How can a sentence (not a formula) describe a set?

Question

What does this sentence mean?

Let $M$ be a model and $S$ be a set described by a sentence in the language of $M$. In Paper

Page 128, part 2.4.

Answer 1

As Eric Wofsey comments, "sentence" should be "formula." I wouldn't call this a typo however; I think it's a deliberate choice to increase readability for a non-logic audience. E.g. consider the line on page $125$:

The first of these principles expresses the fact that ${}^*M$ is a model of $M$.

To a logic audience this would be a terrible misuse of the term "model" (since $M$ is a structure, not a theory). However, to a non-logic audience the notion of one structure modelling another might feel pretty natural.

So really I think the situation is this: some liberties are being taken with respect to technical logic language here in the interest of readability.