Throughout, we work in $\mathsf{ZF}+$ "There is an infinite Dedekind-finite set."
Say that a Dedekind-finite cardinality $\kappa$ is $\Sigma^1_1$-isolated iff there is some first-order formula $\varphi$ such that the Dedekind-finite cardinalities of models of $\varphi$ are exactly the Dedekind-finite cardinalities $\ge\kappa$ (= admitting an injection from $\kappa$), and weakly $\Sigma^1_1$-isolated iff $\varphi$ has a model of cardinality $\kappa$ but no Dedekind-finite model of cardinality $<\kappa$.
I believe it's not hard to produce models in which no infinite Dedekind-finite cardinality is weakly $\Sigma^1_1$-isolated (this happens in Cohen's model, if I'm not mistaken). I'm interested in the opposite extreme:
Is it consistent that every Dedekind-finite cardinality is $\Sigma^1_1$-isolated, or at least weakly $\Sigma^1_1$-isolated?
I'm especially interested in what we can achieve when the Dedekind-finite cardinalities are linearly ordered (in which case weak and strong isolation coincide); this is consistent by an old result of Sageev, and by an older result of Ellentuck in such a situation the arithmetic of Dedekind-finite cardinalities parallels true arithmetic to a high degree.
