Hello,
I looked through MathOverflow's existing entries but couldn't find a satisfactory answer to the following question:
What is the relationship between No, Conway's class of surreal numbers, and V, the Von Neumann set-theoretical universe?
In particular, does V contain all the surreal numbers? If so, then is there a characterization of the surreal numbers as sets in V? And does No contain large cardinals?
I came across surreal numbers recently, but was surprised by the seeming lack of discussion of their relationship to traditional set theory. If they are a subclass of V, then I suppose that could explain why so few people are studying them.
Thank you,
Alex
Best Answer
Yes, every surreal number is also an element of $V$ (at least, once you choose some method of encoding ordered pairs of sets as sets). The (highly recursive) characterization of which elements of $V$ are surreal numbers is precisely the definition of the surreal numbers: an element of $V$ is a surreal number just in case it is an ordered pair $(L, R)$ of sets of surreal numbers, with every element of $L$ being less than every element of $R$, with the definition of the ordering being, etc., etc. (Note, however, that this particular embedding of the surreal numbers into $V$ doesn't respect surreal number equivalence; it depends on the particular selection of $L$ and $R$ sets defining the surreal number.)
Conversely, while not every element of $V$ is a surreal number, there is at least a natural way of encoding every ordinal (and thus every cardinal, under the usual Axiom of Choice-based identification of cardinals with their initial ordinals) as a surreal number: the encoding of the ordinal $\alpha$ is the surreal number specified with an empty right set and a left set consisting of the encodings of all the ordinals less than $\alpha$. Thus, if there are large cardinals in $V$, there are corresponding elements in $No$.