Since Cohen  showed the independence of the axiom of choice from the other axioms of ZF, a number of people have used his forcing technique to show how badly choice can fail while certain vestiges of it remain. Many of these arguments involve the introduction of a generic sequence or generic set of generic sets of ordinals into a countable standard model of ZF + V = L. The desired results then follow by the use of clever permutation and forcing arguments. (See Cohen [2, pp. 136–142].)
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