I didn't say anything that depends on the continuum hypothesis. (I mean, I guess exactly which ordinal I'm referring to when I say "the first ordinal of size 2^

I didn't say anything that depends on the continuum hypothesis. (I mean, I guess exactly which ordinal I'm referring to when I say "the first ordinal of size 2^aleph-0" depends on CH, but unless I'm confused I didn't say anything whose truth depends on CH.) But your original question gave the impression that you were a bit fuzzy on the difference between ordinals and cardinals, and I thought "thing the size of the reals" a more friendly notion than "first uncountable ordinal". Of course, I then goofed, so maybe now you think I am a bit fuzzy on the difference between ordinals and cardinals :-). So, to be quite precise, and hopefully correct all errors and apparent errors so far: If you construct the surreals "day by day" then each ordinal alpha is born on day alpha. Every ordinal appears; no two ordinals are equal. Cardinals play no special role in the surreal numbers, so far as I am aware. There are whatever large cardinals there are in the underlying set theory; these too play no particular role in the surreal numbers, so far as I am aware. (Though I assume that e.g. inaccessible cardinals give you set models of the surreal numbers, just as they give you set models of ZFC.) It is true that 2^omega appears on day 2^omega but since 2^omega=omega this is just the same as saying that omega appears on day omega.