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 ${\rm cf}(\kappa)>\aleph_0$, 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. Later Shelah showed that $\lozenge$ can be eliminated from the proof. Nevertheless one can still ask, as Hodges did in 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.

References:

Kaufmann, Matt, A rather classless model. Proc. Amer. Math. Soc. 62 (1977), no. 2, 330–333.

Hodges, Wilfrid, Building models by games. London Mathematical Society Student Texts, 2. Cambridge University Press, Cambridge, 1985