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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