A constructive proof for a theorem on contractive mappings in power sets

Abstract

Recently I proved the following theorem: To every positive integer m there exists a positive integer h such that the following holds: If S is a set of h elements and f a mapping of the power set B of S into B such that f( T)⊆ T for all TϵB, then there exists a strictly increasing sequence T 1⊂…⊂ T m of subsets of S such that one of the following three possibilities holds: (a) All sets f( T i ), i=1,…, m, are equal. (b) For all i=1,…, m we have f( T i )= T i (c) T i = f( T i+1 ) for all i=1,…, m−1. The proof given in [2] was non-constructive. In this paper now we give a constructive proof. By the way, this also yields a solution of a problem of Rado [3, p. 106].