Every Filter is Contained in an Ultrafilter

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Theorem

Let $X$ be a set.

Every filter on $X$ is contained in an ultrafilter on $X$.

Let $\Omega$ be the set of filters on $X$.

From Subset Relation is Ordering, the subset relation "$\subseteq$" makes $\Omega$ a partially ordered set.

If $C \subseteq \Omega$ is a non-empty chain, then $\bigcup C$ is again a filter on $X$ and thus an upper bound of $C$.

For any $\mathcal F \in \Omega$ there is therefore by Zorn's Lemma a maximal element $\mathcal F'$ such that $\mathcal F \subseteq \mathcal F'$.

The maximality of $\mathcal F'$ is in this context equivalent to $\mathcal F'$ being an ultrafilter.

$\blacksquare$

This theorem depends on the Axiom of Choice, by way of Zorn's Lemma.

Because of some of its bewilderingly paradoxical implications, the Axiom of Choice is considered in some mathematical circles to be controversial.

Most mathematicians are convinced of its truth and insist that it should nowadays be generally accepted.

However, others consider its implications so counter-intuitive and nonsensical that they adopt the philosophical position that it cannot be true.