Notes on Laver Tables

TL;DR

This work sharpens the lower bounds for the rapid growth function $F(n)$ arising from large cardinal Laver tables, proving that $F(n)$ dominates every computable function whose totality is PA-provable. It develops a rich combinatorial framework of basic Laver patterns, realizations, and a hierarchy of operations (including $\mathrm{del}$, $\mathrm{Copied}$, $M$, and $E$) to construct increasingly complex patterns and bounds. By linking these constructions to ordinal analysis via patterns up to $\boldsymbol{\varepsilon_0}$ and the Steinhaus–Moser hierarchy, the paper shows how PA-provable limits can be surpassed and connects to Hardy/Weiermann-type hierarchies. Additionally, it introduces Laver Table Yarn (LTY) to weave multiple Laver tables together and outlines future work on ranks, heights, and computable embeddings with potential implications for foundational set theory and beyond.

Abstract

We present some new lower bound estimates for certain numbers in Laver table theory and introduce several related structures of interest.

Figures (25)

• Figure 1: $\theta_1$, $\theta_2$, $\theta_3$ are 3-lined; and $\theta_2$, $\theta_3$, $\theta_4$ are 2-lined.
• Figure 2: Result of applying $l'_2$ to the other embeddings.
• Figure 3: Representation of the conditions in Lemma \ref{['aaab']} for $m=3$.
• Figure 4: Result of applying the last row's embedding to the previous rows.
• Figure 5: A basic Laver pattern with $s = ((0, 1, 2), \dots)$ and $\ell = (1, 1, 2, 2)$.

Theorems & Definitions (90)

• Theorem 1.1
• Proposition 1.2
• Proposition 1.3
• Lemma 2.1
• Definition 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5: Dougherty