Is there an elementary topos which

  1. Is well-pointed
  2. Satisfies the internal axiom of choice
  3. Does not have a natural numbers object; and
  4. Is not the category of finite sets?

Yes: take any non-trivial ultrapower of $\mathbf{FinSet}$. Your properties 1–3 are all expressible in the first-order language of categories, so they are preserved by ultrapowers.


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