**February 26**

**Farmer Schlutzenberg**,
University of Münster

**(Non)uniqueness and (un)definability of embeddings beyond choice**

Work in ZF and let $j:V_\alpha\to V_\alpha$ be an elementary, or partially elementary, embedding. One can examine the degree of uniqueness, definability or constructibility of $j$. For example, is there $\beta<\alpha$ such that $j$ is the unique (partially) elementary extension of $j\upharpoonright V_\beta$? Is $j$ definable from parameters over $V_\alpha$? We will discuss some results along these lines, illustrating that answers can depend heavily on circumstances. Some of the work is due independently and earlier to Gabriel Goldberg.