This will be a talk at the CUNY Logic Workshop on 13 March 2026, held at the CUNY Graduate Center.
Abstract. I shall introduce the elementary theory of surreal arithmetic (SA), a first-order theory that is true in the surreal field when equipped with its birthday order structure. This structure, I shall prove, is bi-interpretable with the set-theoretic universe (V,∈), and indeed the theory of surreal arithmetic SA is bi-interpretable with ZFC. This is joint work in progress with myself, Junhong Chen, and Ruizhi Yang, of Fudan University, Shanghai.
Would this bi-interepretability extend to ZFC + P where P is some large cardinal axioms?
Yes. Any additional axiom in set theory, such as a large cardinal axiom, would translate to a statement in the language of surreal arithmetic, preserving the bi-interpretability between those two theories.