Non-field-type completions do not interpret infinite fields

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Let (R,<,+)(R,<,+) be a dense archimedean ordered abelian group, let R{\mathscr{R}} be a strongly dependent and noiseless expansion, and let N{\mathscr{N}} be a highly saturated elementary extension of R{\mathscr{R}}. Let NSh{\mathscr{N}}^{\mathrm{Sh}} be the Shelah expansion and R□{\mathscr{R}}^{\square} the completion structure. Non-field-type completion conjecture. If R□{\mathscr{R}}^{\square} is not field-type, then NSh{\mathscr{N}}^{\mathrm{Sh}} does not interpret an infinite field. This is a conjectural strengthening of the modular decomposition: the field-type alternative is the only proposed source of an interpreted infinite field in the Shelah expansion.

References

Primary source

Erik Walsberg, “Externally definable quotients and NIP expansions of the real ordered additive group”, arXiv:1910.10572 (2020).

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