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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.

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    $\begingroup$ what is "ETCS"? $\endgroup$
    – T..
    Commented Nov 15, 2010 at 20:43
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    $\begingroup$ Elementary theory of the category of sets. Lawvere's category-theoretic axiomatization of set theory. Well-pointed topos with choice and natural numbers object. $\endgroup$ Commented Nov 15, 2010 at 21:10
  • $\begingroup$ Please try to make the body of your posts self-contained, not relying on the subject line for content. I've edited the question, also taking into account your response above. $\endgroup$ Commented Feb 14, 2011 at 3:21

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A topos with an NNO has intuitionistic internal logic in general (if you assume Choice then Dionescu's theorem tells you that the internal logic is classical), so any proof that relies on proof by contradiction will not work (not to say the results won't, but the proof needs to be fixed, or your concepts altered).

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    $\begingroup$ ok, but what if you assume Choice? $\endgroup$ Commented Sep 16, 2011 at 20:49
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    $\begingroup$ A topos with NNO and Choice but not well-pointed would require arguments with generalised elements. I'm not aware off the top of my head of anything that would break ordinary mathematics. $\endgroup$ Commented Sep 23, 2011 at 2:49
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    $\begingroup$ Note that from the point of view of the internal language, any topos looks like it were well-pointed. Well-pointedness is an external property. Mike Shulman's beautiful stack semantics paper might be of some interest. $\endgroup$ Commented Mar 10, 2014 at 10:34
  • $\begingroup$ I should say 'Internal Choice' in my previous comment, as that is the only thing that makes sense without well-pointedness. $\endgroup$ Commented Mar 11, 2014 at 2:00
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Ah, I forgot about the Yoneda lemma.

Also, I think it's impossible to show that a strictly increasing function into a partial order is monic, because the normal proof uses the fact that monic on global elements implies monic. In fact, this lemma is probably used a lot.

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