What is meant by $HOD(S)$ in this paper?

Spencer Unger wrote a short paper on Solovay's Model, and in this he shows that if $S$ is the class of increasing sequences of ordinals, then the model of $\textsf{ZF+DC}$ that we are after is $HOD(S)$. I'm familiar with $HOD$, but what does it mean to tack on a set $S$? Is this the minimal $HOD$ model which contains all elements of $S$? Here's a screenshot for context:

Solution 1:

No. Unfortunately not. And I say that because I keep falling into that trap repeatedly.

In general, $\mathrm{HOD}(S)$ is the model of all sets hereditarily definable from an ordinal and an element of $S$. Indeed, it is quite possible that even if $S$ is a set, it is not an element of $\mathrm{HOD}(S)$, generally speaking.

This can be presented as a symmetric extension, although in general the forcing might need to be the Boolean completion of your original poset. This was studied by Grigorieff in his very classic paper (which uses another set of confusing notations, from a modern perspective)

Grigorieff, Serge, Intermediate submodels and generic extensions in set theory, Ann. Math. (2) 101, 447-490 (1975). ZBL0308.02060.

Interestingly enough, if your ground model was $L$, then every countable sequence of ordinals in $V[G]$ is constructible from a real, and therefore $\mathrm{HOD}(S)=L(\Bbb R)$ in that case.