Choice Principles for Finitely Supported Structures

Abstract

The validity and the non-validity of choice principles in various models of Zermelo-Fraenkel set theory and of Zermelo-Fraenkel set theory with atoms (including the symmetric models and the permutation models) was investigated in the last century. Actually, choice principles are proved to be independent of the set of axioms for Zermelo-Fraenkel and Zermelo-Fraenkel with atoms, respectively. Since the theory of finitely supported algebraic structures is connected to the related permutation models, it became an open problem to study the consistency of choice principles with this new framework. We prove that many choice principles are inconsistent within finitely supported atomic structures, and so some paradoxes (such as Banach-Tarski paradoxical decomposition of a sphere) are eliminated in the new atomic context. However, no non-atomic result is weakened, meaning that the independence of the choice principles in the non-atomic frameworks of Zermelo- Fraenkel sets is not affected. Proving the inconsistency of choice principles in FSM (i.e. the non-validity of their atomic FSM formulations) is not an easy task because the Zermelo-Fraenkel results between choice principles are not necessarily preserved into this new framework, unless we reprove them with respect to the finite support requirement.