Mathematical Logic: Proof theory, Constructive Mathematics

The workshop “Mathematical Logic: Proof Theory, Constructive Mathematics” was centered around proof-theoretic aspects of current mathematics, constructive mathematics and logical aspects of computational complexit

Implicit Commitment in a General Setting

G\"odel's Incompleteness Theorems suggest that no single formal system can capture the entirety of one's mathematical beliefs, while pointing at a hierarchy of systems of increasing logical strength that make progressively more explicit those \emph{implicit} assumptions. This notion of \emph{implicit commitment} motivates directly or indirectly several research programmes in logic and the foundations of mathematics; yet there hasn't been a direct logical analysis of the notion of implicit commitment itself. In a recent paper, \L elyk and Nicolai carried out an initial assessment of this project by studying necessary conditions for implicit commitments; from seemingly weak assumptions on implicit commitments of an arithmetical system $S$, it can be derived that a uniform reflection principle for $S$ -- stating that all numerical instances of theorems of $S$ are true -- must be contained in $S$'s implicit commitments. This study gave rise to unexplored research avenues and open questions. This paper addresses the main ones. We generalize this basic framework for implicit commitments along two dimensions: in terms of iterations of the basic implicit commitment operator, and via a study of implicit commitments of theories in arbitrary first-order languages, not only couched in an arithmetical language

Um estudo sobre teorias de definições indutivas e admissibilidade

Tese de mestrado em Matemática, apresentada à Universidade de Lisboa, através da Faculdade de Ciências, 2013Esta dissertação é um estudo das definições indutivas e suas teorias sob o ponto de vista da teoria da demonstração. No primeiro capítulo apresentamos as noções básicas das definições indutivas. De seguida definimos as teorias ID1 e ID1(W) e apresentamos uma nova interpretação da primeira na segunda. Na segunda parte da dissertação estudamos a teoria de conjuntos admissíveis KPw. Começámos por apresentar KPw e alguns resultados que nos permitirão interpretar ID1 em KPw. No último capítulo estudamos uma recente interpretação funcional de KPw numa teoria de funcionais de árvore recursivos primitivos de Howard. Esta interpretação permite caracterizar duma forma simples o denominado Σ-ordinal de KPw.This thesis concerns the study of inductive definitions and their theories from a proof-theoretical point of view. In the first chapter, we present the basic notions of inductive definitions. Next we define the theories ID1 and ID1(W) and present a novel interpretation of the former in the latter. In the second part of the thesis, we study the admissible set theory KPw. We start by presenting KPw and some results that will allow us to interpret ID1 in KPw. In the last chapter, we study a recent functional interpretation of KPw in a theory of Howard's primitive recursive tree functionals. This interpretation yields a simple characterization of the so-called Σ-ordinal of KPw

Thinning Operators and Pi 4 -Reflection

Erweitert man das Axiomensystem der Kripke-Platek-Mengenlehre um ein Reflexionsschema für \Pi_4-Formeln, so erhält man das System der \Pi_4-Reflexion. Einen anschaulichen beweistheoretischen Zugang zu diesem bietet die Technik der Ausdünnhierarchien. Diese basiert auf der transfiniten Iteration von Ausdünnoperatoren, welche durch Verkleinern von Klassen von Modellkandidaten gewährleisten, daß im Kollabierungsverfahren Reflexionsregeln eliminiert werden können. Durch Untersuchung von speziellen Herleitungen läßt sich mit diesem Ansatz eine Ordinalzahlanalyse durchführen

Constructing the Constructible Universe Constructively

We study the properties of the constructible universe, L, over intuitionistic theories. We give an extended set of fundamental operations which is sufficient to generate the universe over Intuitionistic Kripke-Platek set theory without Infinity. Following this, we investigate when L can fail to be an inner model in the traditional sense. Namely, we show that over Constructive Zermelo-Fraenkel (even with the Power Set axiom) one cannot prove that the Axiom of Exponentiation holds in L.Comment: 26 pages. Revised following referee's recommendation

Pure Σ2\Sigma_2-Elementarity beyond the Core

We display the entire structure ${\cal R}_2$ coding $\Sigma_1$- and $\Sigma_2$-elementarity on the ordinals. This will enable the analysis of pure $\Sigma_3$-elementary substructures.Comment: Extensive rewrite of the introduction. Mathematical content of sections 2 and 3 unchanged, extended introduction to section 2. Removed section 4. Theorem 4.3 to appear elsewhere with corrected proo