Hereditary size and H_kappa

Let $\kappa$ be an infinite initial ordinal. In ZF set

$H_{\kappa }=\{x:\mathrm{\exists }\lambda <\kappa \mathrm{\exists }j(j:\mathrm{TC}(\{x\})\hookrightarrow \lambda )\}\mathrm{.}$

This initially defines a class. Its root-inclusive transitive closure contains $x$ as an element. When $\mathrm{TC}(\{x\})$ is well-orderable its hereditary cardinality means the least ordinal equinumerous with it; under Choice this exists for every set. The injection formulation above is used without Choice.

For infinite $\kappa$, replacing $\mathrm{TC}(\{x\})$ by $\mathrm{TC}(x)$ gives the same class. Indeed $\mathrm{TC}(\{x\})=\{x\}\cup \mathrm{TC}(x)$ by the finite-stage formula. Adding one point to a set injecting into finite $\lambda$ gives an injection into $\lambda +1<\kappa$; for infinite $\lambda$, keep indices at least $\omega$, shift natural indices by one, and use index zero for the added point, obtaining an injection into $\lambda$. Restriction gives the converse. We retain the root-inclusive convention throughout.

Conventions and prerequisites: Minimality and closure laws of TC, Cardinal (initial ordinal) and cardinality.