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 \[3, 5, 7, \cdots, 3 \cdot 2, 5 \cdot 2, 7\cdot 2, \cdots, \] \[3 \cdot 2^2, 5 \cdot 2^2, 7 \cdot 2^2, \cdots, 3\cdot 2^3, \cdots, 2^3, 2^2, 2^1, 2^0\] where \(a\le b\) if \(a\) appears before \(b\) in the sequence or \(a = b\).

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