In descriptive set theory, there is a significant number of results that have been established using forcing; typically, dichotomy theorems such as Silver's (a $\Pi^1_1$ equivalence relation has only countably many classes, or else there is a perfect set of pairwise inequivalent points).

Harrington used what is now called the Gandy-Harrington topology to eliminate the use of forcing from these arguments, replacing it instead with an appeal to effective'' techniques. So, for example, Silver's result can actually be stated as: If $E$ is a $\Pi^1_1$-in-the-parameter-$a$ equivalence relation, then either every $E$-class is itself $\Pi^1_1(a)$ (so, in particular, there are only countably many), or else there are perfectly many inequivalent classes.

For a long while, we actually thought these uses of forcing or effective descriptive set theory were essential to the theory. Benjamin Miller recently transformed the field by showing how derivative'' arguments can eliminate just about all these uses.

The latest twist is that Richard Ketchersid and I have been studying structural properties of models of determinacy, and have shown that the descriptive set theoretic dichotomies hold in this context. This is more general than what Miller's technique can establish. Once again, our arguments make essential use of forcing (and ultrapower constructions).

For example, we have shown that the $G_0$-dichotomy of Kechris-Solecki-Todorcevic holds in models of ${\sf AD}^+$ of arbitrary graphs on reals: Any such graph either can be colored by ordinals (so that points connected by an edge receive different colors) or else, there is a continuous homomorphism of the graph $G_0$ into $G$. (See for example these slides from a recent talk for details and complete definitions).

Ben has shown how Baire category arguments allow one to deduce most other dichotomies from appropriate versions of the $G_0$-dichotomy. Using this, Richard and I have deduced some interesting global dichotomies in these models (meaning, they hold of all sets, not just sets of reals). For example, in the presence of large cardinals, $L({\mathbb R})$ is a model of determinacy, and for any $X\in L({\mathbb R})$, either $X$ can be well-ordered inside $L({\mathbb R})$, or else, there is in $L({\mathbb R})$ an injection of ${\mathbb R}$ into $X$. In short: containing a copy of the reals is the only obstacle to being well-orderable. (This is a strong version of the statement that one cannot well-order the reals "definably".) --- Actually, Richard and I first established this directly, via a forcing argument, but it can now be deduced from our version of the $G_0$-dichotomy.

(There is another, subtle use of forcing in the context of determinacy, via the theory of generic codings.)