This will be a talk for the Logic Seminar at the University of Notre Dame, Tuesday 18 November 20215 2pm 125 Hayes-Healy Building.
Abstract. I shall introduce what I call the first-order elementary theory of surreal arithmetic, a 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 very new joint work in progress with myself, Junhong Chen, and Ruizhi Yang, of Fudan University, Shanghai.
Hello Joel, perhaps you and your co-authors have noticed this already, but I thought it worthwhile to point out that the bi-interpretation between ZFC and surreal arithmetic provides another pair of natural theories that are bi-interpretable, but not definitionally equivalent (since there are models of ZFC that do not have a global definable linear order, whereas every model of surreal arithmetic has a definable linear order).
The failure of definitional equivalence between these two theories, in turn can be used to show that in contrast to ZFC (and even ZF), surreal arithmetic does not “eliminate imaginaries” (since, if it did, by the Friedman-Visser theorem, it would be definitionally equivalent to ZFC, contradiction).
I recently noticed that SO (the second order theory of ordinals) does not eliminate imaginaries with a similar reasoning (answering a question of Mateusz Łełyk, who had asked if elimination of imaginaries is preserved by bi-interpretations)
Thanks for the comment! Very interesting points.