In this paper, we prove results concerning the existence of proper end extensions of arbitrary models of fragments of Peano arithmetic (PA). In particular, we give alternative proofs that concern (a) a result of Clote (Fundam Math 127(2):163–170, 1986); (Fundam Math 158(3):301–302, 1998), on the end extendability of arbitrary models of \(\Sigma _n\)-induction, for \(n{\ge } 2\), and (b) the fact that every model of \(\Sigma _1\)-induction has a proper end extension satisfying \(\Delta _0\)-induction; although this fact was not explicitly stated before, it follows by earlier results of Enayat and Wong (Ann Pure Appl Log 168:1247–1252, 2017) and Wong (Proc Am Math Soc 144:4021–4024, 2016).

在本文中，我们证明了有关Peano算法（PA）的任意片段模型的适当末端扩展的存在的结果。特别是，我们给出了与（a）Clote的结果有关的替代证明（Fundam Math 127（2）：163-170，1986）；（Fundam Math 158（3）：301–302，1998），关于\（\ n Sigma_n \）归纳的任意模型对于\（n {\ ge} 2 \）和（b）每个\（\ Sigma _1 \）归纳模型都有一个满足\（\ Delta _0 \）归纳的适当末端扩展；尽管之前没有明确说明这一事实，但随后是Enayat和Wong（Ann Pure Appl Log 168：1247–1252，2017）和Wong（Proc Am Math Soc 144：4021–4024，2016）的早期结果。