A Realizability Approach to Constructing Higher Types via Classifiers

Failure of Higher-Order Truth within Intuitionistic Propositional Logic

We answer the question whether all Heyting algebras can appear as the lattice of subterminal objects of an elementary topos in the negative. Concretely, we show that the free Heyting algebra on two generators, hence also on N generators for every N greater than 2, cannot be such a Heyting algebra. The mathematical resu...

Improving the Expressivity of Structural Type Theory

Structural type theory is an extension of simple type theory that incorporates types for any kind of mathematical structure and allows isomorphic structures to be identified within them. In order to guarantee consistency, the system incorporates several syntactic restrictions. Modelling structural type theory in Tars...

Indiscernible Extraction in Theories with Bounded $k$-splitting

Hanson introduced the class of theories with bounded $k$-splitting, a higher-arity notion of stability, and proved an indiscernible extraction result for those theories assuming the existence of a $k$-ineffable cardinal. In this article, we show that the conclusion of Hanson's theorem is equivalent to the weaker assert...

On the Logic of LGB Type Structures. Part II: Reentrancy Structures

Counterexamples to a conjecture of Kamenetsky on OEIS A173419

Define a(n) as the number corresponding to the shortest computation using only addition, subtraction, or multiplication, as A173419 states on the related OEIS page. In a comment to the sequence, D. Kamenetsky conjectured that a(p) $\ge$ a(p -1) for every prime. We show that the conjecture is false through four countere...

A counterexample to Kanalas'problem of continuously realising types

We give an example of a coherent theory, a topological space, and a continuous assignment of the (model-theoretic) types of the theory to the points of the space, such that there is no sheaf model of the theory over the space with the property that the fibre over a point, as a set-based model, realises the correspondin...