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Suppose in an arithmetic universe we have two predicates and ψ for natural numbers, satisfying a base case (0)→ψ(0) and an induction step that, for generic n, the hypothesis (n)→ψ(n) allows one to deduce (n+1)→ψ(n+1). Then it is already true in that arithmetic universe that (n)((n)→ψ(n)). This is substantially harder than in a topos, where cartesian closedness allows one to form an exponential (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 open and the closed, and it is isomorphic to the free Boolean algebra over the distributive lattice of subobjects of 1 in the arithmetic universe.

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本文标签： inductionsignaturearithmeticunivPrinciple

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