Historically it was proved that the ultrafilter lemma is independent from the axiom of choice by showing that there is a model in which there is an infinite Dedekind-finite set of real numbers, but every filter can be extended to an ultrafilter. Where an infinite Dedekind-finite set is an infinite set which does not have a countably infinite subset.
The existence of infinite Dedekind-finite sets negates not only the axiom of choice, but also the [much] weaker axiom of countable choice. These sets cannot be well-ordered, and since the real numbers have such subset they cannot be well-ordered themselves in such model.
The proof was given by Halpern and Levy in 1964.
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