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.

It can feel like that, though I’m sure it’s (mostly) not deliberate. Also the sudden jump from straightforward to incomprehensible, accompanied by a comment from the author along the lines of “well duh”

All maths papers are like this

I thought that too as there is 1 planet too many, but now I think Pluto is the extra. The one between Mars and Neptune is Mercury

If you haven’t seen them you’ve been lucky. They are definitely out there

Reminds me of this: https://en.m.wikipedia.org/wiki/Benford’s_law