Mostowski's Fundamental Operations - Part II
Grzegorz Bancerek, Andrzej Kondracki · 1991
In the chapter II.4 of his book [17] A.Mostowski introduces what he calls fundamental operations: A1(a, b) = {{〈0, x〉, 〈1, y〉} : x ∈ y ∧ x ∈ a ∧ y ∈ a}, A2(a, b) = {a, b}, A3(a, b) = ⋃ a, A4(a, b) = {{〈x, y〉} : x ∈ a ∧ y ∈ b}, A5(a, b) = {x ∪ y : x ∈ a ∧ y ∈ b}, A6(a, b) = {x y : x ∈ a ∧ y ∈ b}, A7(a, b) = {x ◦ y : x ∈ a ∧ y ∈ b}. He proves that if a non-void class is closed under these operations then it is predicatively closed. Then he formulates sufficient criteria for a class to be a model of ZF set theory (theorem 4.12). The article includes the translation of this part of Mostowski’s book. The fundamental operations are defined (to be precise, not these operations, but the notions of closure of a class with respect to them). Some properties of classes closed under these operations are proved. At last it is proved that if a non-void class X is closed under the operations A1−A7 then DH(a) ∈ X for every a in X and every H being formula of ZF language (DH(a) consists of all finite sequences with terms belonging to a which satisfy H in a).