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D'oh! Of course you're right. What I should have said: The first ordinal of cardinality 2^aleph_0 -- call it alpha -- appears on day alpha. (But I definitely d

by gjm11 5y ago

D'oh! Of course you're right. What I should have said: The first ordinal of cardinality 2^aleph_0 -- call it alpha -- appears on day alpha. (But I definitely didn't mean to say that anything appears "on day aleph-1"; day numbers are ordinals.)

skissane 5y ago
It would be better to say omega-1 appears on day omega-1, and leave the cardinality of power sets (2^x) out of it. What you are trying to say does not depend on the continuum hypothesis, so it is better to choose a way of saying it which does not depend on the continuum hypothesis either.
- gjm11 5y ago
  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.

by gjm11 5y ago

skissane 5y ago

It would be better to say omega-1 appears on day omega-1, and leave the cardinality of power sets (2^x) out of it. What you are trying to say does not depend on the continuum hypothesis, so it is better to choose a way of saying it which does not depend on the continuum hypothesis either.

gjm11 5y ago
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.