The gap between ${\rm I}_3$ and the wholeness axiom
Volume 179 / 2003
Abstract
$\exists\kappa\, {\rm I}_3(\kappa)$ is the assertion that there is an elementary embedding $i:V_\lambda\to V_\lambda$ with critical point below $\lambda$, and with $\lambda$ a limit. The Wholeness Axiom, or $\mathop{\rm WA}\nolimits$, asserts that there is a nontrivial elementary embedding $j: V\to V$; $\mathop{\rm WA}\nolimits$ is formulated in the language $\{\in,{\bf j}\}$ and has as axioms an Elementarity schema, which asserts that ${\bf j}$ is elementary; a Critical Point axiom, which asserts that there is a least ordinal moved by ${\bf j}$; and includes every instance of the Separation schema for ${\bf j}$-formulas. Because no instance of Replacement for ${\bf j}$-formulas is included in $\mathop{\rm WA}\nolimits$, Kunen's inconsistency argument is not applicable. It is known that an ${\rm I}_3$ embedding $i: V_\lambda\to V_\lambda$ induces a transitive model $\langle V_\lambda, \in, i\rangle $ of $\mathop{\rm ZFC}+\mathop{\rm WA}\nolimits$. We study here the gap in consistency strength between ${\rm I}_3$ and $\mathop{\rm WA}\nolimits$. We formulate a sequence of axioms $\langle {\rm I}_4^{n}: n\in\omega\rangle$ each of which asserts the existence of a transitive model of $\mathop{\rm ZFC}+\mathop{\rm WA}\nolimits$ having strong closure properties. We show that ${\rm I}_3$ represents the “limit” of the axioms ${\rm I}_4^{n}$ in a sense that is made precise.
