This will be a talk for the Conference on the occasion of Jörg Brendle’s 60th birthday at Kobe University in Kobe, Japan, 2-5 September 2025.
Many years ago, I was a JSPS Fellow at Kobe University, at the same time that Jörg first took up his position in Japan, a time when Philip Welch also had his professorship there.
Title The elementary theory of surreal arithmetic is bi-interpretable with set theory
Speaker Joel David Hamkins, O’Hara Professor of Logic, University of Notre Dame
Abstract I shall introduce the first-order elementary theory of surreal arithmetic, a theory that is true in the surreal field when equipped with its birthday order. This structure is bi-interpretable with the set-theoretic universe (V,∈), and indeed the theory of surreal arithmetic SA is bi-interpretable with ZFC. This is a preliminary report on very new joint work in progress with myself, Junhong Chen, and Ruizhi Yang, both of Fudan University, Shanghai.
Do you have a link to a preprint? This sounds amazingly interesting.
In particular, how do you axiomatize the theory of the surreal field?
Because the surreal operations can be defined in ZFC, representing every surreal number by a function from its birthday ordinal to an arbitrary 2-element set {+,-}, a sufficient R.E. axiomatization would be “every theorem that ZFC proves holds in the surreal field”, and that can be transformed into a recursive axiomatization by standard methods, and does not run into set-class problems because the first order theory of the surreal field can only talk about finitely many surreal numbers at a time. But that isn’t a very “nice” axiomatization!
The preprint is not yet available—this is very new work. Our theory is much nicer and more natural than the theory you suggest. For example, we have what we call the saturation scheme, asserting that if A and B are definable (allowing parameters) classes of numbers whose birthdays are bounded, and every a in A is below every b in B, then there is a number z with A < z < B. And we have the eternity axiom, which asserts that if one can define a unique number for every number simpler than a given number, then there is a birthday beyond all those numbers.
If your language for the surreal field includes only the ordered field operations and relations, how do you define “birthday” and simpler”? (You only need to define one of the two, of course.)
Did you just mean that the relation “x was born earlier than y” is a primitive that is given at the start? Then I see how everything can work.
In the language of ordered fields only, it is impossible, since it is a real-closed field and hence a decidable theory. Yes, our theory is expressed in a language augmented with the birthday order. (Alternatively, we adopt the left-prior and right-prior relations, which are very natural from the point of view of the axioms and the construction.)
Do you have, for some n, an easily describable mapping of sets to n-tuples of surreals such that the membership relation translates to a definable 2n-ary relation in an easily describable way?
Yes, see the slides, which explains a bit of it. The interpretation doesn’t use all surreals, but rather defines a domain on which the interpretation lives.
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