### 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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