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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 ?

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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 $2$.

JDH
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  • Thanks , but in one article about surreal numbers I read x={xL|xR} then x is average of xL and xR. Is this wrong?! –  Jun 14 '17 at 21:37
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    Yes, that is wrong. It is true in many natural cases, however, and I think one can prove that it is true when there isn't another better way to describe the cut. – JDH Jun 14 '17 at 21:40
  • Thanks so much.. –  Jun 14 '17 at 21:56