Inner-model reflection principles. (English) Zbl 1481.03058

Barton, Neil; Caicedo, Andrés Eduardo; Fuchs, Gunter; Hamkins, Joel David; Reitz, Jonas; Schindler, Ralf

Stud. Log. 108, No. 3, 573-595 (2020).

Summary: We introduce and consider the inner-model reflection principle, which asserts that whenever a statement \(\varphi (a)\) in the first-order language of set theory is true in the set-theoretic universe \(V\), then it is also true in a proper inner model \(W \subsetneq V\). A stronger principle, the ground-model reflection principle, asserts that any such \(\varphi (a)\) true in \(V\) is also true in some non-trivial ground model of the universe with respect to set forcing. These principles each express a form of width reflection in contrast to the usual height reflection of the Lévy-Montague reflection theorem. They are each equiconsistent with ZFC and indeed \(\Pi_2\)-conservative over ZFC, being forceable by class forcing while preserving any desired rank-initial segment of the universe. Furthermore, the inner-model reflection principle is a consequence of the existence of sufficient large cardinals, and lightface formulations of the reflection principles follow from the maximality principle MP and from the inner-model hypothesis IMH. We also consider some questions concerning the expressibility of the principles.

MSC:

03E45	Inner models, including constructibility, ordinal definability, and core models
03E35	Consistency and independence results
03E55	Large cardinals
03E65	Other set-theoretic hypotheses and axioms

Keywords:

inner-model reflection principle; ground-model reflection principle

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