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It is my understanding that the surreal numbers form a class rather than a set, because their collection is larger than any set. Thus it would seem to follow that for any cardinality, such as $\aleph_n$ or $\beth_n$ for a fixed $n$, a set of surreal numbers can be defined with that cardinality. Is explicitly defining such a set something that can be easily done?

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Yes: Since every ordinal is a surreal number, the sets you're looking for can be taken to be the initial ordinals that represent $\aleph_n$, $\beth_n$, and so forth.

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