Why is not some numbers in nth days?

Question

In surreal numbers : In second day we have 2 and 1/2 and ... but why in third day we don't have {1/2|2} =5/4 ?

Answer 1

The surreal number $\{A\mid B\}$ is the first-born number $x$ for which $a<x$ for all $a\in A$ and $x<b$ for all $b\in B$. That is, while there will be many surreal numbers filling that cut, provided $a<b$ for all $a\in A$ and $b\in B$, nevertheless there is exactly one first-born such surreal number filling the cut (since for any two surreals born on the same day, there is an earlier-born surreal number between them).

Thus, $\{\frac12\mid 2\}=1$, since $1$ is the first-born number that is larger than $\frac12$ and less than $2$.