User blog comment:Edwin Shade/Surreals and the Nature of Number/@comment-1605058-20171215082950

As Hyp Cos mentions, \(*=\{\{|\}|\{|\}\}\) is not a surreal numbers - in the formal definition, it is required that surreals to the left of \(|\) must be all smaller than all the surreals on the right, whereas here we have \(\{|\}\) on both sides.

However, \(*\) is still a perfectly valid game. The class of all games also has some number-like properties, but other properties are not as intuitive, like the ordering. The ordering is a bit like the ordering on the set of complex numbers where \(a<b\) iff \(\mathrm{Re}\,a<\mathrm{Re}\,b\), and \(*\) behaves like \(i\). Note that the complex numbers don't carry any order at all, so this one is about as good as any other - what's crucial is that not all elements are comparable.

As for what should be considered a number - in my opinion we are best not defining the notion of a "number" and leaving it either context dependent or, preferably, not ever using the standalone term "number". There are many systems which bear the name "numbers" which behave in very different ways and are very much incompatible, for example the complex numbers, ordinal numbers, cardinal numbers, p-adic numbers, surreal numbers. I don't know of a single notion which would encompass all of the above together with all their structure.