^{*}y\textrm{ if and only if }\mathcal{M}\models f(x)\subseteq^{\mathcal{M}}y.$$

  Therefore $\langle M,\in^{*}\rangle\models\mathrm{BAS}$. But, for all $x,y\in M$,

  $$x\in^{*}y\textrm{ if and only if }\{x\}\subseteq^{\mathcal{M}}y\textrm{ if and only if }x\in^{\mathcal{M}}y.$$

  Therefore $\langle M,\in^{\mathcal{M}}\rangle\models\mathrm{BAS}$. □

In this section we show every $2$-stratified $\mathcal{L}$-sentence, $\phi$, can be translated into an $\mathcal{L}_{\mathrm{po}}$-sentence, $\phi^{\tau}$, such that if $\mathcal{M}=\langle M,\in^{\mathcal{M}}\rangle$ is a model of $\mathrm{BAS}$, then $\phi$ holds in $\mathcal{M}$ if and only if $\phi^{\tau}$ holds in $\langle M,\subseteq^{\mathcal{M}}\rangle$. Since the axioms of $\mathrm{BAS}$ completely determine the theory of $\langle M,\subseteq^{\mathcal{M}}\rangle$, this shows that $\mathrm{BAS}$ decides every $2$-stratified sentence.

We begin with the observation that by replacing the order $\sqsubseteq$ by the set-theoretic $\subseteq$ order in an $\mathcal{L}_{\mathrm{po}}$-formula one obtains a $2$-stratified $\mathcal{L}$-formula.

• Proof

  The $\mathcal{L}$-formulae $x=y$ and $x\subseteq y$ admit $2$-stratifications that assigned the value $1$ to both the variables $x$ and $y$. The theorem now follows by a straightforward induction on the structural complexity of $\phi(x_{1},\ldots,x_{n})$. □

We now turn to defining a translation of $2$-stratified $\mathcal{L}$-formulae into $\mathcal{L}_{\mathrm{po}}$-formulae.

Note that $\tau$ treats variables that appear on the left-hand side of the membership relation differently from variables that appear on the right-hand side of the membership relation. This prevents this translation from being generalised to formulae which contain two atomic subformulae that contain the same variable appearing on different sides of the membership relation.

• Proof

  For all $a,b\in M$,

  $$\mathcal{M}\models a\in b\textrm{ if and only if }\langle M,\subseteq^{\mathcal{M}}\rangle\models\{a\}\subseteq b.$$

  The theorem then follows by a straightforward induction on the structural complexity of $\phi(x_{1},\ldots,x_{n},y_{1},\ldots,y_{m})$. □

Combined with the fact that the theory of the subset relation the set theory $\mathrm{BAS}$ is a complete theory, this result shows that $\mathrm{BAS}$ decides every $2$-stratified sentence.

• Proof

  Let $\phi$ be a $2$-stratified $\mathcal{L}$-sentence. Since $\mathrm{Mer}$ is complete 2.1,

  $$\mathrm{Mer}\vdash\phi^{\tau}\textrm{ or }\mathrm{Mer}\vdash\neg\phi^{\tau}.$$

  It follows from Theorems 3.1 and 4.2 that either $\phi$ holds in all models of $\mathrm{BAS}$ or $\neg\phi$ holds in all models of $\mathrm{BAS}$. □

Gogol [Gog78] shows that $\mathrm{ZFC}$ decides every sentence in prenex normal form that is prefixed by a block of universal quantifiers followed by a single existential quantifier. H. Friedman [Fri] extending the work of Gogol [Gog79] shows that a weak subsystem of $\mathrm{ZF}$ decides every sentence with only three quantifiers.

• Proof

  The algorithm that tests whether $\phi$ is a $2$-stratified $\mathcal{L}$-sentence and then simultaneously searches for a proof of $\phi$ or $\neg\phi$ from the axioms of $\mathrm{BAS}$ decides this set and will always halt. □

In particular, $\mathrm{ZF}$ decides every $2$-stratified sentence. We conclude this section by observing that this result cannot be extended to $3$-stratified sentences.

• Proof

  The following $\mathcal{L}$-sentence asserts that for every non-empty set of non-empty disjoint sets $X$, there exists a set $C$ such that for all $x\in X$, $x\cap C$ is a singleton:

  $$\forall X\left(\begin{array}[]{c}(\forall x\in X)\exists w(w\in x)\land(\forall x,y\in X)(x\neq y\Rightarrow\neg\exists w(w\in x\land w\in y))\Rightarrow\\ \exists C\left(\begin{array}[]{c}(\forall z\in C)(\exists x\in X)(z\in x)\land(\forall x\in X)(\exists z\in C)(z\in x)\\ \land(\forall z,w\in C)(\forall x\in X)(z\in x\land w\in x\Rightarrow z=w)\end{array}\right)\end{array}\right).$$

  This sentence is $3$-stratified and equivalent to the Axiom of Choice. □

This section extends the results of sections 3 and 4 to class theory. We begin by showing that $\mathrm{BAC}$ is the minimal subsystem of $\mathrm{NBG}$ guaranteeing that the definable subset relation is an infinite atomic Boolean algebra.

• Proof

  Work inside $\mathcal{M}$. Since $\mathcal{M}$ satisfies $\mathrm{Subset}$, $\mathcal{S}^{\mathcal{M}}$ is closed downwards by $\subseteq^{\mathcal{M}}$ in $\langle M,\subseteq^{\mathcal{M}}\rangle$. Since $\langle\mathcal{S}^{\mathcal{M}},\in^{\mathcal{M}}\rangle\models\mathrm{BAS}$, it follows immediately from Theorem 3.1 that $\langle\mathcal{S}^{\mathcal{M}},\subseteq^{\mathcal{M}}\rangle$ is a model of $\mathrm{Mer}$ with the same number of atoms as elements. Therefore, $\mathcal{S}^{\mathcal{M}}$ is a proper ideal of $\langle M,\subseteq^{\mathcal{M}}\rangle$ with the same number of atoms as elements. It is clear that the atoms of $\langle M,\subseteq^{\mathcal{M}}\rangle$ all belong to $\mathcal{S}^{\mathcal{M}}$. We are left to show that $\langle M,\subseteq^{\mathcal{M}}\rangle$ is an infinite atomic Boolean algebra. Note that $\langle M,\subseteq^{\mathcal{M}}\rangle$ is reflexive and transitive, and $\mathrm{CExt}$ ensures that $\langle M,\subseteq^{\mathcal{M}}\rangle$ is antisymmetric. The axiom $\mathrm{Emp}$ ensures that $\emptyset$ exists and this set is the least element of $\langle M,\subseteq^{\mathcal{M}}\rangle$. The axioms $\mathrm{Emp}$ and $\mathrm{CComp}$ ensure that there exists a class, $V$, that contains every set, and this class is the greatest element of $\langle M,\subseteq^{\mathcal{M}}\rangle$. In the order $\langle M,\subseteq^{\mathcal{M}}\rangle$, $X\cup Y$ is the least upper bound of $X$ and $Y$, and $X\cap Y$ is the greatest lower bound of $X$ and $Y$. Since for all $X$, $Y$ and $Z$,

  $$X\cap(Y\cup Z)=(X\cap Y)\cup(X\cap Z)\textrm{ and }X\cup(Y\cap Z)=(X\cup Y)\cap(X\cup Z),$$

  $\langle M,\subseteq^{\mathcal{M}}\rangle$ is a distributive lattice. If $X$ is a class and $x$ is a set with $x\in X$, then $\{x\}$ is an atom that sits below $X$ in $\langle M,\subseteq^{\mathcal{M}}\rangle$. Therefore, $\langle M,\subseteq^{\mathcal{M}}\rangle$ is atomic. Let $X$ be a class. The axiom $\mathrm{CComp}$ ensures that $V-X$ is a class and

  $$V=X\cup(V-X)\textrm{ and }\emptyset=X\cap(V-X).$$

  Therefore, $\langle M,\subseteq^{\mathcal{M}}\rangle$ has complements. The fact that $\mathcal{S}^{\mathcal{M}}$ is a proper ideal of $\langle M,\subseteq^{\mathcal{M}}\rangle$ shows that $\langle M,\subseteq^{\mathcal{M}}\rangle$ has infinitely many atoms. □