II.5b: Structure of the Surreal Numbers

In addition to ONAG, some of the claims in this post are coming from Foundations of Analysis over Surreal Number Fields.

Ordinals
An “ordinal” is a Number equivalent to one with no right options. There are a few basic facts you should know about ordinal numbers.

  • If G is an ordinal, G={ordinals H<G |} (in particular, with this set up the collection of left options really is a set if we only pick equivalence class representatives of this standard form).
  • Any non-empty class (for example, a nonempty set) of ordinals has a least ordinal.
  • For any set of ordinals S, {S|} is an ordinal bigger than any member of S.
  • The nonnegative integers are ordinals, and ω={0,1,2,…|} is the next biggest ordinal after those.

If you happen to have a sense of ordinal arithmetic already, Surreal Number arithmetic works differently, in general.

Birthdays
With the ordinals in hand, we can give names to the steps we take to build all the Numbers (in fact, all the Conway positions). Before we have any positions, there’s nothing we can assign to be left or right options, so we say “on day 0, 0={|} was born”. Then on day 1, 1={0|} and -1={|0} (and *={0|0}) were born, etc. In general: the Numbers (or Conway positions) made by day α (α being an ordinal) are defined to be all the Numbers you get where left options and right options were “made by day β” for β<α. With this setup, each Number (or Conway position) has a “birthday”: the first ordinal α such that it was “made by day α”.

By convention, we usually only worry about the equivalence class of a position: {*|*}=0, so we’d usually say that {*|*} was born on day 0, not day 2. Also, as a consequence of the simplicity rule from II.5, a Number is equal to the unique one of earliest birthday that fits between the sets of left and right options.

Infinite days and real Numbers
Dyadic fractions were born on finite days. What Numbers were born on day ω? Well, the infinite ω={0,1,2,…|}=”LLLL…” was born (and its negative). So was the infinitesimal 1/ω={0|1,1/2,1/4,…}=”LRRR…” (and its negative) and dyadic translations of it like 1+1/ω={1|1+1,1+1/2,1+1/4,…}.

The other things that were born on day ω are every real number that’s not a dyadic! Like 1/3={0,1/4,1/4+1/16,…|1/2,1/4+1/8,1/4+1/16+1/32,…} and π={3,3+1/8,3+1/8+1/64,…|4,3+1/2,3+1/4,3+1/8+1/16,…}. If you’re familiar with Dedekind cuts, you can probably figure out how to make an explicit definition of the reals this way (you can do it without the dyadic restriction, but then you won’t get all of the reals as early as possible).

On day ω+1 you get things like ω-1={0,1,2,…|ω}, 1/3+1/ω, and 1/(2ω)={0|1/ω}. If you want ω/2, you actually have to wait until day 2*ω; ω/2={0,1,2,…|ω,ω-1,ω-2,…}.

However, there’s an implicit definition of the reals that’s rather elegant. A Number G is said to be “real” if 1. There are some integers m and n such that m<G<n (so its value isn’t infinite) and 2. G={G-q|G+q} where q ranges over the positive rationals (or positive dyadics, if you prefer). ω isn’t a real number because it’s not less than any integer. 1/ω={0|1,1/2,1/4,…} isn’t a real number because 1/ω-1/n for any positive integer n is negative, so {G-q|G+q}=0 when G=1/ω.

Infinite Hackenstrings
The game of Hackenstrings actually covers all of the Surreal Numbers. But it’s a little complicated to see carefully. Given a Number G with birthday α, we can form the day-β approximation to it Gβ (for ordinal β≤α) by giving Gβ all Numbers born before day β that are less than G as left options, and similarly for the right options. Note that G0=0 for all Numbers G.

With these approximations, we can build a (possibly infinite; it’s really a function from the left set of some ordinal in standard form to {“L”,”R”}) Hackenstring equal to G in the following way. For each ordinal β less than the birthday of G, give the βth spot an “L” if G>Gβ, and an “R” otherwise. (We can’t ever have equality by the definition of “birthday” and day-β approximation.) Every one of these Hackenstrings is in a distinct equivalence class: to compare two distinct Hackenstrings, read them until the first spot they differ, and then “L”>blank>”R”.

Scattered Closing Remarks

There are lots of different equivalent (up to FIELD isomorphism, I guess) ways to axiomatize the Surreals. There are more technical versions of statements like “they’re the unique FIELD with all the ordered fields in it”, or “they’re the unique FIELD where ordinal birthdays make sense”. You can adjoin i and get the Surcomplex Numbers (which are algebraically closed). You can define things like infinite series and something like a limit and lots of other crazy things that could fill at least one whole book.

Everything in the FIELD On2 was equivalent to a Nim pile: basically a Hackenstring except all the letters were “E”s, which can be taken by either player. Everything in the FIELD of Surreals was equivalent to a Hackenstring with no “E”s, just “L”s and “R”s. You could mix them up and get weird games like “EL”, but there’s no nice arithmetic for those. And I must emphasize: not every Conway position is equivalent to a Hackenstring.

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