The wholeness axioms, proposed by Paul Corazza {% cite Corazza2000 Corazza2003 %}, occupy a high place in the upper stratosphere of the large cardinal hierarchy, intended as slight weakenings of the [](Kunen_inconsistency.md), but similar in spirit. The **wholeness axioms** are formalized in the language $\{\in,j\}$, augmenting the usual language of set theory $\{\in\}$ with an additional unary function symbol $j$ to represent the [](Elementary%20embedding.md). The base theory ZFC is expressed only in the smaller language $\{\in\}$. Corazza's original proposal, which we denote by $\text{WA}_0$, asserts that $j$ is a nontrivial amenable elementary embedding from the universe to itself, without adding formulas containing $j$ to the separation and replacement axioms. Elementarity is expressed by the scheme $\varphi(x)\iff\varphi(j(x))$, where $\varphi$ runs through the formulas of the usual language of set theory; nontriviality is expressed by the sentence $\exists x j(x)\not=x$; and amenability is simply the assertion that $j\upharpoonright A$ is a set for every set $A$. Amenability in this case is equivalent to the assertion that the separation axiom holds for $\Delta_0$ formulae in the language $\{\in,j\}$. The **wholeness axiom** $\text{WA}$, also denoted $\text{WA}_\infty$, asserts in addition that the full separation axiom holds in the language $\{\in,j\}$. Those two axioms are the endpoints of the hierarchy of axioms $\text{WA}_n$, asserting increasing amounts of the separation axiom. Specifically, the wholeness axiom $\text{WA}_n$, where $n$ is amongst $0,1,\ldots,\infty$, consists of the following: 1. (elementarity) All instances of $\varphi(x)\iff\varphi(j(x))$ for $\varphi$ in the language $\{\in,j\}$. 2. (separation) All instances of the Separation Axiom for $\Sigma_n$ formulae in the full language $\{\in,j\}$. 3. (nontriviality) The axiom $\exists x\,j(x)\not=x$. Clearly, this resembles the [](Kunen_inconsistency.md). What is missing from the wholeness axiom schemes, and what figures prominantly in Kunen's proof, are the instances of the replacement axiom in the full language with $j$. In particular, it is the replacement axiom in the language with $j$ that allows one to define the critical sequence $\langle \kappa_n\mid n\lt\omega\rangle$, where $\kappa_{n+1}=j(\kappa_n)$, which figures in all the proofs of the Kunen inconsistency. Thus, none of the proofs of the Kunen inconsistency can be carried out with $\text{WA}$, and indeed, in every model of $\text{WA}$ the critical sequence is unbounded in the ordinals. The hiearchy of wholeness axioms is strictly increasing in strength, if consistent. {% cite Hamkins2001 %} If $j:V_\lambda\to V_\lambda$ witnesses a [](Rank_into_rank.md) cardinal, then $\langle V_\lambda,\in,j\rangle$ is a model of the wholeness axiom. **Axioms $\mathrm{I}_4^n$** for natural numbers $n$ (starting from $0$) are an attempt to measure the gap between $\mathrm{I}_3$ and $\mathrm{WA}$. Each of these axioms asserts the existence of a transitive model of $\mathrm{ZFC} + \mathrm{WA}$ with additional, stronger and stronger properties. Namely, $\mathrm{I}_4^n(\kappa)$ holds if and only if there is a transitive model $(M,\in,j)$ of $\mathrm ZFC+WA$ with $V_{j^n(\kappa)+1}\subseteq M$ and $\kappa$ the critical point of $j$. $\mathrm{I}_3(κ)$ is equivalent to the existence of an $\mathrm{I}_4(κ)$-coherent set of embeddings. On the other hand, it is not known whether the $\mathrm{I}_4^n$ axioms really increase in consistency strength and whether it is possible in $\mathrm{ZFC}$ that $\forall _{n\in\omega} \mathrm{I}_4^n(κ) \land \neg \mathrm{I}_3(κ)$.{% cite Corazza2003 %} If the wholeness axiom is consistent with $\text{ZFC}$, then it is consistent with $\text{ZFC+V=HOD}$.{% cite Hamkins2001 %}