This paper has its origins in a question by Neil Barton on Math.SE, “What is the consistency strength of width-reflection?”
Start with one of the foundational relationships of set theory, that of height-reflection — any property true in the universe $V$ is true in some initial segment $V_\kappa$. Then “rotate your head ninety degrees” and consider the corresponding question along a different axis — whether any property true in $V$ is also true in some proper inner model, a model with the same height as $V$ but lacking some of its bulk, hence width-reflection. As this principle directly addresses inner models of $V$, it requires something more than pure set theory to state and analyze, so you will see will us delve into second-order set theories. My own contributions concern the restriction of our attention to ground models, those inner models $W$ of which $V$ is a set forcing extension — this provides a uniformly definable collection of inner models that can be treated in a purely first-order fashion.
Abstract. We introduce and consider the inner-model reflection principle, which asserts that whenever a statement $\phi(a)$ 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 $\phi(a)$ 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.
Inner-model reflection principles, with Neil Barton, Andres Caicedo, Gunter Fuchs, Joel David Hamkins and Ralf Schindler, Stud Logica (2019). https://doi.org/10.1007/s11225-019-09860-7. pdf, arXiv
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