### Vive la diff\'erence III

by Shelah. [Sh:509]

Israel J Math, 2008

We show that, consistently, there is an ultrafilter F on omega
such that if N^ell_n=(P^ell_n cup Q^ell_n, P^ell_n,Q^ell_n,R^ell_n)
(for ell =1,2, n< omega), P^ell_n
cup Q^ell_n subseteq omega, and prod limits_{n< omega} N^1_n/
F equiv prod limits_{n< omega}N^2_n/ F are
models of the canonical theory t^ind of the strong
independence property, then every isomorphism from
prod limits_{n< omega} N^1_n/ F onto prod limits_{n<
omega} N^2_n/ F is a product isomorphism.

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