Zermelo ’ s Theorem 1

The articles [4], [3], [5], [2], and [1] provide the notation and terminology for this paper. For simplicity we follow the rules: a, x, y will be arbitrary, B, D, N , X, Y will denote sets, R, S, T will denote relations, F will denote a function, and W will denote a relation. We now state several propositions: (1) x ∈ fieldR if and only if there exists y such that 〈x, y〉 ∈ R or 〈y, x〉 ∈ R. (2) R ∪ S is a relation. (3) If X 6= ∅ and Y 6= ∅ and W = [:X, Y :], then fieldW = X ∪ Y . (4) If y = R, then y is a relation. (5) For all a, T holds x ∈ T−Seg(a) if and only if x 6= a and 〈x, a〉 ∈ T . In the article we present several logical schemes. The scheme R Separation deals with a set A, and a unary predicate P, and states that: there exists B such that for every relation R holds R ∈ B if and only if R ∈ A and P[R] for all values of the parameters. The scheme S Separation deals with a set A, and a unary predicate P, and states that: there exists B such that for every set X holds X ∈ B if and only if X ∈ A and P[X] for all values of the parameters. The following four propositions are true: (6) For all x, y, W such that x ∈ fieldW and y ∈ fieldW and W is well ordering relation holds if x / ∈ W−Seg(y), then 〈y, x〉 ∈ W .