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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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3 Answers 3

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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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    $\begingroup$ Ah, of course it does. Just learning about the classical independence/forcing models, so it didn't jump out at me. $\endgroup$
    – David Roberts ♦
    Nov 2, 2011 at 22:46
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If you're looking for a purely topos-theoretic model, I think you don't need to go through set theory (even though the end result may end up being basically equivalent). Look at the topos of continuous actions of the pro-completion of the integers, which is to say, the category of sets equipped with an automorphism all of whose orbits are finite. Here we have an N-indexed family of objects (one orbit of each cardinality) which are all inhabited, but whose product is empty -- hence the NNO is not internally projective.

Have you read P. Freyd's paper "The Axiom of Choice"?

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    $\begingroup$ No I haven't - though I've got Blass and Scedrov. Thanks for the reminder. $\endgroup$
    – David Roberts ♦
    Nov 6, 2011 at 0:34
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Sheaves on the unit interval also break countable choice (and also weak countable choice, where you only require at most one of the sets to not be a singleton, which is weaker than both CC and LEM).

In particular, the Dedekind reals and the Cauchy reals fail to be the same in that topos (the former are continuous Real functions on the interval, the latter are just the reals from the category of sets), while they are equal in any topos where Countable Choice or LEM holds.

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    $\begingroup$ More generally, if $X$ is a topological space which is sober and locally connected and such that continuous real-valued functions are not necessarily locally constant, countable choice fails because the Cauchy reals are represented by the locally constant real-valued functions and the Dedekind reals by the continuous real-valued functions (see example D.4.7.12(a) and thereabouts in Johnstone's Elephant). $\endgroup$
    – Gro-Tsen
    Nov 5, 2022 at 19:22
  • $\begingroup$ @Gro-Tsen I'm kind of interested by which parts of this generalizes to locally connected Grothendieck topoi and which don't. For a locally connected topos over set, is it true that the cauchy reals are always a locally constant sheaf? And the Dedekind reals are equal to the Cauchy reals iff they are a locally constant sheaf? $\endgroup$
    – saolof
    Nov 9, 2022 at 22:54
  • $\begingroup$ In the Kleene-Vesley topos, every function from $S^1$ to itself is continuous, but it admits not only Countable Choice but Dependent Choice. Also, it's not yet known whether Johnstone's TT admits CC or not. $\endgroup$
    – wlad
    Dec 26, 2022 at 13:11
  • $\begingroup$ You can't use Countable Choice to go from the existence of an embedding $\mathbb N \to S$ to an embedding $S \to S$. The circle $S^1$ in the KV topos is your first counterexample. $\endgroup$
    – wlad
    Dec 26, 2022 at 13:23
  • $\begingroup$ @wlad Deleted the second part of the answer for now until I understand that example better. Does embedding $\mathbb{N} \to S$ implies Dedekind-infinite fail for that topos despite having CC (DC)? Or do we just have that the circle can have continuous injections that are not homeomorphisms? $\endgroup$
    – saolof
    Dec 26, 2022 at 19:42

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