Locality of Predicative Formulas: Truth Depends Only on the Values of the Free Variables

In the setting of Sets and Maps: Ordinary Notation. This item belongs to the metatheory. Formulas are those of Formulas of the Language of Class Theory, Free Variables, Predicative Formulas and Abbreviations (ii), with $\vee$, $\Rightarrow$, $\Leftrightarrow$, $\forall$ and $\notin$ read as the strings they abbreviate there. An expression written with abbreviations as in Formulas of the Language of Class Theory, Free Variables, Predicative Formulas and Abbreviations (vi) stands for the formula obtained by eliminating them, and the clauses apply to that formula; in particular the conditions on the free set variables and class variables of $\varphi$ below are conditions on that formula.

Let $n\ge0$, $p\ge0$ and $m\ge0$ be metatheoretic numbers. Let $x_{1},\dots,x_{n}$ be distinct set variables, and let $y_{1},\dots,y_{p}$ be distinct set variables, each different from $x_{1},\dots,x_{n}$. Let $Z_{1},\dots,Z_{m}$ be distinct class variables, and let $C_{1},\dots,C_{m}$ be classes, assigned to $Z_{1},\dots,Z_{m}$ in every reading below and left unwritten. Let $\varphi$ be a predicative formula whose free set variables are among $x_{1},\dots,x_{n}$ and whose class variables are among $Z_{1},\dots,Z_{m}$. For sets $a_{1},\dots,a_{n}$, let $\varphi(a_{1},\dots,a_{n})$ be the statement obtained, as in Formulas of the Language of Class Theory, Free Variables, Predicative Formulas and Abbreviations (iv), by reading $\varphi$ as $\varphi(x_{1},\dots,x_{n},Z_{1},\dots,Z_{m})$ with $a_{i}$ assigned to $x_{i}$; and for further sets $b_{1},\dots,b_{p}$, let $\varphi(a_{1},\dots,a_{n},b_{1},\dots,b_{p})$ be the statement obtained in the same way by reading $\varphi$ as $\varphi(x_{1},\dots,x_{n},y_{1},\dots,y_{p},Z_{1},\dots,Z_{m})$ with $a_{i}$ assigned to $x_{i}$ and $b_{j}$ to $y_{j}$.

(i) Assigning sets to the set variables, with values agreeing on $x_{1},\dots,x_{n}$, gives $\varphi$ the same truth value: for all sets $a_{1},\dots,a_{n}$, $b_{1},\dots,b_{p}$ and $c_{1},\dots,c_{p}$, the statement $\varphi(a_{1},\dots,a_{n},b_{1},\dots,b_{p})$ holds if and only if $\varphi(a_{1},\dots,a_{n},c_{1},\dots,c_{p})$ holds.

(ii) For all sets $a_{1},\dots,a_{n}$ and $b_{1},\dots,b_{p}$, the statement $\varphi(a_{1},\dots,a_{n},b_{1},\dots,b_{p})$ holds if and only if $\varphi(a_{1},\dots,a_{n})$ holds.