A categorical proof shows that the natural numbers object is projective in the free topos, corresponding to countable choice in intuitionistic higher-order logic.
Review of ‘From sets and types to topology and analysis—towards practicable foundations for constructive mathematics’, ed. Crosilla, Schuster
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Makkai's lost proof of projectivity of N in the free topos
A categorical proof shows that the natural numbers object is projective in the free topos, corresponding to countable choice in intuitionistic higher-order logic.