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 $U$
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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.