I'm having some difficulty making this question precise, so bear with me. I have read that some "small objects" can have consequences on the larger set theoretic universe, such as showing that certain kinds of large cardinals exist, but I don't understand how this process works.
I can see how you can have constructions on the natural numbers that have high consistency strength (i.e. a subset of $\Bbb N$ that is a countable model of ZFC), but can this process be "reversed" to actually get the large cardinal? For example, I might know that ZFC + an inaccessible cardinal is consistent given a countable model of such, but I don't think that implies that $V$ contains an inaccessible cardinal.
I often see $0^\#$ discussed in these terms, but I'm utterly baffled by that definition and it's not clear to me whether you can actually extract large cardinals from it.