Laver table

A Laver table is a table of integers that gives 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:

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

(The latter condition could be interpreted to mean "self-distributive.")

The periodicity of the function \(a \mapsto 1 \star a\) is a function of \(n\), which we will denote as \(p(n)\). Outputs of \(p 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, some of the largest cardinals known.

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. The first few values of \(p^{-1}\) are \(0, 2, 3, 5, 6\). The existence of \(p^{-1}(5)\) has not even been confirmed, but if it does, it is at least \(A(9,A(8,A(8,255)))\). It is probably reasonable to say that \(p^{-1}(6)\) is far greater than TREE(3), SCG(13), or even Loader's number.