Is the axiom of choice a necessary condition for the application of "back and forth construction" in model theory?
2026-04-04 08:43:00.1775292180
Back and forth and the axiom of choice
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As with many other things, the axiom of choice is not needed if the domain of discourse is well-ordered. So for example, without the axiom of choice countable atomless Boolean algebra still has a unique model.
The generality of the question, however, seem to imply the answer is yes. You make a lot of choices, and since you haven't put any limitations on the size of the models, or otherwise, then the answer is yes.