I know, that there aren't many results on order types of arithmetic fragments. E.g. there are some basic results which one can find in texts of Kaye and Bovykin. But does anyone know, if there is something like an overview of the possible order types of (weak) arithmetic fragments like $Q$, $IOpen$, $I\Delta_0$ or $I\Sigma_1$ as far as they have been discovered until today?
2026-03-25 06:07:11.1774418831
Is there an overview of possible order types of fragments of first-order arithmetic?
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You might be interested in "The metamathematics of first-order logic" by Hajek and Pudlak: http://www.amazon.com/Metamathematics-First-Order-Arithmetic-Perspectives-Mathematical/dp/354063648X.