In set theory you sometimes read statements like "hereditarily finite" or "hereditarily well founded", in the presence of the axiom of foundation Wikipedia says ordinals are sets that are "hereditarily transitive".
I haven't been able to find a good introductory definition of what being hereditory means. It seems to revolve around a property being inherited by the subsets of a set?
Can someone give some basic definitions and direction please, hopefully by explaining what "$A$ is the class of hereditarily $\varphi $ sets" means, where $\varphi $ is some property.
A set $x$ is hereditarily $\varphi$ if and only if every member of its transitive closure (including $x$ itself) has property $\varphi$.
For instance, being hereditarily finite means not just that $x$ is finite, but also every member of $x$ is finite, and every member of every member of $x$, and so on.
(Note that in the absence of foundation, this is a bit peculiar. For instance, if $x=\{x\}$, then $x$ is hereditarily finite, although it does not belong to $V_\omega$.)