Fixed Point Algebras

Abstract

Algebra rarely has anything deep to say about logic. Most applications of algebra to logic are fairly shallow, the exceptions being applications of representation theory, where one is really using the non-algebraic properties of the representations. Nonetheless, algebra does offer another perspective and a convenient language or framework in which to work. The purpose of the present chapter is the presentation of such an algebraic framework-language in which to place the results already discussed, in which goals like those of the last chapter can be expressly delineated, and in which we can state and prove a theorem that delimits, not very convincingly, the boundaries for such successful generalisations of Solovay’s Completeness Theorems as were obtained in the last chapter and explains, again not convincingly, the necessarily close relation between these results and Solovay’s results for PRL.