= Solution
Here $ZF^*$ denotes <ZF without foundation>. <Transfinite recursion> on <ordinals> defines the <cumulative hierarchy> by
$$
\boxed{V_0=\varnothing,\qquad V_{\alpha+1}=\mathcal P(V_\alpha),\qquad V_\lambda=\bigcup_{\beta<\lambda}V_\beta\quad(\lambda\text{ a nonzero limit}).}
$$
The associated class is $V=\{x:\exists\alpha\in\operatorname{Ord}\ (x\in V_\alpha)\}$. Each level is a set, while $V$ is a definable class. Without <Foundation> it need not contain every ambient set; it consists of the sets built by this <well-founded> hierarchy.
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