# Example of a topos that violates countable choice

At this nLab page we have the line

In contrast, any topos that violates countable choice, of which there are plenty, must also violate internal COSHEP.

It doesn't give an example, and neither does the page on countable choice. So, what are these all-so-common examples?

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One sort of examples consists of the topoi of sets and functions obtained from models of ZF that violate countable choice. The original Cohen model is among these, and so are many others. Perhaps easier to understand are permutation models of ZFA (the variant of ZF that allows for atoms (= urelements)). The basic Fraenkel model, the second Fraenkel model, and Mostowski's linearly ordered model (probably the three best-known permutation models --- see Chapter 4 of Jech's book "The Axiom of Choice") all have infinite Dedekind-finite sets and therefore violate countable choice.

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Ah, of course it does. Just learning about the classical independence/forcing models, so it didn't jump out at me. –  David Roberts Nov 2 '11 at 22:46