> On the second day, there are two gaps: “between nothing and zero” and “between zero and nothing”. Two numbers spawn in those two gaps. Call them –1 and 1.

That's a mistake. They should've written "empty set of surreal numbers" and not "nothing".

It is a constructive theory, like sets/ordinals. For ordinals you can use ∅, { }, ∪ and you construct

  ∅, {∅}, {{∅},∅}, ... (von Neumann ordinals). 

For surreal numbers you use the form { A | B } where A and B are sets of surreal numbers. Some restrictions apply so not all of these forms will be surreal numbers.

You build up surreal numbers as

  {∅|∅}, {∅|{∅|∅}}, {{∅|∅}|∅}

and so on.

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Edit: A happy accident: I denoted the "empty set of surreal numbers" with the symbol ∅. It works, and it gives you the surreal numbers. But if you think of ∅ as the empty set in set theory, then the same construction (using an ordered-pair construction) gives you the surreal numbers as sets!