Abstract
AbstractWe present a large collection of properties of the set of atoms, of its (finite or cofinite) powerset and of its (finite) higher-order powerset in the world of finitely supported algebraic structures. Firstly, we prove that atomic sets have many specific FSM properties (that are not translated from ZF). We can structure these specific properties into five main groups, presenting the relationship between atomic and non-atomic sets, specific finiteness properties of atomic sets, specific (order) properties of cardinalities in FSM, surprising fixed point properties of self-mappings on the (finite) powerset of atoms, and the inconsistency of various choice principles for specific atomic sets. Other properties of atoms are obtained by translating classical (non-atomic) ZF results into FSM, by replacing ‘non-atomic object’ with ‘atomic finitely supported object’. Furthermore, we also proved that the powerset of atoms satisfies some choice principles such as prime ideal theorem and ultrafilter theorem, although these principles are generally not valid in FSM. Ramsey theorem for the set of atoms and Kurepa antichain principle for the powerset of atoms also hold, and admit constructive proofs. The properties presented in this chapter are also valid in the frameworks of nominal sets and in the framework of Fraenkel-Mostowski sets. This chapter is based on [11, 15].
Published Version
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call