The Cardinality of the Surreal Numbers and Games
By John Thompson
I've been learning about John H Conway's surreal numbers, which are a rich superset of the reals. It's clear that the real numbers are dense in the middle section of the surreals, with the surreals extending much further toward the left and right.
I'm wondering about the cardinality of the surreals. Can we put the surreal numbers into a relationship with the power set of the reals, for instance?
The surreals form a proper subset of another JH Conway discovery, the games. Is anyone aware of the cardinality of these games?
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$\begingroup$They have no "cardinality" as they form a proper class: every ordinal number is also a surreal number. So they are not in 1-1 correspondence to any set, in ZFC. If you have a set theory with proper classes, the surreal numbers are probably as large as the whole universe.
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