Critical points in an algebra of elementary embeddings

Abstract

Given two elementary embeddings from the collection of sets of rank less than $λ$ to itself, one can combine them to obtain another such embedding in two ways: by composition, and by applying one to (initial segments of) the other. Hence, a single such nontrivial embedding $j$ generates an algebra of embeddings via these two operations, which satisfies certain laws (for example, application distributes over both composition and application). Laver has shown, among other things, that this algebra is free on one generator with respect to these laws. The set of critical points of members of this algebra is the subject of this paper. This set contains the critical point $κ_0$ of $j$, as well as all of the other ordinals $κ_n$ in the critical sequence of $j$ (defined by $κ_{n+1} = j(κ_n)$). But the set includes many other ordinals as well. The main result of this paper is that the number of critical points below $κ_n$ (which has been shown to be finite by Laver and Steel) grows so quickly with $n$ that it dominates any primitive recursive function. In fact, it grows faster than the Ackermann function, and even faster than a slow iterate of the Ackermann function. Further results show that, even just below $κ_4$, one can find so many critical points that the number is only expressible using fast-growing hierarchies of iterated functions (six levels of iteration beyond exponentials).