Let $U$ be a free ultrafilter on a set $X$.
I want to prove that the cardinality of every element $u\in U$ is equipotent to $X$. Is that true? Or does it lack some hypothesis?
2026-04-06 16:37:46.1775493466
What is the cardinality of an element of an free ultrafilter?
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No, this is not provable because it can be false.
Consider an ultrafilter on $\Bbb N$, say $\cal F$. You can show that $\{A\subseteq\Bbb R\mid A\cap\Bbb N\in\cal F\}$ is a free ultrafilter on the real numbers.
The property that you are looking for is called uniform ultrafilter.