Do surreal numbers contain all other number systems?

Question

I know that some operations are defined for surreal numbers like addition or multiplication.

Are all operations for all "number systems" defined for surreal numbers in a way that they give same result as respective operations defined for those "number systems"?

Welcome to Mathematics Stack Exchange. Consider $p$-adic numbers — J. W. Tanner, May 24 '20 at 16:31
@J.W.Tanner Can't you define "closeness" and absolute value in two different ways on surreal numbers? Like "real closeness" and "p-adic closeness" and have surreal numbers still cover both real and p-addic systems? Would you be so kind to upvote my question? — Kamil Szot, May 24 '20 at 16:38
"$p$-adic closeness" is defined on rational numbers, not (sur)real numbers — J. W. Tanner, May 24 '20 at 16:42

score 3 · Accepted Answer · answered May 24 '20 at 16:39

No, surreal numbers do not contain all other number systems. They contain real numbers, which are a completion of rational numbers using the ordinary absolute value as a metric, but not $p$-adic numbers, which are a completion of rational numbers using the $p$-adic metric, where numbers are smaller if they are divisible by a higher power of $p$.

Do surreal numbers contain all other number systems?

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