Topological representations of Post algebras of order ω⁺ and open theories based on ω⁺-valued Post logic

Abstract

Post algebras of order ω⁺ as a semantic foundation for ω⁺-valued predicate calculi were examined in [5]. In this paper Post spaces of order ω⁺ being a modification of Post spaces of order n≥2 (cf. Traczyk [8], Dwinger [1], Rasiowa [6]) are introduced and Post fields of order ω⁺ are defined. A representation theorem for Post algebras of order ω⁺ as Post fields of sets is proved. Moreover necessary and sufficient conditions for the existence of representations preserving a given set of infinite joins and infinite meets are established and applied to Lindenbaum-Tarski algebras of elementary theories based on ω⁺-valued predicate calculi in order to obtain a topological characterization of open theories.