*We work in $\mathsf{ZFC+V=L}$.*

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Define a **plausible theory** to be a theory $T\subseteq\mathcal{L}_{\omega_1,\omega}$ in an $\omega_1$-finite language which is $\omega_1$-c.e. and $\omega_1$-finitely satisfiable; [that is](https://projecteuclid.org/ebooks/perspectives-in-logic/Higher-Recursion-Theory/toc/pl/1235422631), $T$ is a theory in a countable language which is $\Sigma_1$-definable over $L_{\omega_1}$ (with parameters) and every countable subtheory of $T$ is satisfiable. 

Since $L_{\omega_1}$ is uncountable, [Barwise compactness](https://www.sciencedirect.com/science/article/pii/S0049237X0870689X) does not apply and plausible theories need **not** be satisfiable. That said, every plausible theory is "generically" satisfiable: they all become satisfiable after forcing with $\mathit{Col}(\omega,\omega_1)$, since in the generic extension we can apply Barwise compactness. 

I'm interested in measuring the difficulty of "satisfiabilizing" a plausible theory via forcing. Specifically, set $T_0\trianglelefteq T_1$ iff $T_0$ is satisfiable in every generic extension in which $T_1$ is satisfiable. *(Note that by absoluteness of $\mathcal{L}_{\omega_1,\omega}$-semantics we don't have to worry about theories becoming unsatisfiable.)* As usual, this preorder induces an equivalence relation $\approx$ and a corresponding poset $\mathcal{Plaus}$ of **plausibility degrees**.

I'm generally interested in the "shape" of $\mathcal{Plaus}$, but in particular the following question seems particularly natural:

> **Question.** Does $\mathcal{Plaus}$ have *coatoms*?

(See below for a proof that $\mathcal{Plaus}$ does have a top element.) I strongly suspect that the answer is negative, but I don't see how to prove that.

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Here are some easy observations:

 - The already-satisfiable plausible theories constitute the least degree $\mathbf 0$, and there is a greatest degree $\mathbf 1$ as well: we can whip up a plausible theory $T_{\mathit{max}}$ describing a structure which (*i*) is a countable linear order and (*ii*) has each countable ordinal as an initial segment, and in order to make $T_{\mathit{max}}$ satisfiable we have to make $\omega_1$ countable. 

 - There are also intermediate degrees. For example, we can whip up a plausible theory describing $(\omega;<)$ equipped with a unary predicate which does not correspond to any constructible real, which becomes satisfiable exactly when we add a non-constructible real. 

 - The "$\omega$-with-a-predicate" trick can be extended to reasonably-simple forcing notions to get a lot more examples — e.g. there are plausible theories corresponding to the existence of a sufficiently Cohen generic real and to the existence of a sufficiently Sacks generic real, the pair of which show that $\trianglelefteq$ is not total and the latter of which shows that $\mathcal{Plaus}$ has atoms (= minimal nonzero degrees). It's also not hard to show that there are exactly $\omega_1$-many plausibility degrees.

Unfortunately, none of this seems particularly relevant to the coatom question (except the existence of $\mathbf 1$ which is needed to pose it in the first place).