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 closeds, and it is isomorphic to the free Boolean algebra over the distributive lattice of subobjects of 1 in the arithmetic universe.