Tarski’s axiom

Tarski proposed the following axiom for set theory:

For every set $S$, there exists a set $U$ which enjoys the following properties:

• $S$ is an element of $U$

• For every element $X\in U$, every subset of $X$ is also an element of $U$.

• For every element $X\in U$, the power set of $X$ is also an element of $U$.

• Every subset of $U$ whose cardinality is less than the cardinality of $U$ is an element of $U$.

This axiom implies the axiom of choice. It also implies the existence of inaccessible cardinal numbers.