The Kaufmann--Clote question on end extensions of models of arithmetic and the weak regularity principle

Abstract: We investigate the end extendibility of models of arithmetic with restricted elementarity. By utilizing the restricted ultrapower construction in the second-order context, for each $n\in\mathbb{N}$ and any countable model of $\mathrm{B}\Sigma_{n+2}$, we construct a proper $\Sigma_{n+2}$-elementary end extension satisfying $\mathrm{B}\Sigma_{n+1}$, which answers a question by Clote positively. We also give a characterization of countable models of $\mathrm{I}\Sigma_{n+2}$ in terms of their end extendibility similar to the case of $\mathrm{B}\Sigma_{n+2}$. Along the proof, we will introduce a new type of regularity principles in arithmetic called the weak regularity principle, which serves as a bridge between the model's end extendibility and the amount of induction or collection it satisfies.