Does set $\mathbb{R}^+$ include zero?

You will often find $ \mathbb R^+ $ for the positive reals, and $ \mathbb R^+_0 $ for the positive reals and the zero.

It depends on the choice of the person using the notation: sometimes it does, sometimes it doesn't. It is just a variant of the situation with $\mathbb N$, which half the world (the mistaken half!) considers to include zero.

I write, e.g., $\mathbb R_{>0}$, $\mathbb R_{\geq0}$, $\mathbb N_{>0}$.