Given a countable o-minimal theory T , we characterize the Borel complexity of isomorphism for countable models of T up to two model-theoretic invariants. If T admits a nonsimple type, then it is shown to be Borel complete by embedding the isomorphism problem for linear orders into the isomorphism problem for models of T . This is done by constructing models with specific linear orders in the t...

In historical discussions of twentieth-century logic, it is typically assumed that model theory emerged within the tradition that adopted first-order logic as the standard framework. Work within the type-theoretic tradition, in the style of Principia Mathematica, tends to be downplayed or ignored in this connection. Indeed, the shift from type theory to first-order logic is sometimes seen as in...

Most current research on automated theorem proving is concerned with proving theorems of first-order logic. We discuss two examples which illustrate the advantages of using higher-order logic in certain contexts. For some purposes type theory is a much more convenient vehicle for formalizing mathematics than axiomatic set theory. Even theorems of first-order logic can sometimes be proven more e...

Given a class of linear order types C, we identify and study several different classes of trees, naturally associated with C in terms of how the paths in those trees are related to the order types belonging to C. We investigate and completely determine the set-theoretic relationships between these classes of trees and between their corresponding first-order theories. We then obtain some general...

We connect and solve two longstanding open problems in quite different areas: the model-theoretic question of whether SOP2 is maximal in Keisler’s order, and the question from general topology/set theory of whether p = t, the oldest problem on cardinal invariants of the continuum. We do so by showing these problems can be translated into instances of a more fundamental problem which we state an...

A structure M is pregeometric if the algebraic closure is a pregeometry in all M ′ elementarily equivalent to M . We define a generalisation: structures with an existential matroid. The main examples are superstable groups of U-rank a power of ω and d-minimal expansion of fields. Ultraproducts of pregeometric structures expanding a field, while not pregeometric in general, do have an unique exi...

We prove that the theory of open projective planes is complete and strictly stable, infer from this Marshall Hall's free ( π n : 4 ⩽ ω ) are all elementary equivalent their common stable decidable, being in fact planes. further characterize substructure relation class planes, show particular an chain. then does not have a prime model, it has elimination quantifiers down to Boolean combinations ...