Philosophy is fun too!

so couldn’t any set become interesting (in the common sense) just as soon as someone becomes interested? e.g. by developing a theorem that incorporates it?

Yes, I think so, especially if the theorem is itself profound somehow. It doesn’t even have to be in a theorem, if it has a really simple definition that non-obviously leads to a finite set that would be enough for me in this question.

it doesn’t feel like it adds anything to qualify an expression as mathematically interesting because it’s the creativity of the mathematician that makes the interest.

I’m reminded of an argument I heard once that there are no uninteresting positive reals, because being the smallest such number would itself be interesting. That seems faulty to me, it just means it’s a set with no minimal element, which can exist even in an interval. The infimum would have to be 0.

maybe it’s just a social ask: “have it be interesting to you, or else keep it to yourself”, heh.

Stepping into linguistics for a moment, have you heard of pragmatics? Not sharing irrelevant information is a common unwritten rule in conversations. I specified it here because someone might incorrectly assume I’m unaware you can build arbitrary sets of natural numbers.