Dougherty, R., Critical points in an algebra of elementary embeddings, Annals of Pure and Applied Logic 65 211-241.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 the other. Hence, a single such nontrivial embedding j generates an algebra of embeddings via these two operations, which satisfies certain laws . 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 κ0 of j, as well as all of the other ordinals κn in the critical sequence of j ). But the set includes many other ordinals as well. The main result of this paper is that the number of critical points below κn 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 κ4, one can find so many critical points that the number is only expressible using fast-growing hierarchies of iterated functions. (shrink)

One of the standard ways of postulating large cardinal axioms is to consider elementary embeddings,j, from the universe,V, into some transitive submodel,M. See Reinhardt–Solovay [7] for more details. Ifjis not the identity, andκis the first ordinal moved byj, thenκis a measurable cardinal. Conversely, Scott [8] showed that wheneverκis measurable, there is suchjandM. If we had assumed, in addition, that, thenκwould be theκth measurable cardinal; in general, the wider we assumeMto be, the largerκmust be.