Diagonalization in Formal Mathematics (BestMasters)

Description

In this book, Paulo Guilherme Santos studies diagonalization in formal mathematics from logical aspects to everyday mathematics. He starts with a study of the diagonalization lemma and its relation to the strong diagonalization lemma. After that, Yablo’s paradox is examined, and a self-referential interpretation is given. From that, a general structure of diagonalization with paradoxes is presented. Finally, the author studies a general theory of diagonalization with the help of examples from mathematics. Acknowledgment Contents 1 Preliminaries 2 Introduction 3 Two uses of the Diagonalization Lemma 3.1 Formal Limitations of Predicates Related to Self-Reference 3.2 The Diagonalization Lemma and the Strong Diagonalization Lemma: A Separation Result 4 Yablo’s Paradox and Self-Reference 4.1 A Minimal Theory to Express Yablo’s Paradox 4.2 Linear Time Temporal Logic and Yablo’s Paradox 5 Smullyan’s Theorem, Lӧb’s Theorem, and a General Approach to Paradoxes 5.1 Smullyan’s Theorem 5.2 The Paradoxes and Lӧb’s Theorem 5.3 The Considered Languages and Diagonalization 5.4 Curry Systems and Lӧb Systems 5.5 General Systems for Paradoxes 5.6 Models of Curry Systems, and Consistency 6 General Theory of Diagonalization 6.1 Towards the Study of Diagonalization in Mathematics 7 Mathematical Examples 7.1 Lawvere’s Diagonal Argument 7.2 Knaster-Tarski Theorem 7.3 General Fixed Point Theorem 7.3.1 Fixed point Lemma for Normal Functions 7.3.2 Banach Fixed Point Theorem 8 Conclusions and Future Work 8.1 Conclusions 8.2 Future Work Bibliography