Rigid models
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There are $\aleph_1$-like rigid models of $PA$. Two different constructions are given in Kossak; Schmerl, Minimal satisfaction classes with an application to rigid models of Peano arithmetic. Notre Dame J. Formal Logic 32 (1991), no. 3, 392–398, and Kossak; Kotlarski Game approximations of satisfaction classes and the problem of rather classless models. Z. Math. Logik Grundlag. Math. 38 (1992), no. 1, 21–26.
Problem: Are there rigid recursively saturated $M\models PA$ such that ${\rm cf}(M)=\aleph_0$?
