In this paper we introduce the Wholeness Axiom (WA), which asserts that there is a nontrivial elementary embedding from V to itself. We formalize the axiom in the language {∈; j}, adding to the usual axioms of ZFC all instances of Separation, but no instance of Replacement, for j-formulas, as well as axioms that ensure that j is a nontrivial elementary embedding from the universe to itself. We show that WA has consistency strength strictly between I3 and the existence of a cardinal that is… CONTINUE READING