An induction principle for consequence in arithmetic universes

Suppose in an arithmetic unverse we have two predicates φ and ψ for natural numbers, satisfying a base case φ(0) → ψ(0) and an induction step that, for generic n, the hypothesis φ(n) → ψ(n) allows one to deduce φ(n+ 1) → ψ(n+ 1). Then it is already true in that arithmetic universe that (∀n)(φ(n) → ψ(n)). This is substantially harder than in a topos, where cartesian closedness allows one to form an exponential φ(n) → ψ(n). The principle is applied to the question of locatedness of Dedekind sections. The development analyses in some detail a notion of “subspace” of an arithmetic universe, including open or closed subspaces and a boolean algebra generated by them. There is a lattice of subspaces generated by the opens and the closed, and it is isomorphic to the free Boolean algebra over the distributive lattice of subobjects of 1 in the arithmetic universe.