Recall that $L$, Godel's constructible universe is constructed by defining the following hierarchy:

$L_0=\varnothing$, for a limit ordinal $\delta$, $L_\delta=\bigcup_{\alpha<\delta}L_\alpha$, and $L_{\alpha+1}=\operatorname{Def}^1(\langle L_\alpha,\in\rangle)$ (where the $^1$ denotes first-order definability).

On the other end of spectrum, we define $\sf HOD$ the class of hereditary ordinal-definable sets as the class of sets which are ordinal definable, i.e. there is some formula $\varphi(x,\alpha_1,\ldots,\alpha_n)$ such that only a single $x$ satisfies with a given sequence of ordinal parameters, and that their transitive closure is made entire of ordinal definable sets.

But Scott and Myhill showed that if you replace $\operatorname{Def}^1$ by $\operatorname{Def}^2$ (second order definability) in the constructible hierarchy, you get $\sf HOD$ as well. So we can construct $\sf HOD$ in two ways:

- Take a "definability closure" for second-order logic.
- Define a property, then consider those which have this property in a hereditary fashion.

Question.Is there some natural definable class which is similar to $\sf OD$, that when taking the "hereditary" subclass we get $L$?