View source for 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. 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.