Hyperfinite logics and non-standard extensions of Boolean algebras

Abstract

Infinitary propositional logics, i.e., propositional logics with infinite conjunction and disjunction, have some deficiencies, e.g., these logics fail to be compact or complete, in general. Such kind of infinitary propositional logics are introduced, called hyperfinite logics, which are defined in a non-standard framework of non-standard analysis and have hyperfinite conjunctions and disjunctions. They have more nice properties than infinitary logics have, in general. Furthermore, non-standard extensions of Boolean algebras are investigated. These algebras can be regarded as algebraizations of hyperfinite logics, they have several unusual properties. These Boolean algebras are closed under the hyperfinite sums and products, they are representable by hyperfinitely closed Boolean set algebras and they are omega-compact. It is proved that standard Boolean algebras are representable by Boolean set algebras with a hyperfinite unit.