I am very interested in the relation between the axiom of choice and the continuum hypothesis. In 1900 David Hilbert mentioned the continuum hypothesis as the first between the important mathematical problems. Kurt Gödel and Paul Cohen¹ gave the solution in the middle of the twentieth century. We know based on the proof of Gödel and Cohen that the continuum hypothesis is consistent and independent assuming ZF². The same is true of the axiom of choice.

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¹I am now reading the book of Cohen, Set Theory and the Continuum Hypothesis /Dover (2008) unabridged republication of the edition published by W.A. Benjamin, Inc., New York, 1966. Copyright renewed 1994 by Paul J. Cohen.

² Zermelo-Fraenkel Set Theory