TITLE: Does negation of Axiom of Choice imply symmetry?
QUESTION [9 upvotes]: It seems that every construction of a model in which the Axiom of Choice fails involves some kind of symmetry. Is there an example of a construction of a model where AC fails but no argument involving symmetry appears? Is there any result that connects the negation of choice (any kind of choice) to some kind of symmetry?

REPLY [2 votes]: I have an example which is not symmetric, or at least not obviously so. It is a bit advanced, so I advice searching the literature for it.
In the effective topos, which is based on recursive realizability, there are surjective functions that do not have sections. In this topos, all morphisms $\mathbb N \to \mathbb N$ are recursive functions. Therefore, there is a set of algorithms for total functions $\mathbb T$ and a surjection $s:\mathbb T \to \mathbb N^{\mathbb N}$. This surjection cannot have a section for the following reason. We can recursively decide whether two algorithms are equal. An inverse of $s$ would allow us to decide the equality of functions, which is impossible.