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According to a history of topos theory by McLarty, Blass (1989) showed that the existence of an object classifier over a given topos implies that the topos has a natural number object.

What is an object classifier, and how does it imply the existence of a natural number object?

ref:

Blass, A. (1989) Classifying topoi and the axiom of infinity. Algebra Universalis 26, no. 3, 341-345

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object classifier: a classifying topos for the one-sorted geometric theory having no non-logical symbols and no axioms, the theory whose models in any category are just the objects of that category.

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    $\begingroup$ This would be better as an edit to the question, rather than an answer. $\endgroup$ Aug 27 '13 at 3:49

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