Forum:Laver tables

The latest post from our favorite blog alerted me to an interesting function based on. I'll repeat the definition here:


 * For nonnegative integers \(n\), define \(L_n(a, 1) = a + 1 \pmod{2^n}\) and \(L_n(a, L_n(b, c)) = L_n(L_n(a, b), L_n(a, c))\). \(p(n)\) is period of the sequence defined as \(a_0 = 1\) and \(a_{k + 1} \mapsto L_n(1, a_k)\).

The first few values of \(p(n)\) are 1, 1, 2, 4, 4, 8, 8, 8, 8, 16, 16, 16, 16, ... Since we're googologists, we'll invert this for a fast-growing function: let \(p^{-1}\) enumerate the values of \(n\) where \(p(n + 1) > p(n)\), starting with \(p^{-1}(0) = 0\).

Apparently \(p\) is divergent (and therefore \(p^{-1}\) is total) iff there exists a rank-into-rank cardinal, a class of large cardinal of truly gigantic proportions. Any thoughts? FB100Z &bull; talk &bull; contribs 07:38, December 19, 2013 (UTC)