# General cardinal, without the axiom of choice

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In ZF, the axiom of choice is equivalent to the assertion that the cardinals are linearly ordered. This is because for every set $X$, there is a smallest ordinal $\alpha$ that does not inject into $X$, the Hartog number of $X$, and conversely, if $X$ injects into $\alpha$, then $X$ would be well-orderable.
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