On natural number objects, finiteness and Kripke-Platek models in toposes

Abstract

Our aim is to show that if a topos has a natural number object, then this object N can be equipped with a binary structure which makes it a Kripke-Platek model. We first give some results concerning (Kuratowski-) finite parts of N (essentially that these are the complemented and upper-bounded parts of N, and that those which are inhabited have a less element). We also study some related properties, which are not true in any topos. Then we use those results to define the binary expansion of the natural numbers as an isomorphism E between N and the object of its finite parts. This gives rise to a binary relation on N ( mεn if and only if m∈E( n)) , which makes N a transitive Kripke-Platek model.