Proof of Predicative Formulas for σ-Algebras, π-Systems and λ-Systems, and the Sets of Them Containing a Given Family

The formulas. Let $a$, $b$, $c$, $m$, $n$, $u$, $v$, $w$, $w'$ and $z$ be distinct set variables other than $G$, $X$ and $\mathbb{N}$. Let $\mu(a,\mathbb{N},G)$ be the formula of Predicativity under Renaming of Set Variables by Variables, of Formulas with Ordered Pairs, and of Formulas for Relations, Functions, Maps, Injectivity, Surjectivity and Bijectivity (vii) with $a$, $\mathbb{N}$ and $G$ for $F$, $A$ and $B$, and let $(n,b)\in a$ and $(m,b)\in a$ be the formulas of Predicativity under Renaming of Set Variables by Variables, of Formulas with Ordered Pairs, and of Formulas for Relations, Functions, Maps, Injectivity, Surjectivity and Bijectivity (iii). Let

$$\begin{aligned} \beta(G)&:\ \forall w\,\bigl(w\in G\Rightarrow\forall v\,(v\in w\Rightarrow v\in X)\bigr),\\ \kappa(G)&:\ \forall w\,\Bigl(w\in G\Rightarrow\exists z\,\bigl(\forall v\,(v\in z\Leftrightarrow(v\in X\wedge v\notin w))\wedge z\in G\bigr)\Bigr),\\ \iota(G)&:\ \forall w\,\forall w'\,\Bigl((w\in G\wedge w'\in G)\Rightarrow\exists z\,\bigl(\forall v\,(v\in z\Leftrightarrow(v\in w\wedge v\in w'))\wedge z\in G\bigr)\Bigr),\\ \upsilon(a,G)&:\ \exists z\,\Bigl(\forall u\,\bigl(u\in z\Leftrightarrow\exists n\,(n\in\mathbb{N}\wedge\exists b\,((n,b)\in a\wedge u\in b))\bigr)\wedge z\in G\Bigr),\\ \delta(a)&:\ \forall m\,\forall n\,\Bigl((m\in\mathbb{N}\wedge n\in\mathbb{N}\wedge\neg\,m=n)\Rightarrow\forall b\,\forall c\,\bigl(((m,b)\in a\wedge(n,c)\in a)\Rightarrow\neg\exists u\,(u\in b\wedge u\in c)\bigr)\Bigr), \end{aligned}$$

and let $\Sigma(G)$ be $\beta(G)\wedge X\in G\wedge\kappa(G)\wedge\forall a\,(\mu(a,\mathbb{N},G)\Rightarrow\upsilon(a,G))$, $\Pi(G)$ be $\beta(G)\wedge\iota(G)$, and $\Lambda(G)$ be $\beta(G)\wedge X\in G\wedge\kappa(G)\wedge\forall a\,((\mu(a,\mathbb{N},G)\wedge\delta(a))\Rightarrow\upsilon(a,G))$. The free variables of $\Sigma$ and $\Lambda$ are among $G$, $X$ and $\mathbb{N}$, and those of $\Pi$ among $G$ and $X$.

Predicativity. $\mu(a,\mathbb{N},G)$ is predicative by Predicativity under Renaming of Set Variables by Variables, of Formulas with Ordered Pairs, and of Formulas for Relations, Functions, Maps, Injectivity, Surjectivity and Bijectivity (vii), and the pair memberships are predicative by Predicativity under Renaming of Set Variables by Variables, of Formulas with Ordered Pairs, and of Formulas for Relations, Functions, Maps, Injectivity, Surjectivity and Bijectivity (iii). The formulas $\Sigma$, $\Pi$ and $\Lambda$ are built from these and from atomic formulas by connectives and by quantifiers over the set variables listed above only; so no quantifier in them is applied to a class variable, and they are predicative by Predicativity of Expressions with Abbreviations, and Symbol-Free Forms of the Basic Defined Set Symbols (i).

Meaning of the parts. Fix a set $G$. By Predicativity of Expressions with Abbreviations, and Symbol-Free Forms of the Basic Defined Set Symbols (vii), with $X$ and $\mathcal{P}(X)$ for its $u$ and $z$, a set $w$ lies in $\mathcal{P}(X)$ if and only if $\forall v\,(v\in w\Rightarrow v\in X)$; so $\beta(G)$ holds if and only if every element of $G$ lies in $\mathcal{P}(X)$, that is, by Subclasses and Subsets (i), if and only if $G\subseteq\mathcal{P}(X)$. For sets $w$ and $w'$, $X\setminus w$ and $w\cap w'$ are sets by Sets and Maps: Ordinary Notation (iv), and by Predicativity of Expressions with Abbreviations, and Symbol-Free Forms of the Basic Defined Set Symbols (ix) a set $z$ satisfies $\forall v\,(v\in z\Leftrightarrow(v\in X\wedge v\notin w))$ if and only if $z=X\setminus w$, and $\forall v\,(v\in z\Leftrightarrow(v\in w\wedge v\in w'))$ if and only if $z=w\cap w'$. Hence $\kappa(G)$ holds if and only if $X\setminus A\in G$ for every $A\in G$, and $\iota(G)$ if and only if $A\cap B\in G$ for all $A,B\in G$.

The sequences in $G$ are ranged over by the set variable $a$. By Predicativity under Renaming of Set Variables by Variables, of Formulas with Ordered Pairs, and of Formulas for Relations, Functions, Maps, Injectivity, Surjectivity and Bijectivity (vii), a set $a$ satisfies $\mu(a,\mathbb{N},G)$ if and only if $a$ is a map $a:\mathbb{N}\to G$, that is, by Sequences (i), a sequence $(a_{n})$ in $G$; and every sequence in $G$ is a set, by Images of Sets under Functions Are Sets, and a Function with a Set Domain Is a Set (ii), its domain $\mathbb{N}$ being a set. So $\forall a\,(\mu(a,\mathbb{N},G)\Rightarrow\dots)$ states a property of every sequence in $G$. Let $a:\mathbb{N}\to G$ be such a sequence. By Predicativity under Renaming of Set Variables by Variables, of Formulas with Ordered Pairs, and of Formulas for Relations, Functions, Maps, Injectivity, Surjectivity and Bijectivity (iii) and Values, Restriction and Equality of a Map, and the Map $u\mapsto t(u)$ as a Defined Set Symbol (i), for $n\in\mathbb{N}$ and every set $b$, $(n,b)\in a$ holds if and only if $b=a_{n}$. Hence the formula $\exists n\,(n\in\mathbb{N}\wedge\exists b\,((n,b)\in a\wedge u\in b))$ holds if and only if $u\in a_{n}$ for some $n\in\mathbb{N}$, that is, by Indexed Families of Sets and Their Union, Intersection and Product (ii) and Class Abstraction: the Class of All Sets Satisfying a Predicative Formula (i), $a$ being a family indexed by $\mathbb{N}$, if and only if $u\in\bigcup_{n\in\mathbb{N}}a_{n}$. This union is a set by The Union and Product of a Family of Sets Indexed by a Set, and the Intersection of a Family with an Inhabited Index Class, Are Sets (i), so a set $z$ satisfies the first conjunct of $\upsilon(a,G)$ if and only if $z=\bigcup_{n\in\mathbb{N}}a_{n}$, by Axiom of Extensionality for Classes; thus $\upsilon(a,G)$ holds if and only if $\bigcup_{n\in\mathbb{N}}a_{n}\in G$. In the same way, $\delta(a)$ holds if and only if, for all $m,n\in\mathbb{N}$ with $m\neq n$, no set lies in both $a_{m}$ and $a_{n}$, that is, by The Boolean Operations on Classes, Disjointness, and the Universal Class (ii), $a_{m}$ and $a_{n}$ are disjoint.

Proof of the first sentences of (i), (ii) and (iii). By the meanings above, for every set $G$: $\Sigma(G)$ holds if and only if $G\subseteq\mathcal{P}(X)$, $X\in G$, $X\setminus A\in G$ for every $A\in G$, and $\bigcup_{n\in\mathbb{N}}A_{n}\in G$ for every sequence $(A_{n})$ in $G$, that is, by σ-Algebra and Measurable Space (i), if and only if $G$ is a $\sigma$-algebra on $X$; $\Pi(G)$ holds if and only if $G\subseteq\mathcal{P}(X)$ and $A\cap B\in G$ for all $A,B\in G$, that is, by π-System (i), if and only if $G$ is a $\pi$-system on $X$; and $\Lambda(G)$ holds if and only if $G\subseteq\mathcal{P}(X)$, $X\in G$, $X\setminus A\in G$ for every $A\in G$, and $\bigcup_{n\in\mathbb{N}}A_{n}\in G$ for every sequence $(A_{n})$ in $G$ whose terms are pairwise disjoint, that is, by λ-System (i), if and only if $G$ is a $\lambda$-system on $X$. So $\Sigma$, $\Pi$ and $\Lambda$ express these conditions, as in Conditions Expressed by a Predicative Formula, and Guards (ii), with $G$ for $x_{1}$.

Proof of the second sentences. Let $\Theta$ be one of $\Sigma$, $\Pi$ and $\Lambda$, and let ``$G$ is a $\Theta$-system'' mean the condition it expresses. By The Union Set and the Power Set of a Set (ii), applied twice, $\mathcal{P}(\mathcal{P}(X))$ is a set; take it and $\mathcal{C}$ as further parameters. The formula $\Theta(G)\wedge\forall w\,(w\in\mathcal{C}\Rightarrow w\in G)$ is predicative by Predicativity of Expressions with Abbreviations, and Symbol-Free Forms of the Basic Defined Set Symbols (i), so the class

$$\mathcal{S}=\{G\in\mathcal{P}(\mathcal{P}(X)):\Theta(G)\wedge\forall w\,(w\in\mathcal{C}\Rightarrow w\in G)\}$$

of Class Abstraction: the Class of All Sets Satisfying a Predicative Formula (ii) is a set by Sets and Maps: Ordinary Notation (iii), and $\mathcal{S}\subseteq\mathcal{P}(\mathcal{P}(X))$ by Class Abstraction: the Class of All Sets Satisfying a Predicative Formula (i) and Subclasses and Subsets (i). Let $G$ be a set. By Class Abstraction: the Class of All Sets Satisfying a Predicative Formula (i) and Subclasses and Subsets (i), $G\in\mathcal{S}$ if and only if $G\in\mathcal{P}(\mathcal{P}(X))$, $G$ is a $\Theta$-system and $\mathcal{C}\subseteq G$. A $\Theta$-system $G$ satisfies $G\subseteq\mathcal{P}(X)$, by $\beta(G)$, so $G\in\mathcal{P}(\mathcal{P}(X))$ by The Union Set and the Power Set of a Set (ii); hence $G\in\mathcal{S}$ if and only if $G$ is a $\Theta$-system with $\mathcal{C}\subseteq G$. If $\mathcal{S}'$ is a set with the same property, then $\mathcal{S}'$ and $\mathcal{S}$ have the same elements, and $\mathcal{S}'=\mathcal{S}$ by Axiom of Extensionality for Classes.