= Fichtenholz-Kantorovich independent family
{c}
{title2=$|\mathcal I|=2^{|X|}$}
= Finite-trace independent family construction
{synonym}
For every infinite <set> $X$ there is an <independent family of sets> consisting of $2^{|X|}$ subsets of $X$. One construction uses a set $D$ of pairs $(s,H)$, where $s\subseteq X$ is finite and $H\subseteq\mathcal P(s)$. For each $A\subseteq X$, let $I_A=\{(s,H):A\cap s\in H\}$. Given distinct $A_1,\ldots,A_m$, choose a finite $s$ distinguishing all their traces and choose $H$ to contain precisely the traces prescribed to have value one. This realizes every finite Boolean choice. With the <axiom of choice>, $|D|=|X|$, so a <bijection> transports this family to $X$. Its independent assignments give $2^{2^{|X|}}$ distinct <ultrafilters>.
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