Large ordinals (Q675870)

\(j:(V_\lambda,\in)\longrightarrow(V_\lambda,\in)\) is a non-trivial elementary embedding and \(\kappa\) is its critical point. \(\mathcal A\) is the closure of \(\{j\}\) under \(\cdot\), the multiplication of elementary embeddings introduced by Laver. This multiplication satisfies a left distributive law and Laver proved that \(\mathcal A\) is the free left distributive algebra on one generator. \(\Gamma=\{a(\kappa): a\in{\mathcal A}\}\) is the set of all critical points of all \(a\in{\mathcal A}\). The set \(\Sigma\) of \textit{simple ordinals} is the closure of \(\Gamma\) under the operation \(a''\alpha=\text{sup}\{a(\xi):\xi <\alpha\}\), and \(\Omega\) is the closure of \(\Gamma\) under \(a^-\alpha=\text{min}\{\xi:a(\xi)\geq\alpha\}\) for all \(a\in\mathcal A\). Laver has conjectured that \(\Sigma=\Omega\). The author proves this equality under an assumption on cyclic left distributive algebras called the Threshold Hypothesis (TH). Using TH the author gives a complete description of \(\Omega\). The paper ends with some numerical evidence for TH.