Rather classless models.

$X\subseteq M\models PA$ is a class if for all $a\in M$, $\{x\in X: x<a\}$ is definable (coded) in $M$.

$M$ is rather classless if each class of $M$ is definable.

Since every model of $PA$ has a conservative elementary end extension, for each cardinal $\kappa$ such that $\cf(\kappa)>\aleph_0$, there are $\kappa$-like rather classless models of $PA$. A model is $\kappa$-like is it is of cardinality $\kappa$ and each of its proper initial segments is of smaller cardinality.

Kaufmann, assuming $\lozenge$, proves that there are recursively saturated $\aleph_1$-like rather classless models. Kaufmann, Matt, A rather classless model. Proc. Amer. Math. Soc. 62 (1977), no. 2, 330–333. Later Shelah showed that $\lozenge$ can be eliminated from the proof. Nevertheless one can still ask, as Hodges did in Hodges, Wilfrid, Building models by games. London Mathematical Society Student Texts, 2. Cambridge University Press, Cambridge, 1985: Prove the existence of rather classless recursively saturated models of $PA$ in cardinality $\aleph_1$ without assuming diamond at any stage of the proof.