Say that a **long model** is an $\mathfrak{A}\models\mathsf{I\Sigma_1}$ such that $\mathfrak{A}$ has strictly greater cardinality than each of its proper initial segments (in the case $\vert\mathfrak{A}\vert=\aleph_1$ this is "$\omega_1$-like"ness). If $\mathfrak{A}$ is a long model and $\vert\mathfrak{A}\vert$ is regular, then in fact $\mathfrak{A}\models\mathsf{PA}$ since the bounding scheme is trivially satisfied in $\mathfrak{A}$ (see Lemma 2.15 of Hajek/Pudlak). Moreover, by the MacDowell/Specker theorem (see Kossak, *63 years of the MacDowell-Specker theorem*) the converse holds: if $\kappa$ is an uncountable cardinal and $\varphi$ is true in every long model of $\mathsf{PA}$ of size $\kappa$, then $\mathsf{PA}\models\varphi$ already. (Given $\mathsf{PA}\not\models\varphi$, let $\mathfrak{B}\models\mathsf{PA}+\neg\varphi$ be countable and iterate MD/S $\kappa$-many times.)

Interestingly, the first observation seems to need regularity:

Is there a singular cardinal $\kappa$ such that the common theory of long models of size $\kappa$ is strictly weaker than $\mathsf{PA}$?

Presburgerarithmetic, but that's a rarity in my experience). The standard source on subsystems of $\mathsf{PA}$ is the Hajek/Pudlak book linked in my question. $\endgroup$4more comments