Surreal Numbers and Set Theory

23 07 2005

Here’s a more purely mathematically-oriented post.

John Conway has developed a class of entities he calls “surreal numbers”, which he describes in his book On Numbers and Games, and I believe in some simpler sources as well. He developed them originally to describe combinatorial games, but later noticed that they generalized both the von Neumann construction of ordinals and the Dedekind cut construction of reals. Each surreal number is an ordered pair of sets of surreal numbers. The simplest one has the empty set on both sides, and is treated as the number 0. We can define an ordering on the surreals recursively, saying that x>y if there is z in the left set of x such that z=y or z>y, and we say that x<y if there is z in the right set of x such that z=y or z<y. These orderings aren’t exactly duals of one another, and aren’t always transitive, but they turn out to generalize the standard notions. If we let the right set remain empty, and let the left set be {0,1,…,n}, then we get the number n+1. To get the negative natural numbers, we basically reverse the construction, letting the left set be empty and setting the right set as {0,-1,…,-n} to get -(n+1). We can generalize both of these constructions to the transfinite in the usual way, letting the positive transfinite ordinals be those whose left set contains all smaller ordinals and whose right set is empty, and dually for the negatives. We let two surreal numbers be equal if each has elements in its left set that are at least as large as any element in the left set of the other, and each has elements in the right set at least as small as any element in the right set of the other. Thus, each integer could be represented with a singleton instead of the standard set. We generate rationals whose denominator is a power of 2 by letting ({x},{y}) represent (x+y)/2, whenever x and y differ by 1/2^n for some n. Thus, 1/2=({0},{1}), 3/4=({1/2},{1}), and 17/64=({1/4},{9/32}). From these, we define all the real numbers by the Dedekind cut construction.

It turns out that one can define addition and multiplication in a way that preserves the expected structure. But in addition to all the reals and ordinals, we get a lot of weirder things. For instance, we get omega-1 as ({0,1,2,…},{omega}), since this is the simplest surreal greater than every natural but less than omega. We can use similar constructions to get omega-2, omega-3, and even things like omega/2 and the like. We can also get 1/omega as ({0},{1/2,1/4,1/8,1/16,…}) as the simplest number greater than 0 and less than every positive real. By even odder constructions we get the square root of omega, the omegath root of 2, and all sorts of other crazy structure. I believe the surreal numbers end up being a model of the theory of real numbers under addition, multiplication, and exponentiation, but with the cardinality of a proper class. (This ignores complications arising from surreal numbers like ({0},{0}) which is both greater and less than zero).

Now, whatever these surreal numbers are, the class of all of them is nicely definable. We let S_0={({},{})}, the set containing just 0. Then, we let S_(a+1) be the product of the powerset of S_a with itself, so that we get the set of all ordered pairs of sets of surreal numbers in S_a. We just take unions at limit stages. These sets are definable by transfinite recursion, so any model of ZF containing all the ordinals must contain all of these, assuming the powerset operation is the correct one in this model. Thus, the class S of all surreal numbers bears a resemblance to Gödel’s class L of all definable sets, which is the smallest model of ZF containing all the ordinals. An important object of study in contemporary set theory is the model L(R) which is the smallest model of ZF containing all ordinals and all reals. Since the reals and ordinals can easily be defined from the surreals, any model containing all the surreals must include L(R). But now I wonder if L(R) itself contains all the surreals. This would be a nice characterization of L(R), and might lead to some interesting results.




2 responses

24 07 2005

Looks like a gigantic portion of your post was cut out because you left a less-than sign in your first paragraph, which was treated as html in my browser.

If you edit your post and replace < with &lt; it will print out how you would expect, and all should be well.

24 07 2005
Kenny Easwaran

Thanks for warning me!

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