Working in MK (or some other not-too-strong class theory if you prefer), say that an up-class is a class of structures $\mathfrak{X}$ which is definable in $V$ (allowing parameters from $V$) and such that whenever $\mathcal{A}\in\mathfrak{X}$ and $i:\mathcal{A}\rightarrow\mathcal{B}$ is an embedd...

This was asked and bountied at MSE without success. Throughout, we work in $\mathsf{ZF}$. Say that a set $X$ is $\Pi^1_1$-pseudofinite if for every first-order sentence $\varphi$, if $\varphi$ has a model with underlying set $X$ then $\varphi$ has a finite model. (See here, and the answer and com...