Is the power of set of elements that are less than $x$ little?

Question

Does every set $X$ can be well ordered such that for all element $x$ the power of set of elements that are less than $x$ is less than the power of $X$?

I saw this idea in proof of some problem. Can you show me why it is true?

Answer 1

Yes. Initial ordinals are exactly the order types of such well-orders.

And we can alternatively define an initial ordinal as the smallest ordinal of a given cardinality. To see why, if $\alpha$ is an initial ordinal, every proper initial segment (i.e. the set of all those which are smaller than some $x$) is isomorphic to a strictly smaller ordinal. Since $\alpha$ was minimal in its cardinality, that smaller ordinal must be smaller also in cardinality.

If so, given a set $X$, if we can well-order $X$ (which we can, assuming the axiom of choice holds), then we can well-order it with a minimal order-type, which is an initial ordinal.