The spectrum of elementary embeddings j:V→Vj:V→V

In 1970, K. Kunen, working in the context of Kelley–Morse set theory, showed that the existence of a nontrivial elementary embedding j:V→Vj:V→V is inconsistent. In this paper, we give a finer analysis of the implications of his result for embeddings V→VV→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 ωω-Erdös cardinal, but stronger than nn-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),j2(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-nn-huge for every nn, 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, ωω Woodin cardinals, and nn-huge cardinals. We also determine precisely which combinations of axioms, of the form ZFC+BTEE+Σm-Separationj+Σn-Replacementj result in inconsistency.