I cannot currently find the original, but if memory serves, Goedel once speculated that there might be a "large-cardinal completeness theorem for $V.$" This theorem would state:

***Theorem.** For every first-order sentence in the language of set theory, that sentence is decided by $\mathrm{ZFC}+\lambda$ for some large cardinal axiom $\lambda$.

Anyway, I think the plausibility of such a theorem has been steadily declining for many decades now, since even our strongest large cardinal axioms cannot decide $\mathrm{CH}$ if they're consistent. Nonetheless, I wonder if there isn't a large-cardinal completeness theorem for Goedel's constructible universe $L.$

Question.Is there a reasonable definition of "large cardinal axiom for $L$" such that the following hold?

- Every large cardinal axiom for $L$ is consistent with $\mathrm{ZFC}+(V=L).$
- Letting $\Lambda$ denote the set of all large cardinal axioms for $L$, there exists a linearly ordered set $(\Lambda,\leq)$ such that $\lambda \leq \mu$ implies that $\mathrm{ZFC}+(V=L)+\mu \vdash \lambda$ for all $\lambda,\mu \in \Lambda$.
- $\mathrm{ZFC}+(V=L)+\Lambda$ is a maximal consistent first-order theory.

Alternatively, is there any reason to think that a "large-cardinal completeness theorem for $L$" cannot exist?