Models and Satissability

The article includes schemes of deening by structural induction, and deenitions and theorems related to: the set of variables which have free occurrences in a ZF-formula, the set of all valuations of variables in a model, the set of all valuations which satisfy a ZF-formula in a model, the satissability of a ZF-formula in a model by a valuation, the validity of a ZF-formula in a model, the axioms of ZF-language, the model of the ZF set theory. and 2] provide the notation and terminology for this paper. In this article we present several logical schemes. The scheme ZFsch ex deals with a binary functor F yielding a set, a binary functor G yielding a set, a unary functor H yielding a set, a binary functor I yielding a set, a binary functor J yielding a set, and a ZF-formula A; and states that: There exist a, A such that (i) for all x, y holds h hx=y; (iii) for all H, a such that h hH; ai i 2 A holds if H is an equality, then a = F(Var 1 (H); Var 2 (H)) and if H is a membership, then a = G(Var 1 (H); Var 2 (H)) and if H is negative, then there exists b such that a = H(b) and h h Arg(H); bi i 2 A and if H is conjunctive, then there exist b, c such that a = I(b; c) and h h LeftArg(H); bi i 2 A and h h RightArg(H); ci i 2 A and if H is universal, then there exist b, x such that x = Bound(H) and a = J (x; b) and h h Scope(H); bi i 2 A for all values of the parameters. The scheme ZFsch uniq deals with a binary functor F yielding a set, a binary functor G yielding a set, a unary functor H yielding a set, a binary functor I yielding a set, a binary functor J yielding a set, a ZF-formula A; a set B; and a set C; and states that: B = C provided the parameters satisfy the following conditions: There exists A such that (i) for all x, y holds h hx=y; (iii) for all H, a such that h hH; ai i 2 A holds if H is an equality, then a =