Reifying dynamical algebra: Maximal ideals in countable rings, constructively

The existence of a maximal ideal in a general nontrivial commutative ring is tied together with the axiom of choice. Following Berardi, Valentini and thus Krivine but using the relative interpretation of negation (that is, as “implies 0 = 1”) we show, in constructive set theory with minimal logic, how for countable rings one can do without any kind of choice and without the usual decidability assumption that the ring is strongly discrete (membership in finitely generated ideals is decidable). By a functional recursive definition we obtain a maximal ideal in the sense that the quotient ring is a residue field (every noninvertible element is zero), and with strong discreteness even a geometric field (every element is either invertible or else zero). Krull’s lemma for the related notion of prime ideal follows by passing to rings of fractions. By employing a construction variant of set-theoretic forcing due to Joyal and Tierney, we expand our treatment to arbitrary rings and establish a connection with dynamical algebra: We recover the dynamical approach to maximal ideals as a parametrized version of the celebrated double negation translation. This connection allows us to give formal a priori criteria elucidating the scope of the dynamical method. Along the way we do a case study for proofs in algebra with minimal logic, and generalize the construction to arbitrary inconsistency predicates. A partial Agda formalization is available at an accompanying repository.