Recently arxiv submitted a new paper (Andrej Bauer, James E. Hanson, _[The Countable Reals][1]_) claiming an incredible theorem that Dedekind reals are not sequence-avoiding, and furthermore obtaining a topos that makes Dedekind reals countable.

I'm wondering if the constructive Eudoxus reals (see R. D. Arthan's _[The Eudoxus Real Numbers](https://arxiv.org/abs/math/0405454)_) are sequence-avoiding?


Also, if “all functions $\mathbb{Z} \to \mathbb{Z}$ are computable” holds, what will be the behaviour of Eudoxus reals? Would it become computable?



  [1]: https://arxiv.org/abs/2404.01256