> Take all the numbers you have so far. Then, “spawn” a new number in every gap between the numbers you already have (crucially, “to the left of all” and “to the right of all” also count as “gaps”). Apply this step forevermore, and you’ll get surreal numbers.

I'm not a mathematician. Can someone explain to me how this approach gets you beyond the rational numbers?

Also, this was formatted as a blockquote, but as far as I can see, this blog post is the only instance of this formulation online.