TITLE: Ultrafilters vs Well-orderings
QUESTION [14 upvotes]: This question was actually asked by John Stillwell in a comment to an answer to this question.  I thought I would advertise it as a separate question since no one has yet answered and I am also curious about it.
Question: Is the existence of a non-principal ultra-filter on $\omega$ a weaker assumption than the existence of a well-ordering of $\mathbb{R}$?

REPLY [4 votes]: 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.