 Solution

A transitive class $M$ satisfies $x\in M\Rightarrow x\subseteq M$. Its membership structure therefore does not omit elements of its members. In particular bounded membership statements are absolute between a transitive model and the ambient universe, and its membership relation is the actual well-founded relation, rather than an arbitrary externally ill-founded coding.

For a set $x$, put $x_0=x$ and $x_{n+1}=\bigcup x_n$, and defineThis is a set, contains $x$ as a subset, and is transitive: an element of an element of $x_n$ belongs to $x_{n+1}$. Any transitive set containing $x$ as a subset contains every $x_n$ by induction, so this construction is the least such set. Shifting the union index givesThese formulas use the specified subset convention for transitive closure.

Work in the usual ambient set theory with Choice. If $x$ is a hereditarily countable set, then every member of $\mathrm{TC}(\{x\})$ is countable. Iterated countable unions show that this entire closure is countable. Conversely a countable transitive closure makes each of its members countable. This equivalent criterion makes the following closure arguments explicit.

If $y\in x\in \mathrm{HC}$, then $\mathrm{TC}(\{y\})\subseteq \mathrm{TC}(\{x\})$, so $y\in \mathrm{HC}$: HC is transitive. If $x\in \mathrm{HC}$, its union is countable and its descendant sets are descendants of $x$, so $\bigcup x\in \mathrm{HC}$. The actual union therefore witnesses the Axiom of union inside $(\mathrm{HC},\in )$.

For axiom schema of separation, take any formula with parameters in HC and a set $x\in \mathrm{HC}$. Form the subset of $x$ consisting of the elements that satisfy that formula with all quantifiers relativized to HC. Ambient Separation supplies this set. It is a subset of a countable set, and all its descendants are inherited from $x$, so it belongs to HC. No claim that the formula itself is absolute is needed.

The axioms satisfied by hereditarily countable sets also include Extensionality, Empty Set, Pairing, Infinity, Foundation and Replacement. Extensionality and Foundation follow from transitivity and ambient well-foundedness. Empty Set and $\omega$ belong to HC, and a pair of HC sets has countable transitive closure. For Replacement, the domain is countable; a definable image of its elements is a countable family of HC sets, whose combined transitive closures form a countable union of countable sets, so the range again belongs to HC. Ambient Choice additionally gives Choice internally: a well-order of a countable HC set has a hereditarily countable relation and hence lies in HC.

Power Set fails. Every subset of $\omega$ is hereditarily countable, so an internal power-set witness would have to contain all ambient subsets of $\omega$. That collection is uncountable by Cantor theorem, whereas every HC set is countable. Thus HC satisfies the other ZF axioms and, in this ambient setting, Choice, but not the Axiom of power set.