How Many Well/Total Orders?

Let $A$ be the number of totally-ordered sets below, and let $B$ be the number of well-ordered sets below. What is $A+B$ ?

• $S = \mathbb{R}$ with the standard order.

• $S = 2^{\mathbb{R}}$ , the power set of $\mathbb{R}$ , ordered by set inclusion, i.e. $A \le B$ iff $A \subseteq B$ .

• $S = \mathbb{R}^2$ with the dictionary order.

• $S = \mathbb{N}$ , with the order given by $\begin{array} { l l l l l l } 3 \times 2^0, & 3 \times 2^1 , & 3 \times 2^2, & \ldots, \\ 5 \times 2^0, & 5 \times 2^1 , & 5 \times 2^2, & \ldots, \\ 7\times 2^0, & 7 \times 2^1 , & 7 \times 2^2, & \ldots, \\ 9 \times 2^0, & 9 \times 2^1 , & 9 \times 2^2, & \ldots, \\ \vdots& \\ \ldots, & 1 \times 2^2, & 1 \times 2^1 , & 1 \times 2^0. \end{array}$