Logic Workshop

One of the most important constructions is model theory is the ultrapower construction. We will discuss a computability-theoretic analogue of an ultrapower where non-principal ultrafilters, which are non-constructive, are replaced by cohesive sets. A cohesive set is a set that cannot be spit by any computably enumerable set. The resultant structure, known as the cohesive power, manages to be computable and constructive while maintaining analogues of the properties of ultrapowers. We will review recent results applying the cohesive power construction to algebraic extensions of Q.

In 1961 Abraham Robinson solved a centuries-old problem by developing rigorous foundations for infinitesimal calculus. His model-theoretic approach was criticized for its dependence on the axiom of choice and its lack of categoricity. I will argue that the axiomatic approach can overcome these objections.

Starting with the ideas that can be found in the writings of Leibniz and other early infinitesimalists, I will present the theory SPOT, a conservative extension of ZF, that is capable of developing elementary analysis via infinitesimals. Natural generalizations then lead to theories that enable techniques covering almost the entire spectrum of nonstandard analysis. The final theory in the sequence, BST, is 'categorical over ZFC.' Similar results are obtained for theories with multiple levels of standardness. A further extension of the language of these theories allows for a simple presentation of recent results of R. Jin and M. Di Nasso; I will give Jin’s proof of Ramsey’s theorem as an example.

The set-theoretic view of the Leibnizian continuum presents a challenge to traditional set theory, as the existence of infinitesimals entails the existence of unlimited ('infinite') natural numbers. I will indicate how the above theories can be formulated from an 'external' point of view, in terms of an embedding of the standard universe into the internal universe.

This is joint work with Mikhail G. Katz.

The 'unknown order property' is a combinatorial dividing line identified as part of the 'tame hypergraph regularity' program. Tame graph regularity tells us that we can understand some classic model theoretic dividing lines, like stability, NIP, and distality, as describing when binary relations can be roughly approximated by rectangles (with the different properties leading to different kinds of approximations). Tame hypergraph regularity looks at higher arity relations (ternary relations are usually sufficient) and tries to find combinatorial properties which tell us when a relation can be approximated by lower-arity information ('cylinder intersection sets', the higher-arity generalization of rectangles).

The unknown order property is one natural way to generalize stability to ternary relations, but has proven to be more complicated to understand - the name refers to the fact that it still has no known combinatorial characterization. We will describe the motivation for this notion and the current state of the art, including the original results by Terry and Wolf identifying key counterexamples to the unknown order property, and recent results by Chernikov and Towsner showing that there cannot be any combinatorial characterization analogous to the excluded half-graph characterization of stability, but also showing that a big family of natural examples must have this property

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 $(V,\in)$, 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.