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Suppose $\kappa$ is a cardinal and we want to guess if $\kappa$ is a large cardinal, and if so what kind, by looking at the large cardinal status of a selection of cardinals below $\kappa$.

The selected cardinals are revealed and we guess the large cardinal nature of $\kappa$. For example, say $\omega$ many cardinals have been revealed and in the selection there are infinitely many inaccessible cardinals. We might guess that $\kappa$ is 1-inaccessible, a limit of inaccessible cardinals. But we could be wrong and $\omega$ many revelations is not necessarily a lot of information, so we might allow even $\kappa$ many cardinals to be revealed.

The cardinals to be revealed are selected somewhat randomly. But suppose we had a little control over where the selection comes from, we can guide it a little, would there be a strategy to help out our chances to make a good guess? What is the least amount of cardinals needed to make a good guess?

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    $\begingroup$ I mean, if your cardinal is required to be regular, then you need to guess a cofinal sequence; even then, to guess Mahlo you'd need to guess a stationary (or even a club) of points. So I don't see how you'd be able to guess with anything less than that. $\endgroup$
    – Asaf Karagila ♦
    Commented Aug 16, 2023 at 8:17
  • $\begingroup$ @AsafKaragila I think so too there need to be $\kappa$ many revealed, and even then do you think one could reveal $\kappa$ many and still hide what $\kappa$ is? Would there be a good "place" to take the selection from to get the best sample? $\endgroup$
    – Erin Carmody
    Commented Aug 16, 2023 at 8:22
  • $\begingroup$ A good place to start looking at is the limit cardinals bellow $κ$, as most reflecting principle will reflect to some stationary subset of those cardinals $\endgroup$
    – Holo
    Commented Aug 16, 2023 at 18:41
  • $\begingroup$ @Holo that's a good idea. Do you think it would be beneficial to (if you have to choose) reveal a solid block of cardinals so as not to miss anything (at least in that block) or that it would be better to reveal limit cardinals (if you had to choose between these two options)? $\endgroup$
    – Erin Carmody
    Commented Aug 16, 2023 at 19:23
  • $\begingroup$ If the block is unbounded, then certainly the block, otherwise the limits $\endgroup$
    – Holo
    Commented Aug 16, 2023 at 19:29

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