Alas, I have nothing substantial to say, except that in reading about the “pathological” results of both the Axiom of Choice and its alternatives, the controversies between them all lie in counterintuitive effects of infinity — and discussing these various alternatives is merely a discussion of what you might expect infinities could or could not do.

I suspect that none of these will fit well with physical reality one way or the other, if only because we mortal humans can’t measure in infinite precision — but I can’t rule out the possibility that some day we’ll find one physical phenomenon that fits with AoC, and another that fits with one of its negations.

I am also now wondering: is it possible to “embed” AoC into the logic system of its various negations, and vice versa? Hyperbolic and Spherical geometries can be embedded into Euclidean easily enough, and for a sufficient “zooming in”, Euclidean is a very good approximation of either geometry!

But it is late, I’m tired, and I may forget to search for such “embeddings” tomorrow … my intuition, for what it’s worth, suggests it may be possible …

The inability to writing it down or memorizing is a weak argument, unless you are an ultrafinitist. We cannot write down any countably infinite set either.

So there seem to be paradoxes involved even before any prisoner gets the chance to invoke the axiom of choice.