Related papers: Extensional realizability for intuitionistic set t…
We describe a realizability framework for classical first-order logic in which realizers live in (a model of) typed {\lambda}{\mu}-calculus. This allows a direct interpretation of classical proofs, avoiding the usual negative translation to…
We investigate completeness and parametricity for a general class of realizability semantics for System F defined in terms of closure operators over sets of $\lambda$-terms. This class includes most semantics used for normalization…
Goodman's theorem (1976) states that intuitionistic finite-type arithmetic plus the axiom of choice plus the axiom of relativized dependent choice is conservative over Heyting arithmetic. The same result applies to the extensional variant.…
Natural philosophy integrates scientific observation with abstract frameworks, often using a mathematical Ansatz to hypothesise about physical phenomena. Exploring the possibility of other universes, however, challenges assumptions that…
In recent years the question of whether adding the limited principle of omniscience, LPO, to constructive Zermelo-Fraenkel set theory, CZF, increases its strength has arisen several times. As the addition of excluded middle for atomic…
We extend the usual internal logic of a (pre)topos to a more general interpretation, called the stack semantics, which allows for "unbounded" quantifiers ranging over the class of objects of the topos. Using well-founded relations inside…
We consider the theory of algebraically closed fields of characteristic zero with multivalued operations $x\mapsto x^r$ (raising to powers). It is in fact the theory of equations in exponential sums. In an earlier paper we have described…
We introduce the concept of inverse powerset by adding three axioms to the Zermelo-Fraenkel set theory. This extends the Zermelo-Fraenkel set theory with a new type of set which is motivated by an intuitive meaning and interesting…
This is the author's PhD thesis. It is a contribution to categorical logic, in particular to the theory of realizability toposes. While the tools of categorical logic have proven very successful in analyzing and organizing proof theoretic…
An element of a finitely generated non-Abelian free group F(X) is said to be filling if that element has positive translation length in every very small action of F(X) on an $\mathbb{R}$-tree. We give a proof that the set of filling…
This paper introduces Relational Type Theory (RelTT), a new approach to type theory with extensionality principles, based on a relational semantics for types. The type constructs of the theory are those of System F plus relational…
We extend the theory of infinite-exponent partition relations to arbitrary linear order types, with a particular focus on the real number line. We give a complete classification of all consistent partition relations on the real line with…
We develop an untyped framework for the multiverse of set theory. $\mathsf{ZF}$ is extended with semantically motivated axioms utilizing the new symbols $\mathsf{Uni}(\mathcal{U})$ and $\mathsf{Mod}(\mathcal{U, \sigma})$, expressing that…
We introduce an extension of the propositional calculus to include abstracts of predicates and quantifiers, employing a single rule along with a novel comprehension schema and a principle of extensionality, which are substituted for the…
Choice functions constitute a simple, direct and very general mathematical framework for modelling choice under uncertainty. In particular, they are able to represent the set-valued choices that typically arise from applying decision rules…
We survey the logical structure of constructive set theories and point towards directions for future research. Moreover, we analyse the consequences of being extensible for the logical structure of a given constructive set theory. We…
Ordinary and transfinite recursion and induction and ZF set theory are used to construct from a fully interpreted object language and from an extra formula a new language. It is fully interpreted under a suitably defined interpretation.…
Although Zermelo-Fraenkel set theory (ZFC) is generally accepted as the appropriate foundation for modern mathematics, proof theorists have known for decades that virtually all mainstream mathematics can actually be formalized in much…
Let m>2 be an integer. We show that ZF + "For every integer n, Every countable family of non-empty sets of cardinality at most n has an infinite partial choice function" is not strong enough to prove that every countable set of m-element…
We introduce a generalization of stationary set reflection which we call "filter reflection", and show it is compatible with the axiom of constructibility as well as with strong forcing axioms. We prove the independence of filter reflection…