Clearly the artist believes in the axiom of choice

Not an expert but I’d guess that is Tycho crater, and the stripes are called its ray system https://en.wikipedia.org/wiki/Ray_system

Nice work!

I always found it funny to read it as Sharon, and now you’re telling me it’s named after someone called Charlene? This keeps getting better

You could say something like “the image of exponentiation over…” to mean the set of values created by applying the function once, but it sounds slightly clunky.

Looks like there aren’t really very many sets of mostly transcendental numbers that have names. Computational numbers and periods are two of them, I’d guess that both probably contain your set, so you could compare with those to see where it gets you.

Fun question! I don’t know the answer other than to say it’s not just the algebraics because of the Gelfond-Schneider constant

Are you sure this is well-defined? You say that a and b are algebraic but “closure” implies that they could also be any members of S. This might mess up your proof that it’s not all the reals if you do mean the closure.