Number of ultrafilters on an infinite set ($|\mathrm{Ult}(X)|=2^{2^{|X|}}$)

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 $\mathcal{P}(X)$.