The Jacobson radical for an inconsistency predicate

Abstract

As a form of the Axiom of Choice about relatively simple structures (posets), Hausdorff’s Maximal Chain Principle appears to be little amenable to computational interpretation. This received view, however, requires revision: maximal chains are more reminiscent of maximal ideals than it seems at first glance. The latter live in richer algebraic structures (rings), and thus are readier to be put under computational scrutiny. Exploiting this, and of course the analogy between maximal chains and maximal ideals, the concept of Jacobson radical carries over from a ring to an arbitrary set with an abstract inconsistency predicate: that is, a distinguished monotone family of finite subsets. All this makes possible not only to generalise Hausdorff’s principle, but also to express it as a syntactical conservation theorem. The latter, which encompasses the desired computational core of Hausdorff’s principle, is obtained by a generalised inductive definition. The over-all setting is constructive set theory.

Maximal ideals in countable rings, constructively

“…The constructively rarer geometric fields-those kinds of fields for which every element is either invertible or zero-are required to ensure, for instance, that kernels of matrices are finite dimensional and that bilinear forms are diagonalizable 5. This notion of a maximal ideal, together with the corresponding one of a complete theory in propositional logic, has been generalized to the concept of a complete coalition[53,55] for an abstract inconsistency predicate.…”