Makkai's lost proof of projectivity of N in the free topos

Abstract

We give a categorical proof of the projectivity of $N$ in the free topos -- in proof-theoretic terms, the rule of countable choice for intuitionistic higher-order logic -- based on the unpublished proof of Michael Makkai (c.1980). The presentation aims to be self-contained and accessible to any reader acquainted with elementary toposes and their logic.

45 pages. v2: Regularised notation for choice rules; other minor style and exposition edits; updated MSC classes; theorem numbering unchanged