= Solution
\b[True in ZF.] For a set $A$, the singleton map injects $A$ into its <power set>. No map $f:A\to\mathcal P(A)$ is onto: the diagonal set
$$
D=\{a\in A:a\notin f(a)\}
$$
differs from $f(a)$ at the element $a$ for every $a$. Thus there is no <bijection> between $A$ and its <power set>. The <injection> together with this failure proves $\boxed{\kappa<2^\kappa}$.
If strict comparison is formulated as <injection> in one direction but not the other, the <Cantor-Schröder-Bernstein theorem>, which requires no choice, excludes an <injection> of the <power set> back into $A$. This is <Cantor's theorem> in a choice-free form.
Back to article page
