**May 11**

Alfredo Roque Freire,
Universidade Estadual de Campinas

**On bi-interpretations in Set Theory that preserve isomorphism**

In this presentation I will show that no different set theories extending ZF can be bi-interpretable. This result was achieved by Joel Hamkins and I this February 2018, but it was later discovered that Ali Enayat have already proved the result in 2017. I will also present what I believe to be a solid strengthening of the result: for any two models $\mathcal{M}$ and $\mathcal{N}$ of Zermelo set theory (Z: ZF without replacement), if they are mutually interpretable trough interpretations $I$ and $J$, there is a isomorphism $b$ from $\mathcal{M}$ to ${\mathcal{M}^I}^J$ and replacement is valid in $\mathcal{M}$ for the function $b$, then $\mathcal{M}$ and $\mathcal{N}$ are isomorphic. Along the way I will introduce some definitions, intending to bring clarity to how meaningful this result is; Moreover, I will partially answer the opposite question: what makes it possible for two different theories to be bi-interpretable? The answer to this question will take the form of describing classes of theories for which these property holds.