Periodicity in the cumulative hierarchy

Goldberg Gabriel, Schlutzenberg Farmer

Zusammenfassung

We investigate the structure of rank-to-rank elementary embeddings, working in ZF set theory without the Axiom of Choice. Recall that the levels $V_\alpha$ of the cumulative hierarchy are defined via iterated application of the power set operation, starting from $V_0=\emptyset$, and taking unions at limit stages. Assuming that \[ j:V_{\alpha+1}\to V_{\alpha+1}\] is a (non-trivial) elementary embedding, we show that the structure of $V_\alpha$ is fundamentally different to that of $V_{\alpha+1}$. We show that $j$ is definable from parameters over $V_{\alpha+1}$ iff $\alpha+1$ is an odd ordinal. Moreover, if $\alpha+1$ is odd then $j$ is definable over $V_{\alpha+1}$ from the parameter \[ j`` V_{\alpha}=\{j(x)\bigm|x\in V_\alpha\},\] and uniformly so. This parameter is optimal in that $j$ is not definable from any parameter which is an element of $V_\alpha$. In the case that $\alpha=\beta+1$, we also give a characterization of such $j$ in terms of ultrapower maps via certain ultrafilters. Assuming $\lambda$ is a limit ordinal, we prove that if $j:V_\lambda\to V_\lambda$ is $\Sigma_1$-elementary, then $j$ is not definable over $V_\lambda$ from parameters, and if $\beta