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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.
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