Where is the well-pointedeness assumption of the Elementary theory of the category of sets (Lawvere's category-theoretic axiomatization of set theory) used in everyday math?
Specifically, if you have a topos with natural numbers object (assume choice if you want to), what familiar theorems don't hold? I've heard that showing the Dedekind reals are the same as the Cauchy reals is one. Where in the arguments is well-pointedness used? It seems hard to find examples of this.