Given two real, nonzero algebraic numbers a and b, with a > 0 (so that it excludes complex numbers), is there any named subset of the reals S such that (a^b) belongs to S forall a,b? I know it’s not all the reals since there should be countably many a^b’s, since a,b are also countable.
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.
My mistake, in that case it’s not the closure what I mean. But then how are those kinds of sets called?