Applied Mathematics without Numbers

In this paper, I develop a "safety result" for applied mathematics. I show that whenever a theory in natural science entails some non-mathematical conclusion via an application of mathematics, there is a counterpart theory that carries no commitment to mathematical objects, entails the same conclusion, and the claims of which are true if the claims of the original theory are "correct": roughly, true given the assumption that mathematical objects exist. The framework used for proving the safety result has some advantages over existing nominalistic accounts of applied mathematics. It also provides a nominalistic account of pure 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

Level theory, part 1: Axiomatizing the bare idea of a cumulative hierarchy of sets

The following bare-bones story introduces the idea of a cumulative hierarchy of pure sets: ‘Sets are arranged in stages. Every set is found at some stage. At any stage S: for any sets found before S, we find a set whose members are exactly those sets. We find nothing else at S.’ Surprisingly, this story already guarantees that the sets are arranged in well-ordered levels, and suffices for quasi-categoricity. I show this by presenting Level Theory, a simplification of set theories due to Scott, Montague, Derrick, and Potte

Pravdivost mezi syntaxí a sémantikou

Sir s m c lem t eto pr ace je vyjasnit vztah mezi syntax a s emantikou, zejm ena pokud jde o jazyky s p resn e speci kovanou strukturou. Hlavn ot azky, kter ymi se zab yv ame, jsou: Co cin s emantick y pojem s emantick ym? Co zp usobuje, ze je pouh a s emantick a anal yza takov eho pojmu nedostate cn a? Co je t m rozhoduj c m krokem, kter y mus me u cinit, abychom pronikli k v yznamov e str ance jazyka? T emito ot azkami se nezab yv ame p r mo, ale prost rednictv m anal yzy typick eho s emantick eho pojmu, a sice pravdivosti. Na s hlavn ot azkou tedy je: Jak e pojmov e prost redky jsou nezbytn e pro uspokojivou de nici pravdivosti? Ke zkoum an pojmu pravdivosti a jednotliv ych zp usob u, jak jej lze de- novat, jsme si vybrali t ri konkr etn syst emy: kumulativn verzi Russellovy rozv etven e teorie typ u, Zermelovu druho r adovou teorii mno zin a Carnapovu logickou syntax. Ka zd y syst em je podroben d ukladn emu studiu. P redkl adan a pr ace je tedy souborem t r v ce m en e samostatn ych studi , je z popisuj mo znosti explicitn de nice pravdivosti a nezbytn eho pojmov eho z azem . Poznamenejme, ze na s m c lem nen historicky v ern a prezentace uveden ych syst em u, n ybr z snaha o dal s rozvinut toho cenn eho, co nab zej , ve sv etle sou casn ych poznatk u. Obecn ym z av erem, k n emu z dosp ejeme na z...The broad aim of this thesis is to clarify the relationship between syntax and semantics, mainly in connection with languages with exactly speci ed structure. The main questions we raise are: What is it that makes a semantic concept genuinely semantic? What exactly makes a merely semantic characterization of such a concept inadequate? What is the decisive step we have to make if we want to start speaking about the meaning-side of language? We approach these questions indirectly: via an analysis of a typically semantic concept, namely that of truth. Our principal question then becomes: What conceptual resources are required for a satisfactory de nition of truth? To investigate the concept of truth and di erent ways in which it can be de ned, we have chosen three individual systems: (a cumulative version of) Russell's rami ed theory of types, Zermelo's second-order set theory and Carnap's logical syntax. Each of the systems is studied in considerable detail. The presented thesis is, in e ect, a collection of three case-studies into the ways in which the concept of truth is explicitly de nable and into the requisite conceptual background, each study forming a more or less closed unity. It should be noted that we are not interested in a historically faithful representation of these systems; our goal is to get...Institute of Philosophy and Religious StudiesÚstav filosofie a religionistikyFilozofická fakultaFaculty of Art

Models Of Second-Order Zermelo Set Theory

The paper discusses models of second-order versions of Zermelo set theory that are not given by certain initial segments of the cumulative hierarchy. These models show that common versions of infinity do not, absent replacement, guarantee the existence of the first transfinite stage of the cumulative hierarchy. Another construction shows that a version of second-order Zermelo set theory that results when infinity is strengthened to deliver the existence of that stage is satisfied in non-well-founded models. A variant of second-order Zermelo set theory is considered all of whose models are given by certain initial segments of the hierarchy