Substructure Lattices of Recursively Saturated Models
The CUNY Graduate Center
Previously, it was known (Kossak-Schmerl 1995) that given two arithmetically saturated models with isomorphic substructure lattices, their standard systems must be the same. In Schmerl’s new paper (2015?), he analyzes this proof a little more carefully: in particular, if M is recursively saturated, and X subset omega, then there is an ideal in the substructure lattice of M corresponding to X if and only if X is Turing computable from the jump of some Y in SSy(M). We will go over this proof and, time permitting, how this is used in the proof of the main theorem: if M and N are countable arithmetically saturated models with isomorphic automorphism groups, then the jumps of their theories are Turing equivalent.