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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?

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up vote 9 down vote accepted

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.

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Surely Gödel of all people must have been aware of this, so presumably he had in mind something else than an r.e. list of axioms? –  Emil Jeřábek Jun 3 at 16:08
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I believe the OP is taking liberties with and exaggerating Goedel's view. He wasn't aiming at a complete theory, but rather speculating that some of the big open questions, such as CH, might be settled by axioms that are abundant and unifying in their consequences, and he pointed to the large cardinal axioms as natural candidates. –  Joel David Hamkins Jun 3 at 16:12
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See Goedel's remarks at books.google.com/…. –  Joel David Hamkins Jun 3 at 16:13
    
If we want a mathematical theorem about completeness of ZF+(V=L)+LC, how could it be stated unless LC is r.e.? –  Monroe Eskew Jun 3 at 16:17
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@MonroeEskew, in much the same way that $\mathrm{Th}(\mathbb{N},0,+,\times)$ can be proven (in ZFC) to be a maximal consistent set of sentences, despite that it is not even arithmetically definable! –  goblin Jun 3 at 16:21

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