Laver table

A Laver table is a table of integers that give rise to a fast-growing function.

For \(n \geq 0\), a size-n Laver table is a binary operator \(\star\) over \(\mathbb{Z}_{2^n}\), with the following properties:

\[a \star 1 = a + 1 \pmod 2^n\] \[a \star (b \star c) = (a \star b) \star (a \star c)\]

(That is, \(\star\) is "distributive over itself.")

The periodicity of the function \(a \mapsto 1 \star a\) is a function of \(n\), which we will denote as \(p(n)\). The first few values of \(p(n)\) are \(1, 1, 2, 4, 4, 8, 16, 16, 16, 16, \ldots\), a slow-growing function. \(p\) might be divergent, but this relies on the existence of rank-into-rank cardinals; the divergence of \(p\) is unprovable in ZFC+I0.

If we let \(p^{-1}\) enumerate the points at which \(p\) changes, we get a fast-growing function that is total iff \(p\) is divergent.