Throughout, we work in $\mathsf{ZF}$. Let $[n]=\{1,...,n\}$. Given a set $X$ and a first-order sentence $\varphi$, let $M_X(\varphi)$ be the set of isomorphism types of models of $\varphi$ with underlying set $X$. For simplicity, all filters contain the Frechet filter.
For $\Gamma$ a set of first-order sentences and $\mathscr{F}$ a filter on $\omega$, say that an infinite set $X$ is $\Gamma$-species-reflecting to $\mathscr{F}$ iff the following are equivalent for each $\varphi,\psi\in\Gamma$:
$\vert M_X(\varphi)\vert\le\vert M_X(\psi)\vert$.
$\{n: \vert M_{[n]}(\varphi)\vert\le\vert M_{[n]}(\psi)\vert\}\in\mathscr{F}$.
Say that $X$ is $\Gamma$-species-reflecting iff $X$ is $\Gamma$-species-reflecting to some filter. Here are some quick observations about this notion:
If $X$ is $\Gamma$-species-reflecting to an ultrafilter, then the cardinalities (in the choiceless sense) of the $M_X(\varphi)$s for $\varphi\in\Gamma$ are linearly ordered. Since this seems rather neat, say that $X$ is strongly $\Gamma$-species-reflecting iff $X$ is $\Gamma$-species-reflecting to some ultrafilter.
For "reasonably large" $\Gamma$, $\Gamma$-species-reflecting implies Dedekind-finiteness (since Dedekind-infinite sets make "too many" $M_X(\varphi)$s coincide).
If $\Gamma$ is large enough for the previous bulletpoint to apply and contains the $\{R\}$-sentence $\lambda$ saying that $R$ is a linear order with a first element in which every non-maximal element has an immediate predecessor, then $\mathsf{ZF}$ proves that there are no $\Gamma$-species-reflecting sets at all; this is because any infinite model of $\lambda$ must have an initial segment of ordertype $\omega$.
Let $\eta$ be the $\{E,F\}$-sentence asserting that $E,F$ are equivalence relations, $E$ has two classes, and $F$ yields a bijection between one $E$-class and either the other $E$-class or the other $E$-class minus one element. Then if $X$ is $\{\eta,\perp\}$-species reflecting, $X$ is not amorphous: every finite set supports more models of $\eta$ than of $\perp$, and no infinite model of $\eta$ is amorphous.
I generally have a bunch of questions about species-reflection, but to start with I'd like to get a handle on what sorts of $\Gamma$ can reasonably hope to support species-reflection.
My first candidate is fairly simple. Let $\Gamma_{er}$ be the set of sentences $\varphi$ in a language consisting of only binary relation symbols such that for each symbol $E$ occurring in $\varphi$ we have $\varphi\vdash$ "$E$ is an equivalence relation." The sentence $\eta$ above is of this type, as is $\perp$.
Question 1: Is it consistent with $\mathsf{ZF}$ that there is a $\Gamma_{er}$-species-reflecting set? What about a strongly $\Gamma_{er}$-species-reflecting set?
My second candidate is rather ambitious, but may ultimately be better structured for this question. Say that a sentence $\varphi$ is persistently finitary iff for every sequence $(\mathcal{X}_i)_{i\in\omega}$ of finite models of $\varphi$ of increasing size and every ultrafilter $\mathscr{U}$ on $\omega$, there is a symmetric extension $V\subseteq W\subseteq V[G]$ of the universe and a $\mathcal{A}\in W$ such that $W\models$ "$\mathcal{A}$ is Dedekind-finite" and $V[G]\models\mathcal{A}\cong\prod_{i\in\omega}\mathcal{X}_i/\mathscr{U}$. (Note that this condition can be rephrased in terms of the automorphism groups of such ultraproducts; however, it's not clear to me that that actually clarifies things.) For example, the $\lambda$ mentioned above is not persistently finitary, while the sentence $\eta$ mentioned above is. Let $\Gamma_{pf}$ be the set of persistently finitary sentences.
Question 2: Is the existence of a (strongly) $\Gamma_{pf}$-species-reflecting set consistent with $\mathsf{ZF}$?
More generally, I'm interested in any information about when the existence of a (strongly) $\Gamma$-species-reflecting set is consistent with $\mathsf{ZF}$.
