We study reflection principles in Kelley-Morse set theory with urelements (KMU). We first show that First-Order Reflection Principle is not provable in KMU with Global Choice. We then show that KMU + Limitation of Size + Second-Order Reflection Principle is mutually interpretable with KM + Second-Order Reflection Principle. Furthermore, these two theories are also shown to be bi-interpretable with parameters. Finally, assuming the existence of a κ+-supercompact cardinal κ in KMU, we construct a model of KMU + Second-Order Reflection Principle (...) where Limitation of Size fails. (shrink)

In this paper we analyse in the framework of constructive mathematics (BISH) the validity of Farkas' lemma and related propositions, namely the Fredholm alternative for solvability of systems of linear equations, optimality criteria in linear programming, Stiemke's lemma and the Superhedging Duality from mathematical finance, and von Neumann's minimax theorem with application to constructive game theory.

We study Kim-independence over arbitrary sets. Assuming that forking satisfies existence, we establish Kim's lemma for Kim-dividing over arbitrary sets in an NSOP1 theory. We deduce symmetry of Kim-independence and the independence theorem for Lascar strong types.

In this paper we study predimension inequalities in differential fields and define what it means for such an inequality to be adequate. Adequacy was informally introduced by Zilber, and here we give a precise definition in a quite general context. We also discuss the connection of this problem to definability of derivations in the reducts of differentially closed fields. The Ax-Schanuel inequality for the exponential differential equation (proved by Ax) and its analogue for the differential equation of the j-function (established (...) by Pila and Tsimerman) are our main examples of predimensions. We carry out a Hrushovski construction with the latter predimension and obtain a natural candidate for the first-order theory of the differential equation of the j-function. It is analogous to Kirby's axiomatisation of the theory of the exponential differential equation (which in turn is based on the axioms of Zilber's pseudo-exponentiation), although there are many significant differences. In joint work with Sebastian Eterović and Jonathan Kirby we have recently proven that the axiomatisation obtained in this paper is indeed an axiomatisation of the theory of the differential equation of the j-function, that is, the Ax-Schanuel inequality for the j-function is adequate. (shrink)

The fluted fragment is a fragment of first-order logic (without equality) in which, roughly speaking, the order of quantification of variables coincides with the order in which those variables appear as arguments of predicates. It is known that this fragment has the finite model property. We consider extensions of the fluted fragment with various numbers of transitive relations, as well as the equality predicate. In the presence of one transitive relation (together with equality), the finite model property is lost; nevertheless, (...) we show that the satisfiability and finite satisfiability problems for this extension remain decidable. We also show that the corresponding problems in the presence of two transitive relations (with equality) or three transitive relations (without equality) are undecidable, even for the two-variable sub-fragment. (shrink)