A number of results that people use that require the axiom of choice (i.e. do not follow from ZF alone) are known to actually imply the axiom of choice. Therefore, one might naturally wonder whether ...
Is it possible to show that an infinite set has a countable (infinite) subset, without using the Axiom of Choice?
Let X be an infinite set. Is it possible to show the existence of a countably infinite subset of X without using the Axiom of Choice?
Does the Axiom of Choice (or any other “optional” set theory axiom) have real-world consequences? [closed]
Or another way to put it: Could the axiom of choice, or any other set-theoretic axiom/formulation which we normally think of as undecidable, be somehow empirically testable? If you have a particular ...
What would be the consequence of requiring that any choice function be computable; i.e. using as the foundational basis ZF + ACC? Does it make a difference if we admit definable functions? I guess I ...