The Wholeness Axioms and V=HOD

If the Wholeness Axiom WA0 is itself consistent, then it is consistent with v=hod. A consequence of the proof is that the various Wholeness Axioms are not all equivalent. Additionally, the theory zfc+wa0 is finitely axiomatizable. The Wholeness Axioms, proposed by Paul Corazza, occupy a high place in the upper stratosphere of the large cardinal hierarchy. They are intended as weakenings of the famous inconsistent assertion that there is a nontrivial elementary embedding from the universe to itself, weakenings which, one hopes, are substantial enough to avoid inconsistency but slight enough for them to remain very strong. The Wholeness Axioms are formalized in the language {∈ , j}, augmenting the usual language of set theory {∈} with an additional unary function symbol j to represent the embedding. The base theory zfc is expressed only in the smaller language {∈}. Corazza’s original proposal, which I will denote by wa0, asserts that j is a nontrivial amenable elementary embedding from the universe to itself. Elementarity is expressed by the scheme φ(x) ↔ φ(j(x)), where φ runs through the formulas of the usual language of set theory; nontriviality is expressed by the sentence ∃x j(x) 6= x; and amenability is simply the assertion that j ↾A is a set for every set A. One can easily see that amenability in this case is equivalent to the assertion that the Separation Axiom holds for Σ0 formulae in the language {∈ , j}. Using this idea and hankering for a stronger assumption, Corazza finally settled on the version of the Wholeness Axiom that I will here denote by wa∞, which asserts in addition that the full Separation Axiom holds in the language {∈ , j}. As I hope my notation suggests, these two axioms are the important endpoints of a natural hierarchy of axioms wa0, wa1, wa2, . . . ,wa∞, which I will refer to My research has been supported in part by a grant from the PSC-CUNY Research Foundation and a fellowship from the Japan Society for the Promotion of Science. And I would like to thank my gracious hosts at Kobe University for their generous hospitality.