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I recall reading somewhere that there is a topos in which the Dedekind reals are exactly the measurable functions.

Now vector spaces are prominently characterised by dimensionality. This prompted the thought as to whether there is a topos in which the natural numbers object are the finite-dimensional vector spaces (upto isomorphism); and if there is such a topos, what would the Dedekind reals be there?

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This question doesn't make sense. If you only care about finite-dimensional vector spaces (over a given field) up to isomorphism, then the set of isomorphism classes can be canonically identified with the natural numbers. But the collection of all finite-dimensional vector spaces is not itself a set. –  Zhen Lin Sep 12 '13 at 10:23
    
@Lin: Thanks for pointing that out, I've corrected the question. I'd say its a little harsh to say that this question doesn't make sense when such a small modification makes it correct. If this was a literary contest, I'd say its more akin to a spelling mistake as opposed to being an entirely ungrammatical sentence. –  Mozibur Ullah Sep 14 '13 at 17:37
    
But then the modified question is boring: $\mathbf{Set}$ is such a topos! –  Zhen Lin Sep 14 '13 at 18:21
    
@Lin: That surprises me, I'd have thought the NNO in Set is just the ordinary integers. –  Mozibur Ullah Sep 14 '13 at 20:20
    
Any countable set can be an NNO in $\mathbf{Set}$ – NNOs are only unique up to unique isomorphism. Please learn some category theory. –  Zhen Lin Sep 14 '13 at 21:56

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