Set Theory And Foundations Of Mathematics: An Introduction To Mathematical Logic - Volume I: Set Theory. Douglas Cenzer

Set Theory And Foundations Of Mathematics: An Introduction To Mathematical Logic - Volume I: Set Theory - Douglas Cenzer


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2012 to 2014, and a postdoctoral associate at the University of Florida from 2014 to 2016, before joining Drake University in 2016.

      Jindrich Zapletal is Professor of Mathematics at University of Florida, specializing in mathematical logic and set theory. He received his Ph.D. in 1995 from the Pennsylvania State University, and held postdoctoral positions at MSRI Berkeley, Cal Tech, and Dartmouth College, before joining the University of Florida in 2000.

       Contents

       Preface

       About the Authors

       1. Introduction

       2. Review of Sets and Logic

       2.1 The Algebra of Sets

       2.2 Relations

       2.3 Functions

       2.4 Equivalence Relations

       2.5 Orderings

       2.6 Trees

       3. Zermelo–Fraenkel Set Theory

       3.1 Historical Context

       3.2 The Language of the Theory

       3.3 The Basic Axioms

       3.4 Axiom of Infinity

       3.5 Axiom Schema of Comprehension

       3.6 Axiom of Choice

       3.7 Axiom Schema of Replacement

       3.8 Axiom of Regularity

       4. Natural Numbers and Countable Sets

       4.1 Von Neumann’s Natural Numbers

       4.2 Finite and Infinite Sets

       4.3 Inductive and Recursive Definability

       4.4 Cardinality

       4.5 Countable and Uncountable Sets

       5. Ordinal Numbers and the Transfinite

       5.1 Ordinals

       5.2 Transfinite Induction and Recursion

       5.3 Ordinal Arithmetic

       5.4 Ordinals and Well-Orderings

       6. Cardinality and the Axiom of Choice

       6.1 Equivalent Versions of the Axiom of Choice

       6.2 Applications of the Axiom of Choice

       6.3 Cardinal Numbers

       7. Real Numbers

       7.1 Integers and Rational Numbers

       7.2 Dense Linear Orders

       7.3 Complete Orders

       7.4 Countable and Uncountable Sets of Reals

       7.5 Topological Spaces

       8. Models of Set Theory

       8.1 The Hereditarily Finite Sets

       8.2 Transfinite Models

       9. Ramsey Theory

       9.1 Finite Patterns

       9.2 Countably Infinite Patterns

       9.3 Uncountable Patterns

       Bibliography

       Index

       Chapter 1

       Introduction

      Set theory and mathematical logic compose the foundation of pure mathematics. Using the axioms of set theory, we can construct our universe of discourse, beginning with the natural numbers, moving on with sets and functions over the natural numbers, integers, rationals and real numbers, and eventually developing the transfinite ordinal and cardinal numbers. Mathematical logic provides the language of higher mathematics which allows one to frame the definitions, lemmas, theorems, and conjectures which form the everyday work of mathematicians. The axioms and rules of deduction set up the system in which we can prove our conjectures, thus turning them into theorems.

      Chapter 2 begins with elementary naive set theory, including the algebra of sets under union, intersection, and complement and their connection with elementary logic. This chapter introduces the notions of relations, functions,


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