4 research outputs found
The Category of Iterative Sets in Homotopy Type Theory and Univalent Foundations
When working in Homotopy Type Theory and Univalent Foundations, the
traditional role of the category of sets, Set, is replaced by the category hSet
of homotopy sets (h-sets); types with h-propositional identity types. Many of
the properties of Set hold for hSet ((co)completeness, exactness, local
cartesian closure, etc.). Notably, however, the univalence axiom implies that
Ob(hSet) is not itself an h-set, but an h-groupoid. This is expected in
univalent foundations, but it is sometimes useful to also have a stricter
universe of sets, for example when constructing internal models of type theory.
In this work, we equip the type of iterative sets V0, due to Gylterud (2018) as
a refinement of the pioneering work of Aczel (1978) on universes of sets in
type theory, with the structure of a Tarski universe and show that it satisfies
many of the good properties of h-sets. In particular, we organize V0 into a
(non-univalent strict) category and prove that it is locally cartesian closed.
This enables us to organize it into a category with families with the structure
necessary to model extensional type theory internally in HoTT/UF. We do this in
a rather minimal univalent type theory with W-types, in particular we do not
rely on any HITs, or other complex extensions of type theory. Furthermore, the
construction of V0 and the model is fully constructive and predicative, while
still being very convenient to work with as the decoding from V0 into h-sets
commutes definitionally for all type constructors. Almost all of the paper has
been formalized in Agda using the agda-unimath library of univalent
mathematics
Mechanised metamathematics : an investigation of first-order logic and set theory in constructive type theory
In this thesis, we investigate several key results in the canon of metamathematics, applying the contemporary perspective of formalisation in constructive type theory and mechanisation in the Coq proof assistant. Concretely, we consider the central completeness, undecidability, and incompleteness theorems of first-order logic as well as properties of the axiom of choice and the continuum hypothesis in axiomatic set theory. Due to their fundamental role in the foundations of mathematics and their technical intricacies, these results have a long tradition in the codification as standard literature and, in more recent investigations, increasingly serve as a benchmark for computer mechanisation. With the present thesis, we continue this tradition by uniformly analysing the aforementioned cornerstones of metamathematics in the formal framework of constructive type theory. This programme offers novel insights into the constructive content of completeness, a synthetic approach to undecidability and incompleteness that largely eliminates the notorious tedium obscuring the essence of their proofs, as well as natural representations of set theory in the form of a second-order axiomatisation and of a fully type-theoretic account. The mechanisation concerning first-order logic is organised as a comprehensive Coq library open to usage and contribution by external users.In dieser Doktorarbeit werden einige Schlüsselergebnisse aus dem Kanon der Metamathematik untersucht, unter Verwendung der zeitgenössischen Perspektive von Formalisierung in konstruktiver Typtheorie und Mechanisierung mit Hilfe des Beweisassistenten Coq. Konkret werden die zentralen Vollständigkeits-, Unentscheidbarkeits- und Unvollständigkeitsergebnisse der Logik erster Ordnung sowie Eigenschaften des Auswahlaxioms und der Kontinuumshypothese in axiomatischer Mengenlehre betrachtet. Aufgrund ihrer fundamentalen Rolle in der Fundierung der Mathematik und ihrer technischen Schwierigkeiten, besitzen diese Ergebnisse eine lange Tradition der Kodifizierung als Standardliteratur und, besonders in jüngeren Untersuchungen, eine zunehmende Bedeutung als Maßstab für Mechanisierung mit Computern. Mit der vorliegenden Doktorarbeit wird diese Tradition fortgeführt, indem die zuvorgenannten Grundpfeiler der Methamatematik uniform im formalen Rahmen der konstruktiven Typtheorie analysiert werden. Dieses Programm ermöglicht neue Einsichten in den konstruktiven Gehalt von Vollständigkeit, einen synthetischen Ansatz für Unentscheidbarkeit und Unvollständigkeit, der großteils den berüchtigten, die Essenz der Beweise verdeckenden, technischen Aufwand eliminiert, sowie natürliche Repräsentationen von Mengentheorie in Form einer Axiomatisierung zweiter Ordnung und einer vollkommen typtheoretischen Darstellung. Die Mechanisierung zur Logik erster Ordnung ist als eine umfassende Coq-Bibliothek organisiert, die offen für Nutzung und Beiträge externer Anwender ist