Is the axiom of choice really related to choice?

What are some alternatives to the axiom of choice that don't lead to bizarre results?

• See There he mentions that denial of the axiom of choice also leads to bizarre results. Could there be a weakened version of the axiom of choice that walks the fine line between these mathematical bugs? For example, could the axiom of choice be restricted to countably infinite sets instead of including uncountable sets, too? Or could restrictions be placed on it as to the measurability of the sets it operates on and/or produces?

• Answer:

  Sure. For example: http://en.wikipedia.org/wiki/Axiom_of_countable_choice http://en.wikipedia.org/wiki/Axiom_of_dependent_choice In fact there's a whole jungle of axioms that are weaker than AC but suffice to hold various chunky pieces of mathematics. A very detailed survey of the various dependencies is in Stephen Simpson's http://www.personal.psu.edu/t20/sosoa/ (Subsystems of Second Order Arithmetic).