How similar are large cardinals, over $L$?

Question

EDIT: Joel's answer shows that no $\Sigma_2$ large cardinal property will do the job - however, $\Pi_2$ properties (such as unfoldability and its relatives) may still be useful.

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Throughout this question, I'm working in "$V=L$". So when I say "large cardinal," I mean "small large cardinal".

Informally, I want to ask: How much of the "$L$-theory" of a cardinal does a large cardinal property imply? That is, if I know that $\alpha$ and $\beta$ both have some large cardinal property $(*)$, what sorts of (not necessarily first-order) sentences do I know that $L_\alpha$ and $L_\beta$ agree about?

Formally, fix an ordinal $\gamma$. Say that a large cardinal property $(*)$ is "$\gamma$-decisive" if whenever $\alpha, \beta$ have property $(*)$, we have $$L_{\alpha+\gamma}\equiv L_{\beta+\gamma}.$$ My question is:

What are some examples of $\gamma$-decisive large cardinal properties?

(CAVEAT: Obviously any large cardinal property can be arbitrarily decisive: suppose it has no instances, or exactly one! So really I want to know about large cardinal properties whose decisiveness can be proved in ZFC+V=L, or some strengthening thereof that does not limit the number of relevant large cardinals.)

Note that decisiveness looks a lot (at least to me) like indescribability; however, indescribability doesn't quite seem to do the job, unless I'm missing something.

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I'm also interested in decisiveness with parameters - e.g. for a finite tuple of sets $\overline{c}\in L$, say that a large cardinal property $(*)$ is $\gamma$-decisive over $\overline{c}$ if whenever $\alpha,\beta$ have property $(*)$ with $\overline{c}\in L_\alpha\cap L_\beta$, we have $$(L_\alpha, \overline{c})\equiv(L_\beta,\overline{c}).$$ But understanding the parameter-free version should come first, and besides, I imagine the situations aren't too different.

By contrast, something potentially interesting happens if we allow, not tuples of elements of $L$, but tuples $\overline{C}$ of subsets of $L$ which are in $V$ but not necessarily $L$ - coded as unary predicates, of course. A really overkill example of this would be the $0^\#$ as a unary predicate, but already tamer examples could be interesting. For instance, suppose $\overline{C}$ is a finite sequence of mutually Cohen reals over $L$ - then what are some $\gamma$-decisive over $\overline{C}$ large cardinal properties? But this question seems much broader, and much harder, so I'll merely mention it as a curiosity. (Note that in this context, detrivializing things becomes a bit trickier, so this is a more informal question.)

Answer 1

Updated answer. I claim that none of the familiar large cardinal notions consistent with $V=L$ are provably $\gamma$-decisive for any $\gamma$. This includes the cases of wordly cardinals, inaccessible cardinals, uplifting cardinals, Mahlo cardinals weakly compact cardinals, $\Pi^n_m$-indescribable cardinals, totally indescribable cardinals, unfoldable cardinals and all the others.

First, as a warm-up, let's handle the case of $\Sigma_2$-definable large cardinal notions.

Theorem. If a large cardinal notion LC is $\Sigma_2$-definable and there are a proper class of them in $L$, then there is no $\gamma$ for which they are $\gamma$-decisive.

Proof. To see this, fix any $\gamma$, and let $\kappa$ be the least LC cardinal above $\gamma$. Note that both $\kappa$ and $\gamma$ are definable in $L_{\kappa+\gamma}$, since $\kappa$ is the largest cardinal of this structure, and $\gamma$ is least such that $\kappa+\gamma$ does not exist. Further, since $\kappa$ was the least LC cardinal above $\gamma$, it follows that $L_{\kappa+\gamma}$ will agree that there are no such large cardinals of that type in the interval $(\gamma,\kappa)$, since $\Sigma_2$ definitions express locally verifiable properties (see my blog post about Local properties in set theory). But the corresponding fact will not be true in $L_{\delta+\gamma}$ for some other much larger LC cardinal $\delta$, since once $\delta$ is large enough, then $L_{\delta+\gamma}$ will think that $\kappa$ is an LC cardinal. So $L_{\kappa+\gamma}$ is not elementarily equivalent to all $L_{\delta+\gamma}$, and so the LC large cardinal notion is not $\gamma$-decisive in this situation. QED

Next, with a slightly stronger assumption on consistency, we can handle the $\Pi_2$-definable notions.

Theorem. Assume that a large cardinal notion LC is $\Pi_2$-definable and there are a sufficiently stationary proper class of them in $L$, then there is no $\gamma$ for which they are $\gamma$-decisive.

Proof. Fix any $\gamma$, and assume that the class of LC cardinals in $L$ meets every $\Pi_2$-definable class club. Let $C$ be the club of $\Sigma_2$-correct cardinals $\delta$, those for which $V_\delta\prec_{\Sigma_2} V$. This is $\Pi_2$-definable. Let $\kappa$ be the least LC cardinal in $C$ above $\gamma$. Note that both $\kappa$ and $\gamma$ are definable in $L_{\kappa+\gamma}$. Further, since $\kappa$ is $\Sigma_2$-correct, it must be a beth-fixed-point, and so $L_\kappa=(V_\kappa)^L$. Since $L_\kappa$ is $\Sigma_2$-correct and hence also $\Pi_2$-correct, it is correct about the LC cardinals below $\kappa$. And it is also correct about the $\Sigma_2$-cardinals below $\kappa$. So by the minimality of $\kappa$, the structure $L_\kappa$ can see that there is no $\Sigma_2$-correct LC cardinal below $\kappa$ and above $\gamma$, and this is part of the theory of $L_{\kappa+\gamma}$. But now, if $\delta$ is a larger LC cardinal in $C$, then $L_{\delta+\gamma}$ will be able to see that there is a $\Sigma_2$-correct LC cardinal below $\delta$. And so $L_{\kappa+\gamma}$ and $L_{\delta+\gamma}$ do not have the same theory. So the LC cardinals are not $\gamma$-decisive. QED

Finally, we can use this idea to push through an argument for any first-order definable large cardinal notion.

Theorem. No first-order definable large cardinal notion LC is provably $\gamma$-decisive for any $\gamma$, if it is consistent that there is a stationary proper class of LC cardinals in $L$.

Proof. Suppose that the LC large cardinal notion is $\Sigma_n$-definable, and the class of LC cardinals in $L$ is stationary with respect to all $\Pi_n$-definable class clubs. Fix any $\gamma$, and let $\kappa$ be the least $\Sigma_n$-correct LC cardinal above $\gamma$. There is such a $\kappa$ because the class of $\Sigma_n$-correct ordinals is a $\Pi_n$-definable class club. Both $\kappa$ and $\gamma$ are definable in $L_{\kappa+\gamma}$, and $L_\kappa$ is correct about LC cardinals and about $\Sigma_n$-correctness. So $L_\kappa$ will see no $\Sigma_n$-correct LC cardinals above $\gamma$. But if we use a much larger $\Sigma_n$-correct LC cardinal $\delta$, then $L_\delta$ will see that $\kappa$ is a $\Sigma_n$-correct LC cardinal above $\gamma$. So $L_{\kappa+\gamma}$ and $L_{\delta+\gamma}$ will have different theories. QED