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1. • CommentRowNumber1.
   • CommentAuthorAsgarJ
   • CommentTimeApr 4th 2013
   • (edited Apr 4th 2013)
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   Author: AsgarJ
   Format: TextLet C be an elementary topos with the following properties: (i) the subobjects of each object of C form a complete boolean algebra, (ii) C has a natural numbers object, (iii) C is generated by the subobjects of its terminal object. [the conjunction of (i) and (iii) imply that C satisfies the axiom of choice] Does every result in ordinary mathematics hold in C? If yes, what does this exactly mean? Does this mean that there exists an equivalent formulation of this result in C involving the language of C, and this result can be proved within C, or does this mean that one can "embed" the result from ordinary mathematics into C, and this result can be proved in C? I would be grateful if you could provide me with a reference in case the answer to the first question is yes. Thank you, Asgar

   Let C be an elementary topos with the following properties:
   
   (i) the subobjects of each object of C form a complete boolean algebra,
   (ii) C has a natural numbers object,
   (iii) C is generated by the subobjects of its terminal object.
   
   [the conjunction of (i) and (iii) imply that C satisfies the axiom of choice]
   
   Does every result in ordinary mathematics hold in C? If yes, what does this exactly mean? Does this mean that there exists an equivalent formulation of this result in C involving the language of C, and this result can be proved within C, or does this mean that one can "embed" the result from ordinary mathematics into C, and this result can be proved in C? I would be grateful if you could provide me with a reference in case the answer to the first question is yes.
   
   Thank you,
   Asgar
2. • CommentRowNumber2.
   • CommentAuthorTobyBartels
   • CommentTimeApr 4th 2013
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   Author: TobyBartels
   Format: MarkdownItexCan you briefly explain why (i)&(iii) imply AC? Assuming that, there is a way in which every result of ordinary mathematics holds in $C$. Although $C$ need not be well-pointed, it is (like every topos) internally well-pointed, and it satisfies the internal axiom of choice (since this is implied by the external axiom of choice), so internally it's a model of $ETCS$. And one definition of ‘ordinary mathematics’ is whatever can be formalised in $ETCS$. I don't know that there's a reference for this definition of ‘ordinary mathematics’, although there is literature where people tout $ETCS$ and say, using various terminology, that essentially all of mathematics can be formalised in it. Ultimately, there is no accepted definition of that term.

   Can you briefly explain why (i)&(iii) imply AC?

   Assuming that, there is a way in which every result of ordinary mathematics holds in $C$. Although $C$ need not be well-pointed, it is (like every topos) internally well-pointed, and it satisfies the internal axiom of choice (since this is implied by the external axiom of choice), so internally it’s a model of $ETCS$. And one definition of ‘ordinary mathematics’ is whatever can be formalised in $ETCS$.

   I don’t know that there’s a reference for this definition of ‘ordinary mathematics’, although there is literature where people tout $ETCS$ and say, using various terminology, that essentially all of mathematics can be formalised in it. Ultimately, there is no accepted definition of that term.

3. • CommentRowNumber3.
   • CommentAuthorAsgarJ
   • CommentTimeApr 4th 2013
   • (edited Apr 4th 2013)
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   Author: AsgarJ
   Format: TextThis is Proposition 8, p. 276, Lane and Moerdijk "Sheaves in Geometry and Logic". The topos C under consideration is not well-pointed. What does it mean do be a INTERNAL model of ETCS?

   This is Proposition 8, p. 276, Lane and Moerdijk "Sheaves in Geometry and Logic".
   
   The topos C under consideration is not well-pointed. What does it mean do be a INTERNAL model of ETCS?
4. • CommentRowNumber4.
   • CommentAuthorZhen Lin
   • CommentTimeApr 4th 2013
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   Author: Zhen Lin
   Format: MarkdownItexIn at least one sense the claim is false: for example, it is possible that the "real" $\mathbf{Set}$ satisfies the continuum hypothesis, but by Cohen forcing, one can build a (Grothendieck) topos satisfying your conditions (i) – (iii) in which the continuum hypothesis fails. Thus what is true in the sorts of toposes you are considering is not merely a function of what is true in the "real" set-theoretic universe. As for "ordinary mathematics", Friedman [[1971](http://dx.doi.org/10.1016/0003-4843(71)90018-0)] showed that Borel determinacy (a theorem of ZFC) is not provable in Zermelo set theory with choice. It is known that the category of sets in a model of Zermelo set theory with choice is a model of ETCS and hence is an elementary topos satisfying your conditions. Of course, you might protest that Borel determinacy is not "ordinary mathematics"; personally, I am inclined to agree: I don't even understand what it says! (That probably says more about my undergraduate education than anything important.) As for being internally well pointed: for all objects $X$, $Y$ in an elementary topos, the following formula $$(\forall x \in X^1 . f \circ x = g \circ x) \to f = g$$ where $f$ and $g$ are free variables of type $Y^X$, is true in the internal logic of that topos.

   In at least one sense the claim is false: for example, it is possible that the “real” $\mathbf{Set}$ satisfies the continuum hypothesis, but by Cohen forcing, one can build a (Grothendieck) topos satisfying your conditions (i) – (iii) in which the continuum hypothesis fails. Thus what is true in the sorts of toposes you are considering is not merely a function of what is true in the “real” set-theoretic universe.

   As for “ordinary mathematics”, Friedman [1971] showed that Borel determinacy (a theorem of ZFC) is not provable in Zermelo set theory with choice. It is known that the category of sets in a model of Zermelo set theory with choice is a model of ETCS and hence is an elementary topos satisfying your conditions. Of course, you might protest that Borel determinacy is not “ordinary mathematics”; personally, I am inclined to agree: I don’t even understand what it says! (That probably says more about my undergraduate education than anything important.)

   As for being internally well pointed: for all objects $X$, $Y$ in an elementary topos, the following formula

   $$(\forall x \in X^1 . f \circ x = g \circ x) \to f = g$$

   where $f$ and $g$ are free variables of type $Y^X$, is true in the internal logic of that topos.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeApr 4th 2013
   • (edited Apr 4th 2013)
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   Author: Urs
   Format: MarkdownItexHey Zhen Lin, plenty of good replies. (I also liked your recent comment on toposes [here](http://mathoverflow.net/questions/126466/are-there-non-categorical-notions-in-topos-theory/126486#126486) on MO a lot!) Once apon a time we had this idea that we would not be like MO and other sites here and just throw away good replies just like that, but that instead whenever a little gem of insight was obtained, we'd archive it on the nLab. Or even reply by writing something in an nLab article in the first place.The point being that this way the answer is contained better in a wider web of material and more likely to be found by and be useful to others later. In short, what I am trying to say is: feel invited to write some of the good stuff that you are posting into some nLab entry or other!

   Hey Zhen Lin,

   plenty of good replies. (I also liked your recent comment on toposes here on MO a lot!)

   Once apon a time we had this idea that we would not be like MO and other sites here and just throw away good replies just like that, but that instead whenever a little gem of insight was obtained, we’d archive it on the nLab. Or even reply by writing something in an nLab article in the first place.The point being that this way the answer is contained better in a wider web of material and more likely to be found by and be useful to others later.

   In short, what I am trying to say is: feel invited to write some of the good stuff that you are posting into some nLab entry or other!

6. • CommentRowNumber6.
   • CommentAuthorAsgarJ
   • CommentTimeApr 4th 2013
   • (edited Apr 4th 2013)
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   Author: AsgarJ
   Format: TextThank you for your comments!

   Thank you for your comments!
7. • CommentRowNumber7.
   • CommentAuthorDavidRoberts
   • CommentTimeApr 5th 2013
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   Author: DavidRoberts
   Format: MarkdownItex@Zhen >the “real” set-theoretic universe what is that?

   @Zhen

   the “real” set-theoretic universe

   what is that?

8. • CommentRowNumber8.
   • CommentAuthorZhen Lin
   • CommentTimeApr 5th 2013
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   Author: Zhen Lin
   Format: MarkdownItexWhatever it means in ordinary mathematical practice. It doesn't have to be ZFC, but it had better support what we do!

   Whatever it means in ordinary mathematical practice. It doesn’t have to be ZFC, but it had better support what we do!

9. • CommentRowNumber9.
   • CommentAuthorTobyBartels
   • CommentTimeApr 9th 2013
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   Author: TobyBartels
   Format: MarkdownItexSo the relevant question (as I see it) about Zhen Lin\'s ‘“real” set-theoretic universe’ is whether ordinary mathematics depends on the truth of CH etc. (And I would say No, but some set theorists, at least, might disagree.)

   So the relevant question (as I see it) about Zhen Lin's ‘“real” set-theoretic universe’ is whether ordinary mathematics depends on the truth of CH etc. (And I would say No, but some set theorists, at least, might disagree.)