arXiv subjects

Alexander Van Abel

Publications and source records attributed to Alexander Van Abel.

3 recordsLinked to original sources

arXiv2023

Counting in Uncountably Categorical Pseudofinite Structures

We show that every definable subset of an uncountably categorical pseudofinite structure has pseudofinite cardinality which is polynomial (over the rationals) in the size of any strongly minimal subset, with the degree of the polynomial equal to the Morley rank of the subset. From this fact, we show that classes of finite structures whose ultraproducts all satisfy the same uncountably categorical theory are polynomial $R$-mecs as well as $N$-dimensional asymptotic classes, where $N$ is the Morley rank of the theory.

math.LO↗

arXiv2022

Tame pseudofinite theories with wild pseudofinite dimensions

We construct two pseudofinite theories which are tame from a neostability perspective, yet have pathological fine pseudofinite dimension in all models. These theories serve as counterexamples to potential converses of results by Garcia, Macpherson and Steinhorn relating pseudofinite dimension to tameness. We demonstrate that pseudofinite cardinality in these theories is well behaved with regards to definability, and provide a novel method of proving quantifier elimination using pseudofinite cardinality.

math.LO↗

arXiv2018

Variations on the Feferman-Vaught Theorem, with applications to $\prod_p \mathbb{F}_p$

Using the Feferman-Vaught Theorem, we prove that a definable subset of a product structure must be a Boolean combination of open sets, in the product topology induced by giving each factor structure the discrete topology. We prove a converse of the Feferman-Vaught theorem for families of structures with certain properties, including families of integral domains. We use these results to obtain characterizations of the definable subsets of $\prod_p \mathbb{F}_p$ -- in particular, every formula is equivalent to a Boolean combination of $\exists \forall \exists$ formulae.

math.LO↗