We will show in ZF that a monotonic function on a partial order where all chains have supremums must have a fixed point. With the addition of the Axiom of Choice, this will provide easy proofs of the Hausdorff maximal principle and Zorn's lemma without use of the ordinals. Time permitting, we will also show how to construct nonmeasurable sets of reals while only allowing choice on unordered pairs.
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