5 ms·
Yes, it's true. Each ordinal appears at the day with its number; e.g., omega (first infinite ordinal, kinda corresponds to aleph-zero) appears on day omega, 2^
by gjm11 5y ago
Yes, it's true.
Each ordinal appears at the day with its number; e.g., omega (first infinite ordinal, kinda corresponds to aleph-zero) appears on day omega, 2^omega appears on day 2^omega, etc., etc. There is a day for each ordinal.
There are whatever large cardinals the underlying set theory has. (Though Conway took the view that rather than building the apparatus of surreal numbers on top of a particular set theory like ZFC, you should think of this as its own theory that resembles set theory but has two different kinds of "set membership" (left and right) and someone should prove a theorem saying that such things are always at least as consistent as ZFC...)
I don't think large cardinals play any particular role in the theory of surreal numbers, games, etc.
- skissane 5y ago> 2^omega appears on day 2^omega, etc., etc. There is a day for each ordinal. 2^omega = omega [0] Did you meant to say that omega_1 (the smallest uncountable ordinal number [1]) appears on day aleph_1 (which is the same as day 2^aleph_0 = beth_1 if we assume CH) [0] https://en.wikipedia.org/wiki/Ordinal_arithmetic#Exponentiation https://en.wikipedia.org/wiki/Ordinal_arithmetic#Exponentiat... [1] https://en.wikipedia.org/wiki/First_uncountable_ordinal https://en.wikipedia.org/wiki/First_uncountable_ordinal
- drdeca 5y agoIf you are identifying cardinals with the first ordinal of that cardinality, such that alepha_1 = omega_1, then yes, omega_1 is born on day omega_1 . For each ordinal \alpha, \alpha is born on day \alpha .
- drdeca 5y agoTypo : s/alepha/aleph/ (didn’t realize until today)
- gjm11 5y agoD'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 agoIt 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 agoI 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.