Models of PA: when two elements are necessarily order automorphic

by Shelah. [Sh:924]
Math Logic Quarterly, 2015
We are interested in the question of how much the order of a non-standard model of PA can determine the model. In particular, for a model M, we want to characterize the complete types p(x,y) of non-standard elements (a,b) such that the linear orders {x:x< a} and {x:x < b} are necessarily isomorphic. It is proved that this set includes the complete types p(x,y) such that if the pair (a,b) realizes it (in M) then there is an element c such that for all standard n,c^n < a,c^n < b,a < bc and b < ac . We prove that this is optimal, because if diamondsuit_{aleph_1} holds, then there is M of cardinality aleph_1 for which we get equality. We also deal with how much the order in a model of PA may determine the addition.


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