The spectrum of elementary embeddings

In 1970, K. Kunen, working in the context of Kelley–Morse set theory, showed that the existence of a nontrivial elementary embedding j : V → V is inconsistent. In this paper, we give a finer analysis of the implications of his result for embeddings V → V relative to models of ZFC. We do this by working in the extended language { ∈ , j } , using as axioms all the usual axioms of ZFC (for ∈ -formulas), along with an axiom schema that asserts that j is a nontrivial elementary embedding. Without additional axiomatic assumptions on j, we show that that the resulting theory (denoted ZFC+BTEE) is weaker than an ω -Erdös cardinal, but stronger than n -ineffables. We show that natural models of ZFC+BTEE give rise to Schindler’s remarkable cardinals. The approach to inconsistency from ZFC+BTEE forks into two paths: extensions of ZFC+BTEE+Cofinal Axiom and ZFC+BTEE+¬Cofinal Axiom, where Cofinal Axiom asserts that the critical sequence κ , j ( κ ) , j 2 ( κ ) , … is cofinal in the ordinals. We describe near-minimal inconsistent extensions of each of these theories. The path toward inconsistency from ZFC+BTEE+¬Cofinal Axiom is paved with a sequence of theories of increasing large cardinal strength. Indeed, the extensions of the theory ZFC +“j is a nontrivial elementary embedding” form a hierarchy of axioms, ranging in strength from Con(ZFC) to the existence of a cardinal that is super- n -huge for every n , to inconsistency. This hierarchy is parallel to the usual hierarchy of large cardinal axioms, and can be used in the same way. We also isolate several intermediate-strength axioms which, when added to ZFC+BTEE, produce theories having strengths in the vicinity of a measurable cardinal of high Mitchell order, a strong cardinal, ω Woodin cardinals, and n -huge cardinals. We also determine precisely which combinations of axioms, of the form ZFC + BTEE + Σ m - Separation j + Σ n - Replacement j result in inconsistency.