Logic Workshop

Two countable structures are siblings if each is embeddable into the other (e.g., any two countable non-scattered linear orders are siblings). Clearly if M and N are siblings, they have the same finite substructures, hence have the same universal theories. We characterize the universal theories in a finite, relational language that have a countable model with $2^{\aleph_0}$ siblings. This is joint work with Sam Braunfeld.

Fix languages L and L' (possibly non-disjoint). An structure M in the union of these languages is interpolative if whenever X is an L-definable set in M and X' is an L'-definable set in M, X and X' intersect unless they are separated by disjoint definable sets in the intersection of L and L'. When T is an L theory and T' is an L' theory, we say that a theory T* is the interpolative fusion of T and T' if it axiomatizes the class of interpolative models of the union of T and T'. If T and T' are model-complete, this is exactly the model companion of the union theory. Interpolative fusions provide a unified framework for studying many examples of 'generic constructions' in model theory. Some, like structures with generic predicates, or algebraically closed fields with several independent valuations, are explicitly interpolative fusions, while others, like structures with generic automorphisms, or fields with generic operators, are bi-interpretable with interpolative fusions. In joint work with Erik Walsberg and Minh Tran, we study two basic questions: (1) When does the interpolative fusion exist, and how can we axiomatize it? (2) How can we understand properties of the interpolative fusion T* in terms of properties of the theories T, T', and their intersection?

It is known that the behavior of the Mitchell order substantially changes at the level of rank-to-rank extenders, as it ceases to be well-founded. While the possible partial order structure of the Mitchell order below rank-to-rank extenders is considered to be well understood, little is known about the structure in the ill-founded case. We make a first step in understanding this case by studying the extent to which the Mitchell order can be ill-founded. Our main results are (i) in the presence of a rank-to-rank extender there is a transitive Mitchell order decreasing sequence of extenders of any countable length, and (ii) there is no such sequence of length $\omega_1$. This is joint work with Omer Ben-Neria.

Zermelo (1930) proved the following categoricity result for set theory: Suppose M is a set and E, E’ are two binary relations on M. If both (M, E) and (M, E’) satisfy the second order Zermelo–Fraenkel axioms, then (M,E) and (M, E’) are isomorphic. Of course, the same is not true for first order ZFC. However, we show that if first order ZFC is formulated in the extended vocabulary {E,E’}, then Zermelo’s result holds even in the first order case. Similarly, Dedekind’s categoricity result (1888) for second order Peano arithmetic has an extension to a result about first order Peano.

A positive primitive (henceforth pp) formula is an existentially quantified (i.e., a projection of a) finite system of linear equations (over a given associative ring R). In this talk I am interested exclusively in such formulas with one free variable. I call such a formula high if, in every injective module E, it defines all of E. Note, the high formulas form a filter in the lattice of all unary pp formulas. Long time ago I discovered this dichotomy: every (unary) pp formula is either high or else bounded (but not both), which means that there is a nonzero ring element that annihilates, in every module, all of the set defined by the formula (which is, in fact, an additive subgroup of the module).

I had not given this much further thought until recently, when I discovered, in collaboration with A. Martsinkovsky, that the dual notion of low formula gives rise to a torsion theory, namely injective torsion as introduce by him and J. Russell in recent work. I call a formula low if it vanishes on the flat modules, or, equivalently, on the ring as a module over itself. Note, the low formulas form an ideal in the lattice of all unary pp formulas. I will explain how elementary duality (as introduced by Prest and Herzog) yields at once another dichotomy: every pp formula is either low or else cobounded (but not both), where this means, dually, that the action of some nonzero ring element sends every module into the subgroup defined by that formula.

Interestingly, these two dichotomies are, in general, completely independent. But I will show how their interplay can be used to characterize (not necessarily commutative) domains within the class of all rings, and one-sided Ore domains and also two-sided Ore domains within all domains. (Commutative domains are two-sided Ore.)

If $\kappa$ is measurable and GCH holds, then any ultrapower by a normal measure on $\kappa$ will be missing some subset of $\kappa^+$. On the other hand, Cummings showed that, starting from a $(\kappa+2)$-strong $\kappa$, one can force to a model where $\kappa$ carries a normal measure whose ultrapower captures the entire powerset of $\kappa^+$. Moreover, the large cardinal hypothesis is optimal. I will present an improvement of Cummings' result and show that this capturing property can consistently hold at the least measurable cardinal.

This is joint work with Radek Honzík.

The nice little package is a regular, existentially closed, Henselian local subring of the ring of (formal) power series over your favorite field (R, C, Q?). Moreover, this ring is closed under derivations, anti-derivations, composition, etc. The favorite series (in a single variable, say) include all algebraic functions, all elementary functions, all hypergeometrical functions, all holonomic functions (i.e., solutions of a linear, algebraic ODE), etc.

The way to obtain these is by looking at some non-standard model of the theory of polynomial rings, and then defining its 'catanomials' as the truncations of these functions by only looking at its finite degree terms. In the special case that the non-standard model is an ultrapower of the polynomial ring, the resulting algebra, called the catapower, is just the full power series ring. Whereas the latter may sound less glorious, we can nonetheless do better by taking different non-standard models. Enters the embedded model of PA* of such a model and its standard systems!

Over the last several decades a robust theory of 'tame' expansions of the real field has been developed. Typically tameness in manifested in such situations by the definable sets having some particularly simple topological type. In this talk I will consider how this machinery can be adapted to expansions of Presburger arithmetic in which such topological considerations are largely irrelevant. This is joint work with Chris Miller.

A first-order theory is n-dependent if the edge relation of an infinite generic (n+1)-hypergraph is not definable in any of its models. N-dependence is a strict hierarchy increasing with n, with 1-dependence corresponding to the well-studied class of NIP theories. I will discuss recent joint work with Nadja Hempel on trying to understand which algebraic structures are n-dependent.

Lusin's Theorem, from real analysis, states that for every Borel-measurable function $f$ from $\mathbb R$ to $\mathbb R$, and for every $\epsilon > 0$, there exists a continuous function $g$ on $\mathbb R$ such that $\{ x\in\mathbb R~:~f(x) \neq g(x)\}$ has measure $< \epsilon$. This is proven in most introductory real analysis courses, but here we will give a proof using computability theory and computable analysis. In addition to the theorem itself, the proof will establish an effective way of producing $g$ from $f$ and $\epsilon$, and will pick out, for each $f$, the specific set of troublemakers $x$ in $\mathbb R$ that create all the discontinuities.