Epistemic predicates in the arithmetical context

Abstract

Abstract In this paper, we investigate epistemic predicates in extensions of arithmetic. We use as our case study Kurt Gödel’s 1951 thesis that either the power of the human mind surpasses that of any finite machine or there are absolutely unsolvable problems. Because Gödel also claimed that his disjunction was a mathematically established fact, we must ask the following: what sort of syntactical object should formalize human reason? In this paper, we lay the foundations for a predicate treatment of this epistemic feature. We begin with a very general examination of the Gödel sentence in the arithmetical context. We then discuss two systems of modal predicates over arithmetic. The first, called coreflective arithmetic or ${\textsf{CoPA}}$, extends ${\textsf{PA}}$ with a coreflective modal predicate but does not contain a consistency statement. The second, called doxastic arithmetic or ${\textsf{DA}}$, has as its characteristic feature the consistency statement but does not contain coreflection or its instance, the ${\textsf{4}}$ axiom. We examine the logical properties of, motivations for and criticisms of both systems. We close with a brief comparison of the systems in the context of Gödel’s disjunction.