Let $T$ be a completion of PA. Let ${\rm Ot}(T)$ be the spectrum of order types of nonstandard models of $T$.
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In <cite> friedman1975:onehundred </cite> Harvey Friedman asked: Does ${\rm Ot(T)}$ depend on $T$?
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Shelah has recently announced that the answer in negative, but there is no paper available yet. See Sh[924] for an interesting effort in the opposite direction.