# 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 $\begin{array} { l l l l l l } 3 \times 2^0, & 3 \times 2^1 , & 3 \times 2^2, & \ldots, \\ 5 \times 2^0, & 5 \times 2^1 , & 5 \times 2^2, & \ldots, \\ 7\times 2^0, & 7 \times 2^1 , & 7 \times 2^2, & \ldots, \\ 9 \times 2^0, & 9 \times 2^1 , & 9 \times 2^2, & \ldots, \\ \vdots& \\ \ldots, & 1 \times 2^2, & 1 \times 2^1 , & 1 \times 2^0. \end{array}$

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