I've been thinking about numbers which have not yet been proven nor disproven to be transcendental, such as $e + \pi,\, \pi - e,\, \frac{\pi}{e},\, \gamma,\,\zeta(3),$ etc. Some of these numbers haven't even been proven to be irrational, so it naturally led to me questioning whether these numbers' transcendence could perhaps be independent of $\sf ZF$ or $\sf ZFC$. Do (or can) there exist numbers such that their transcendence or irrationality is independent of $\sf ZF$ or $\sf ZFC$?
I'm aware that this question may be unsolved, so relevent references would be appreciated as well!
Note that by number I mean one which is not defined conditionally using some other independent statement such as the continuum hypothesis or the axiom of choice. (Side question: what happens if we change this definition to demanding the number can be computed to arbitrary precision?)