The Deranged Mathematician: Ultrafilters

What are instances of mathematical objects that are: 1) useful (in that they have been used to prove various interesting results), and 2) for which it is impossible to come up with even a single explicit example?

I know of one: non-principal ultrafilters, which this post is devoted to describing. (Briefly, they are used in topology and model theory, such as for constructing ultraproducts, which can be used to give a rigorous construction of infinitesimals.) If you know of others, please share!

Read the full post (for free) on Substack: [Ultrafilters](https://derangedmathematician.substack.com/p/ultrafilters)