= Number of ultrafilters on an infinite set
{title2=$|\operatorname{Ult}(X)|=2^{2^{|X|}}$}
The <finite-trace independent family construction> gives $2^{|X|}$ independent subsets of an infinite $X$. Each binary assignment generates a proper filter and, in <ZFC>, has an ultrafilter extension. Different assignments force incompatible membership decisions, giving $2^{2^{|X|}}$ distinct ultrafilters. The matching upper bound follows because every ultrafilter is a subset of $\mathcal P(X)$.
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