As a sort of follow up question to a previous question found here, besides the Liouville numbers, are there any other uncountable collections of transcendental numbers that are known? Clearly you could union the Liouville numbers with some other transcendental number and get an uncountable collection. I am seeking to find a distinct set from the Liouville numbers.
There are many examples of countable collections of transcendental numbers:
$e^a$ if $a$ is algebraic and nonzero (by the Lindemann–Weierstrass theorem).
$a^b$ where $a$ is algebraic but not 0 or 1, and $b$ is irrational algebraic (by the Gelfond–Schneider theorem)
$\sin(a), \cos(a)$ and $\tan(a),$ for any nonzero algebraic number $a$ (again by the Lindemann–Weierstrass theorem)
But, as all of these are based on the algebraic numbers, none of these collections are uncountable.