When the axiom of choice is not available, the concept of cardinality is somewhat more subtle, and there is in general no fully satisfactory solution of the cardinal assignment problem. Rather, in ZF one works directly with the equinumerosity relation. In ZF, the [axiom of choice](Axiom_of_choice "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.md) of $X$, and conversely, if $X$ injects into $\alpha$, then $X$ would be well-orderable. ## Dedekind finite sets The *Dedekind finite* sets are those not equinumerous with any proper subset. Although in ZFC this is an equivalent characterization of the finite sets, in ZF the two concepts of finite differ: every finite set is Dedekind finite, but it is consistent with ZF that there are infinite Dedekind finite sets. An *amorphous* set is an infinite set, all of whose subsets are either finite or co-finite. This article is a stub. Please help us to improve Cantor's Attic by adding information.