This volume explores A.P. Morse’s (1911-1984) development of a formal language for writing mathematics, his application of that language in set theory and mathematical analysis, and his unique perspective on mathematics. The editor brings together a variety of Morse’s works in this compilation, including Morse's book A Theory of Sets, Second Edition (1986), in addition to material from another of Morse’s publications, Web Derivatives, and notes for a course on analysis from the early 1950's. Because Morse provided very little in the way of explanation in his written works, the editor’s commentary serves to outline Morse’s goals, give informal explanations of Morse’s formal language, and compare Morse’s often unique approaches to more traditional approaches. Minor corrections to Morse’s previously published works have also been incorporated into the text, including some updated axioms, theorems, and definitions. The editor’s introduction thoroughly details the corrections and changes made and provides readers with valuable insight on Morse’s methods.
A.P. Morse’s Set Theory and Analysis will appeal to graduate students and researchers interested in set theory and analysis who also have an interest in logic. Readers with a particular interest in Morse’s unique perspective and in the history of mathematics will also find this book to be of interest.
Author(s): Robert A. Alps
Publisher: Birkhäuser
Year: 2022
Language: English
Pages: 522
City: Cham
Table of Contents
Preface
Editor's Introduction
I. INTRODUCTORY REMARKS
Biographical Sketch
Background for This Text
Unusual and Unique Features
II. EXPLANATORY COMMENTS
Chapter 0. Language and Inference
Unification: Syntax and Semantics.
Symbols and Expressions.
Schematic Expressions.
Replacement and Schematic Replacement.
Forms, Formulas, and Definitions.
Free and Indicial Variables.
Theory of Notation.
Rules of Inference.
Chapter 1. Logic
Foundations.
Notations.
Theorems.
Chapter 2. Set Theory
Foundations.
Unification of Logic and Set Theory.
Union and Intersection of a Family.
Singletons.
The Classifier.
Ordered Pairs.
Indefinite and Definite Descriptions.
Relations.
Functions and Lonzo Notation.
Ordinals and Inductive Definition.
Other Topics in Set Theory.
Natural Numbers and Sequences.
Equinumerosity, Cardinals, Cardinality, and, Cardinal Arithmetic.
Comparison to ZFC and MK Set Theories.
Chapter 3. Elementary Analysis
Number Systems.
Infinite Numbers and Their Arithmetic.
Extension of Plus and Times.
Infima and Suprema.
Absolute Value.
Runs and Limits.
Summation.
Products.
Chapter 4. Metrics
Linear Spaces.
Chapter 5. Measure
Submeasures.
Metric Measures.
Constructed Measures.
Hulled Measures.
Counting Measure.
Chapter 6. Linear Measure and Total Variation
Chapter 7. Integration
General Methods.
Integration by Refinement.
Measure Integration.
Chapter 8. Product Measures
Infinite product measures.
Chapter 9. Web Derivatives
Blankets.
Webs.
Tiles of Various Sorts.
Notation.
Chapter 10. Classical Differentiation
Functions of Bounded Variation.
Absolutely Continuous Functions.
Appendix A. The Construction of Definitions
Structure of Basic Forms.
Structure of Definitions.
Adherence and Translatability.
Appendix B. The Consistency of the Axiom of Size
Appendix F. Integration with Respect to Addor Functions
Appendix G. The Henstock-Kurzweil Integral
Index of Constants
III. EDITORIAL EXPLANATIONS
Chapter 0. Language and Inference
Chapter 1. Logic
Chapter 2. Set Theory
Chapter 3. Elementary Analysis
Chapter 4. Metrics
Chapter 5. Measure
Chapter 6. Linear Measure and Total Variation
Chapter 7. Integration
Chapter 8. Product Measures
Chapter 9. Web Derivatives
Chapter 10. Classical Differentiation
Appendices
Printing Considerations
IV. COMPUTERIZED PARSING AND PROOF CHECKING
Chapter 0. Language and Inference
INTRODUCTION
FREE VARIABLES AND FORMULAS
INDICIAL AND ACCEPTED VARIABLES
RULES OF INFERENCE; THEOREMS
THEORY OF NOTATION
DEMONSTRATIONS
CHAINS
Chapter 1. Logic
PRELIMINARIES
SENTENCE LOGIC
PREDICATE LOGIC
SUPPLEMENTARY RULES OF INFERENCE
Chapter 2. Set Theory
PRELIMINARIES
SOME ASPECTS OF EQUALITY
CLASSIFICATION
THE ROLE OF REPLACEMENT
SINGLETONS
ORDERED PAIRS
SUBSTITUTION
UNICITY
RELATIONS
FUNCTIONS
ORDINALS
DEFINITION BY INDUCTION
REGULARITY AND CHOICE
MAXIMALITY
WELL ORDERING
NATURAL NUMBERS
SEQUENCES
REITERATION
FIXED SETS AND BIPARTITION
EQUINUMEROSITY
CARDINALS
CARDINALITY
CARDINAL ARITHMETIC
DIRECT EXTENSIONS
FAMILIES OF SETS
TUPLES
Chapter 3. Elementary Analysis
THE NUMBER SYSTEMS
PLUS AND TIMES
RUNS AND LIMITS
FINITE SUMMATION
INFINITE SUMMATION
Summation by Positive Distribution
Summation by Partition
Summation by Finite Partition
Positive Summation by Partition
Summation by Substitution
Summation by Transplantation
Summation by Commutation
Dominated Summation by Distribution
FINITE PRODUCTS
INFINITE PRODUCTS
Chapter 4. Metrics
INITIAL DEFINITIONS
OPEN AND CLOSED SETS
COMPLETENESS
SEPARABILITY
COMPACTNESS
CONTINUITY
METRICS ON THE LINE AND PLANE
LINEAR SPACES
EUCLIDEAN AND HILBERT SPACES
SIMPLE METRICS
CLUSTER AND CONDENSATION POINTS
CATEGORY 1 AND CATEGORY 2
EXERCISES
Chapter 5. Measure
PRELIMINARIES
FUNDAMENTALS
THE MEASURABILITY OF CERTAIN SETS
METRIC FUNDAMENTALS
CONSTRUCTED MEASURES
APPROXIMATIONS
THE LEBESGUE DECOMPOSITION
LEBESGUE MEASURE
Chapter 6. Linear Measure and Total Variation
TOTAL VARIATION
LINEAR MEASURE
Chapter 7. Integration
GENERAL METHODS
INTEGRATION BY REFINEMENT
THE OSCILLATION INTEGRAL
INTEGRATION OF STEP FUNCTIONS
MEASURABLE FUNCTIONS
MEASURE AND MEAN CONVERGENCE
LEBESGUE METRICS
APPROXIMATIONS
COMPLETELY ADDITIVE FUNCTIONS
SPECIAL INTEGRALS
MEASURE AND INTEGRAL SIZE
Chapter 8. Product Measures
GENERAL FUBINI THEORY
TOPOLOGICAL MEASURES
TOPOLOGICAL FUBINI THEORY
GENERAL PRODUCT MEASURES
GENERAL TOPOLOGICAL MEASURES
Chapter 9. Web Derivatives
WEBS
SOME TILE THEORY
DERIVATIVES AND DENSITIES
TILES OF VARIOUS SORTS
THE INTEGRATION OF THE DERIVATIVE
DIFFERENTIATION OF THE INTEGRAL
Chapter 10. Classical Differentiation
PRELIMINARIES
MONOTONE FUNCTIONS
FUNCTIONS OF BOUNDED VARIATION
ABSOLUTELY CONTINUOUS FUNCTIONS
THE DERIVATIVE OF ONE FUCNTION WITH RESPECT TO ANOTHER
APPENDIX A THE CONSTRUCTION OF DEFINITIONS
THE STRUCTURE OF BASIC FORMS
THE STRUCTURE OF DEFINITIONS
ADHERENCE AND TRANSLATABILITY
APPENDIX B THE CONSISTENCY OF THE AXIOM OF SIZE
APPENDIX C SUGGESTED READING
APPENDIX D PUBLICATIONS OF A. P. MORSE
BOOKS
ARTICLES BY MORSE
ARTICLES CO-AUTHORED BY MORSE
APPENDIX E ERRATA TO A THEORY OF SETS, SECOND EDITION
APPENDIX F INTEGRATION WITH RESPECT TO ADDOR FUNCTIONS
APPENDIX G THE HENSTOCK-KURZWEIL INTEGRAL
Index of Constants
Part I. Alphanumeric Constants
Part II. Nonalphanumeric Constants
Part III. Metamathematical Terminology
