Jonsson models
A model B is J\'onsson if |B|>ℵ0 and for every A≺B, if |A|=|B|, then A=B.
Gaifman and Knight independently showed in 1976 that there are Jonsson models of PA.
Jonsson models M of PAofcardinality\aleph_1areeither\aleph_1-likeorare″short″i.e.thereisana\in MsuchthattheSkolemclosureofaiscofinalinM.EachknownJonssonmodelrealizesuncountablymanycompletetypes.Kossakhasasked:Istherean\aleph_1−likeJonssonmodelM\models\PAsuchthat|\{{\rm tp}(a): a\in M\}|=\aleph_0?IfM\models\PAis\aleph_1−likeandrecursivelysaturated,then|\{{\rm tp}(a): a\in M\}|=\aleph_0,butMisnotJonsson.Therefore,anotherrelatedquestionis:Istherea″weaklyJonssonmodel″M\models \PA,i.e.arecursivelysaturatedmodelM\models PAsuchthatforeveryrecursivelysaturatedK\prec M,if|K|=|M|,thenK=M$? Some results related to this question are in Kossak, Roman, Four problems concerning recursively saturated models of arithmetic. Special Issue: Models of arithmetic. Notre Dame J. Formal Logic 36 (1995), no. 4, 519–530.