Assume ZF (without the Axiom of Choice). Let $j:V_\delta\to V_\delta$ be a non-trivial $\Sigma_1$-elementary embedding, where $\delta$ is a limit ordinal. We prove some basic restrictions on the constructibility of $j$ from $V_\delta$; in particular, if $j\in L(V_\delta)$ then $\delta$ has uncountable cofinality. We show that, however, assuming an $I_3$-embedding, with the appropriate $\delta,j$, it is possible to have $j\in L(V_\delta)$. Assuming Dependent Choice and that $\delta$ has countable cofinality (but not assuming $V=L(V_\delta)$), and $j$ is as above, we show that the collection of such embeddings is of high complexity, and that there are "perfectly many" such embeddings. We also show that a ZF theorem of Suzuki, that no elementary $j:V\to V$ is definable from parameters, actually follows from a theory weaker than ZF. The main results rely on a development of extenders under ZF, which we also give.
- Pub Date:
- June 2020
- Mathematics - Logic;
- 28 pages. This version: Changed title, expanded introduction, other small edits. Note: the author has split arXiv:2002.01215v1 into separate components, and this paper constitutes one of those components, and hence draws heavily on those notes