NONSTANDARD ANALYSIS

By PAUL JACKSON

CONTENTS PRELIMINARIES INTRODUCTION GENERAL REMARKS ARCHIMEDEAN FIELDS AND IN?NITESIMALS SUPERSTRUCTURES, SENTENCES, AND INTERPRETATIONS SUPERSTRUCTURES FORMAL LANGUAGE INTERPRETATIONS NONSTANDARD MODELS THE THREE FUNDAMENTAL PRINCIPLES ELEMENTARY EMBEDDINGS AND THE TRANSFER PRINCIPLE THE STANDARD DE?NITION PRINCIPLE THE INTERNAL DE?NITION PRINCIPLE EXISTENCE OF EXTERNAL SETS NONSTANDARD ULTRAPOWER MODELS ULTRA?LTERS ULTRAPOWERS EMBEDDING IN A SUPERSTRUCTURE NONSTANDARD REAL ANALYSIS HYPERREAL NUMBERS HYPERREAL AND HYPERNATURAL NUMBERS INTERPRETATION OF THE STANDARD PART HOMOMORPHISM THE PERMANENCE PRINCIPLE AND ?-?NITE SETS CALCULUS SEQUENCES SETS FUNCTIONS ENLARGEMENTS AND SATURATED MODELS ENLARGEMENTS, SATURATION, AND CONCURRENCY SATURATED MODELS MODELS FOR ENLARGEMENTS COMPACT ENLARGEMENTS POLYSATURATED MODELS FUNCTIONALS, GENERALIZED LIMITS, AND ADDITIVE MEASURES NORMED SPACES LINEAR FUNCTIONALS AND OPERATORS HAHN-BANACH AND BANACH-MAZUR LIMITS ADDITIVE MEASURES NONSTANDARD TOPOLOGY AND FUNCTIONAL ANALYSIS TOPOLOGIES AND FILTERS TOPOLOGICAL SPACES FILTERS IN NONSTANDARD ANALYSIS TOPOLOGIES IN NONSTANDARD ANALYSIS FUNCTIONS IN NONSTANDARD TOPOLOGY UNIFORM STRUCTURES UNIFORM SPACES NONSTANDARD HULLS TOPOLOGICAL VECTOR SPACES MISCELLANEOUS LOEB MEASURES DISTRIBUTIONS