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| − | $X\subseteq M\models PA$ is a ''class'' if for all $a\in M$, $\{x\in X: x<a\}$ is definable (coded) in $M$.
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| − | $M$ is ''rather classless'' if each class of $M$ is definable.
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| − | 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.
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| − | 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
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| − | 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
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| − | models of $PA$ in cardinality $\aleph_1$ without assuming diamond at any stage of the proof.
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