
4Universism and Extensions of VReview of Symbolic Logic 14 (1): 112154. 2021.A central area of current philosophical debate in the foundations of mathematics concerns whether or not there is a single, maximal, universe of set theory. Universists maintain that there is such a universe, while Multiversists argue that there are many universes, no one of which is ontologically privileged. Often modeltheoretic constructions that add sets to models are cited as evidence in favor of the latter. This paper informs this debate by developing a way for a Universist to interpret ta…Read more

315It is often assumed that concepts from the formal sciences, such as mathematics and logic, have to be treated differently from concepts from nonformal sciences. This is especially relevant in cases of concept defectiveness, as in the empirical sciences defectiveness is an essential component of lager disruptive or transformative processes such as concept change or concept fragmentation. However, it is still unclear what role defectiveness plays for concepts in the formal sciences. On the one ha…Read more

192Expanding the notion of inconsistency in mathematics: the theoretical foundations of mutual inconsistencyFrom Contradiction to Defectiveness to Pluralism in Science: Philosophical and Formal Analyses. forthcoming.

399Explanation in Descriptive Set TheoryIn Alastair Wilson & Katie Robertson (eds.), Levels of Explanation, Oxford University Press. forthcoming.

37Models as Fundamental Entities in Set Theory: A Naturalistic and Practicebased ApproachErkenntnis 89 (4): 16831710. 2022.This article addresses the question of fundamental entities in set theory. It takes up J. Hamkins’ claim that models of set theory are such fundamental entities and investigates it using the methodology of P. Maddy’s naturalism, Second Philosophy. In accordance with this methodology, I investigate the historical case study of the use of models in the introduction of forcing, compare this case to contemporary practice and give a systematic account of how settheoretic practice can be said to intr…Read more

11Conceptions of Infinity and Set in Lorenzen’s Operationist SystemIn Gerhard Heinzmann & Gereon Wolters (eds.), Paul Lorenzen  Mathematician and Logician, Springer Verlag. pp. 2346. 2021.In the late 1940s and early 1950s, Lorenzen developed his operative logic and mathematics, a form of constructive mathematics. Nowadays this is mostly seen as a precursor of the betterknown dialogical logic, and one might assume that the same philosophical motivations were present in both works. However, we want to show that this is not everywhere the case. In particular, we claim that Lorenzen’s wellknown rejection of the actual infinite, as stated in Lorenzen, was not a major motivation for …Read more

365Modern Class ForcingIn A. Daghighi A. Rezus M. Pourmahdian D. Gabbay M. Fitting (ed.), Research Trends in Contemporary Logic, College Publications. forthcoming.We survey recent developments in the theory of class forcing for malized in the secondorder settheoretic setting.

414Penelope Maddy’s Second Philosophy is one of the most wellknown ap proaches in recent philosophy of mathematics. She applies her secondphilosophical method to analyze mathematical methodology by reconstructing historical cases in a setting of meansends relations. However, outside of Maddy’s own work, this kind of methodological analysis has not yet been extensively used and analyzed. In the present work, we will make a first step in this direction. We develop a general framework that allows…Read more

476Conceptions of infinity and set in Lorenzen’s operationist systemIn S. Rahman (ed.), Logic, Epistemology, and the Unity of Science, Kluwer Academic Publishers. 2004.In the late 1940s and early 1950s Lorenzen developed his operative logic and mathematics, a form of constructive mathematics. Nowadays this is mostly seen as the precursor to the more wellknown dialogical logic and one could assumed that the same philosophical motivations were present in both works. However we want to show that this is not always the case. In particular, we claim, that Lorenzen’s wellknown rejection of the actual infinite as stated in Lorenzen (1957) was not a major motivation…Read more

211Multiverse Conceptions in Set TheorySynthese 192 (8): 24632488. 2015.We review different conceptions of the settheoretic multiverse and evaluate their features and strengths. In Sect. 1, we set the stage by briefly discussing the opposition between the ‘universe view’ and the ‘multiverse view’. Furthermore, we propose to classify multiverse conceptions in terms of their adherence to some form of mathematical realism. In Sect. 2, we use this classification to review four major conceptions. Finally, in Sect. 3, we focus on the distinction between actualism and pot…Read more

785Universism and extensions of VReview of Symbolic Logic 14 (1): 112154. 2021.A central area of current philosophical debate in the foundations of mathematics concerns whether or not there is a single, maximal, universe of set theory. Universists maintain that there is such a universe, while Multiversists argue that there are many universes, no one of which is ontologically privileged. Often modeltheoretic constructions that add sets to models are cited as evidence in favour of the latter. This paper informs this debate by developing a way for a Universist to interpret t…Read more

17Hyperclass Forcing in MorseKelley Class TheoryIn Carolin Antos, SyDavid Friedman, Radek Honzik & Claudio Ternullo (eds.), The Hyperuniverse Project and Maximality, Birkhäuser. pp. 1746. 2018.In this article we introduce and study hyperclassforcing in the context of an extension of MorseKelley class theory, called MK∗∗. We define this forcing by using a symmetry between MK∗∗ models and models of ZFC− plus there exists a strongly inaccessible cardinal. We develop a coding between βmodels ℳ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsid…Read more

22Multiverse Conceptions in Set TheoryIn Carolin Antos, SyDavid Friedman, Radek Honzik & Claudio Ternullo (eds.), The Hyperuniverse Project and Maximality, Birkhäuser. pp. 4773. 2018.We review different conceptions of the settheoretic multiverse and evaluate their features and strengths. In Sect. 1, we set the stage by briefly discussing the opposition between the ‘universe view’ and the ‘multiverse view’. Furthermore, we propose to classify multiverse conceptions in terms of their adherence to some form of mathematical realism. In Sect. 2, we use this classification to review four major conceptions. Finally, in Sect. 3, we focus on the distinction between actualism and pot…Read more

7Class Forcing in Class TheoryIn Carolin Antos, SyDavid Friedman, Radek Honzik & Claudio Ternullo (eds.), The Hyperuniverse Project and Maximality, Birkhäuser. pp. 116. 2018.In this article we show that MorseKelley class theory provides us with an adequate framework for class forcing. We give a rigorous definition of class forcing in a model \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{69pt} \begin{document} $$$$ \end{document} of MK, the main result being that the Definability Lemma can be proven without r…Read more

30The Hyperuniverse Project and Maximality (edited book)Birkhäuser. 2018.This collection documents the work of the Hyperuniverse Project which is a new approach to settheoretic truth based on justifiable principles and which leads to the resolution of many questions independent from ZFC. The contributions give an overview of the program, illustrate its mathematical content and implications, and also discuss its philosophical assumptions. It will thus be of wide appeal among mathematicians and philosophers with an interest in the foundations of set theory. The Hyperu…Read more

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Areas of Specialization
Science, Logic, and Mathematics 
Philosophy of Mathematics 
Logic and Philosophy of Logic 