The gap between ${\rm I}_3$ and the wholeness axiom

Paul Corazza¹

¹ 310 East Washington Ave. Fairfield, IA 52556, U.S.A.

Fundamenta Mathematicae, Tome 179 (2003) no. 1, pp. 43-60

Résumé

$\exists\kappa\, {\rm I}_3(\kappa)$ 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.

Détail

DOI : 10.4064/fm179-1-4

Keywords: exists kappa kappa assertion there elementary embedding lambda lambda critical point below lambda lambda limit wholeness axiom mathop nolimits asserts there nontrivial elementary embedding mathop nolimits formulated language has axioms elementarity schema which asserts elementary critical point axiom which asserts there least ordinal moved includes every instance separation schema formulas because instance replacement formulas included mathop nolimits kunens inconsistency argument applicable known embedding lambda lambda induces transitive model langle lambda rangle mathop zfc mathop nolimits study here gap consistency strength between mathop nolimits formulate sequence axioms langle omega rangle each which asserts existence transitive model mathop zfc mathop nolimits having strong closure properties represents limit axioms sense made precise