Generators of \(\sigma\)- and \(\pi\)-classes (Q1274075)

This short note presents two simple theorems concerning properties of \(\sigma\)- and \(\pi\)-classes. Namely, it is proved that given a \(\pi\)-class (\(\sigma\)-class) \(X\) and a set-formula \(\varphi({\mathbf x}, {\mathbf y})\) then the property \[ (\forall {\mathbf x}\in X^n)(\exists {\mathbf y}\in X^k)(\varphi({\mathbf x}, {\mathbf y})) \] is valid for \(X\) iff there exists a decreasing (increasing) chain of generators \(X_i\), \(i\in FN\) such that \[ (\forall i>1)(\forall {\mathbf x}\in X^n_i)(\exists {\mathbf y}\in X^k_j)(\varphi({\mathbf x},{\mathbf y})) \] where \(j=i-1\) for the decreasing and \(j=i+1\) for the increasing chain. The paper is clearly, though briefly written and two examples are also provided. The English translation uses incorrectly the term ``open class'' instead of ``revealed class''.