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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    $\begingroup$ The surreal numbers are a proper class, so there are a lot of them. Indeed strictly speaking they are not a subset because they are bigger than any set $\endgroup$
    – Henry
    Feb 26 at 11:32
  • $\begingroup$ Here is a difficulty: The surreals are sometimes represented as a tree whose nodes are pairs of subsets of the rational numbers. This might naively indicate a correspondence between the surreals and the power set of the rationals, thus establishing an equivalence between the and the power set of the rationals. In other words, the cardinality of the surreals is Aleph1, just like the reals. Is this correct? $\endgroup$ Feb 26 at 11:39
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    $\begingroup$ I only heard of the hyper-real numbers which are usually represented as infinite sequences over the natural numbers. In this case, the cardinality is $2^{\aleph_0}$ (which is $\aleph_1$, if we accept the continuum hypothesis). This remains true, if the sequence must contain rational entries. I am not sure however in the case of infinite sequences over the reals. $\endgroup$
    – Peter
    Feb 26 at 11:52
  • $\begingroup$ Where you ask: "is this correct?", the answer is no. Each ordinal can be represented a surreal number, and the ordinals form a proper class, much larger than $\aleph_1$ $\endgroup$
    – Henry
    Feb 26 at 12:29
  • $\begingroup$ Assume there is a smallest ordinal $\beta$ not in the surreal numbers. Then $\{\alpha: \alpha <\beta \mid \}$ is a surreal number, but this is precisely $\beta$. So there is no such $\beta$. $\endgroup$
    – Henry
    Feb 26 at 12:36

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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