"The Axioms of choice" is talking about a mutable set of sets that contain immutable axioms.

The number 4 in a set is immutable, 4 == 4 under all stances, while it can be taken out or added to the selected set. Objective reasoning.

Example: The OS function that applies IP port assignments. Application may utilize one or more port numbers. When the application is done, they are moved back into the unused set. The port numbers themselves are immutable.

Social axioms are mutable because subjective reasoning is being applied.

Example:

Ask a person if they support domestic abuse. They most likely will so no. Ask them if they ever funded a performer that committed acts of domestic abuse after the facts were revealed. There is a high chance they did and write it off by saying, "don't look at their character, look at how well they perform. That was a good show, movie, book, ..."

Ever hear someone use the phrase, "that is different" when describing two equal scenarios that apply to two different people?

Kanye West still selling out shows highlights that people are willing to reject objective reasoning in a social environment.

Or to rephrase, social axioms are mutable because they can collapse and no longer exist.

My point was - mathematicians still argue about whether the axiom of choice is a good idea or not. They probably always will.

So mathematical axioms are in flux.