[1] The wholeness axiom states roughly that there is an elementary embedding j from the Von Neumann universe V to itself.
More specifically, as Samuel Gomes da Silva states, "the inconsistency is avoided by omitting from the schema all instances of the Replacement Axiom for j-formulas".
However, Holmes, Forster & Libert (2012) write that Corrazza's theory should be "naturally viewed as a version of Zermelo set theory rather than ZFC".
[3] If the wholeness axiom is consistent, then it is also consistent to add to the wholeness axiom the assertion that all sets are hereditarily ordinal definable.
[4] The consistency of stratified versions of the wholeness axiom, introduced by Hamkins (2001),[4] was studied by Apter (2012).