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Comparison and Analysis of Constructive Set Theories

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Comparison and Analysis of Constructive Set Theories Comparison and Analysis of Constructive Set Theories DIPLOMARBEIT zur Erlangung des akademischen Grades Diplom-Ingenieur im Rahmen des Studiums Logic and Computation eingereicht von Robert Freiman Matrikelnummer 1225784 an der Fakultät für Informatik der Technischen Universität Wien Betreuung Betreuer/in: Ao.Univ.Prof. Dipl.-Ing. Dr.techn. Christian Fermüller Mitwirkung: Univ.Prof.i.R. Dr.phil. Alexander Leitsch Wien, 09.12.2019 Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. ___________________________ ___________________________ Unterschrift Verfasser/in Unterschrift Betreuer/in Technische Universität Wien Karlsplatz 13 | 1040 Wien | +43-1-58801-0 | www.tuwien.at Comparison and Analysis of Constructive Set Theory WS 2019/20 Erklärung Hiermit erkläre ich, dass ich diese Arbeit selbständig verfasst, die verwendeten Quellen und Hilfsmittel vollständig angegeben und Stellen der Arbeit – einschließlich Tabellen, Karten und Abbildungen –, die anderen Werken oder dem Internet im Wortlaut oder dem Sinn nach entnommen sind, auf jeden Fall unter Angabe der Quelle als Entlehnung kenntlich gemacht habe Ort, Datum Unterschrift Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. ~ 1 ~ Acknowledgements WS 2019/20 Acknowledgements Ich möchte mich hiermit herzlich bei meinen Betreuern, Prof. Christian Fermüller und Prof. Alexander Leitsch, bedanken, die mir während der gesamten Zeit mit positivem Input, konstruktiver Kritik und aufmerksamen Korrekturhinweisen zur Seite standen. Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. ~ 2 ~ Comparison and Analysis of Constructive Set Theory WS 2019/20 Kurzfassung Seit der Entdeckung der Curry-Howard-Korrespondenz kennen wir die tiefliegenden Zusammenhänge zwischen Berechenbarkeit und Beweisbarkeit in intuitionistischer Logik. Die konstruktive Mathematik, die auf intuitionistischer Logik basiert, ist daher ein vielversprechender Ansatz in der Untersuchung, welche Teile der Mathematik computational fassbar sind. Potentielle Anwendungen liegen dabei im Be- reich der automatischen Deduktion und der automatischen Theorembeweiser. Das Ziel der vorliegenden Arbeit ist es, verschiedene Ansätze der konstruktiven Mengenlehre in einer verständlichen und in sich geschlossenen Art und Weise zu präsentieren und zu vergleichen und dadurch das Potential der Anwendbarkeit in automatischer Deduktion und automatischen Theorembe- weisern offenzulegen. Das spezielle Augenmerk auf Mengenlehre ist ihrer historischen Rolle als Grund- lage der gesamten Mathematik geschuldet. Die wichtigsten Theorien, die in dieser Arbeit behandelt werden, ist Brouwers Mengenlehre, die durch die Zermelo-Fraenkelsche Mengenlehre inspirierten Mengenlehren und , sowie Martin-Löfs Mengenlehre . In einem ersten Schritt werden die Theorien und ihre Axiomatisierungen vom kon- 퐈퐙퐅 퐂퐙퐅 struktivistischen Standpunkt aus motiviert. Einige grundlegende Resultate werden abgeleitet, um einen 퐌퐋 Eindruck vom Arbeiten in den Theorien und deren Grenzen zu erhalten. Der Schwerpunkt der Untersu- chung liegt allerdings in der metamathematischen Analyse. Die metamathematische Analyse der konstruktiven Zermelo-Fraenkel Mengenlehren wird durch zwei semantische Methoden durchgeführt: Realisierbarkeit und topologische Semantik. Erstere baut direkt auf Konzepten der Berechenbarkeit auf und ist deshalb geeignet, konstruktivistisch wünschenswerte metamathematische Eigenschaften zu untersuchen. Ein Beweis wird präsentiert, der sich auf die Ver- wendung topologischer Semantik stützt und zeigt, dass das Prinzip der entscheidbaren Bar-Induktion unabhängig von einer auf basierten Variante der Brouwer’schen Mathematik ist. Schlussendlich wird eine sinnbewahrende Interpretation von in – einer Theorie, mit konstruktiv sehr klarem 퐈퐙퐅 und wohlbegründetem Mengenbegriff – diskutiert. Dadurch stellt sich nicht nur besonders geeignet 퐂퐙퐅 퐌퐋 für den mathematischen Alltag, sondern auch als konstruktiv gerechtfertigte Theorie und dadurch als 퐂퐙퐅 vielversprechende Ausgangbasis für Anwendungen heraus. Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. ~ 3 ~ Abstract WS 2019/20 Abstract Since the discovery of the Curry-Howard-correspondence we know of the deep-lying connections be- tween computability and provability in intuitionistic logic. Hence, constructive mathematics, being based on intuitionistic logic, promises to be a fruitful tool in investigating the computational content of classical mathematics with potential applications in the areas of automated deduction and automated theorem proving. The objective of this thesis is to present and compare different approaches to constructive set theories in a comprehensible and self-contained fashion and thereby demonstrate its potential for applications in automated deduction and automated theorem proving. The particular significance placed here on set theory is due to its historically proven relevance in providing the very foundation of mathematics. The main theories discussed in the thesis are Brouwerian set theory, the axiomatic Zermelo-Fraenkel- style set theories and and Martin-Löf’s set theory . In a first step, the theories and their axio- matizations are motivated from the constructive standpoint. Some basic results are inferred to get used 퐈퐙퐅 퐂퐙퐅 퐌퐋 to reasoning and limitations within the respective system. The broadest investigation, however, is con- ducted by means of metamathematical analysis. Metamathematical analysis of constructive Zermelo-Fraenkel set theories is executed by two semantical tools: Realizability and topological semantics. The former builds directly on notions from computability theory and thus allows for an investigation of metamathematical properties that are constructively de- sirable. A proof using topological semantics is presented to obtain an independence proof of the principle of decidable bar induction from a variant of Brouwer’s mathematics formalized within . Finally, a meaning-persevering interpretation of into – a theory that is considered to give a constructively 퐈퐙퐅 clear and well-justified notions of sets – is discussed. This makes not only especially well-suited for 퐂퐙퐅 퐌퐋 mathematical practice, but also vindicates its constructive nature and makes it a promising starting point 퐂퐙퐅 for applications. Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. ~ 4 ~ Comparison and Analysis of Constructive Set Theory WS 2019/20 Contents Erklärung ................................................................................................................................................................. 1 Acknowledgements ................................................................................................................................................ 2 Kurzfassung ............................................................................................................................................................ 3 Abstract .................................................................................................................................................................... 4 Contents ................................................................................................................................................................... 5 1 Introduction ..................................................................................................................................................... 7 2 Set Theory and Constructivism .................................................................................................................. 11 2.1 Informal constructive set theory ......................................................................................................... 11 2.2 Brouwer’s set theory ............................................................................................................................ 16 2.3 Martin-Löf’s set theory ........................................................................................................................ 20 2.4 and ........................................................................................................................................... 23 3 Axiomatic퐈퐙퐅 constructive퐂퐙퐅 set theories – and .................................................................................. 34 3.1 Setting the stage ....................................................................................................................................퐈퐙퐅 퐂퐙퐅 34 3.2 Basic concepts in ........................................................................................................................... 36 3.3 vs or the problem퐁퐒퐓 with trichotomy ........................................................................................ 39 3.4 퐙퐅 vs퐈퐙퐅 or how to live without the powerset operation ............................................................ 42 3.5 퐈퐙퐅Relation퐂퐙퐅 between
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