Large ordinals

Abstract

Let $j$ be an elementary embedding of $V_λ$ into $V_λ$ that is not the identity, and let $κ$ be the critical point of $j$. Let $\Cal A$ be the closure of $\{j\}$ under the operation $a (b)$ of application, and let $Ω$ be the closure of $\{κ\}$ under the operation $\min \{ξ: a(ξ) \ge b(α)\}$. We give a complete description of the set $Ω$ under an assumption (Threshold Hypothesis) on cyclic left distributive algebras.