Surreal Trees

Recall that the surreal numbers may be enumerated using infinite binary trees. After the finite ordinals $1$, $2$, $3$ and so on, we have the first infinite ordinal $\omega$. The surreals also contain an infinitesimal element $\epsilon = 1/ \omega$. Transfinite induction also introduces numbers of the form $\omega^{\omega}$ or $3 \omega^{\omega} + \omega^{7}$, so that the surreals contain an infinite number of infinite trees, where the usual reals are all contained in the first infinite tree. The collection of surreals is far too large to be considered a set, but logos theorists don't need to worry about that. They show that one can enumerate real numbers in a discrete fashion, using a binary tree.

The matrices that we consider in quantum gravity tend to have a restricted set of allowed entries, arising for instance as the characters of a finite field. In principle, a matrix with real or complex entries must be properly constructed from a matrix that is indexed by such continua. But clearly it is easier to index matrices by discrete lists.

The first splitting of the surreal binary tree splits the reals into positive and negative reals. We can thus consider only the positive reals as an infinite binary tree in its own right. This semiring (with an infinity $\omega$) is the domain of tropical geometry, where addition and multiplication are replaced by the binary operations of minimum and addition. The minimum function is idempotent, since the minimum min$(a,a)$ must be $a$ for an ordered set. Dually, one may consider the negative numbers with a maximum operation. Recall that in tropical geometry, lines look more like the trivalent vertices of a binary tree.

Since the surreals contain an infinite number of infinite binary trees, why not use one copy of the surreals to model the complex numbers? We just need a function of $\omega$ to act as our complex phase $i$. The surreals are a little tricky, because there are weird rules for $\omega$, like $2^{\omega} = \omega$. However, $\omega^{\omega}$ is not fixed under exponentiation, so we could consider a function that sends $\omega^{\omega}$ to $i$, so that the positive surreals cover a non standard set of complex numbers using a functorial multiplication rule.

And if this is not enough to index all the matrices of quantum gravity, there are always ternary and $n$-ary analogues of the surreals.