Is there a large-cardinal completeness theorem for $L$?

Question

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?

Answer 1

On the one hand, we cannot easily describe any such list of axioms, since if $\Lambda$ was computably enumerable, then by your third bullet point, the arithmetic consequences of ZFC+(V=L)+$\Lambda$ would be a c.e. completion of PA, contrary to the incompleteness theorem.

On the other hand, meanwhile, we can as a purely formal matter identify a list of axioms $\Lambda$ with your three properties. Let $\varphi_0,\varphi_1,\ldots$ enumerate any particular consistent completion of ZFC+V=L. Let $\mu_n=\varphi_0\wedge\cdots\wedge\varphi_n$, and take $\Lambda$ to include the statements $\mu_n$. These are all consistent with ZFC+(V=L); it is easy to see that $n<m\implies \mu_m\vdash\mu_n$; and they form a maximal consistent first order theory. So they have all three of your desired properties.

Probably someone would object that these are not large cardinal axioms. I reply that we don't actually have a mathematical concept of "large cardinal axiom", and the matter seems to be subjective. Nevertheless, I can make the example more large-cardinal-like, as follows. Assume that the enumerated theory includes the assertions that there are at least $n$ inaccessible cardinals, for each natural number $n$. Now, let $\mu_n^*=\mu_n+$ there are at least $n$ inaccessible cardinals. This is now at least a little large-cardinal-like, and still has all three of your properties.

So the question will come down to: what counts as a reasonable large cardinal axiom? This is no longer really a mathematical question, and my expectation is that there will be no satisfactory answer.