eBook ISBN:  9780821877678 
Product Code:  CONM/176.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
eBook ISBN:  9780821877678 
Product Code:  CONM/176.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 

Book DetailsContemporary MathematicsVolume: 176; 1994; 135 ppMSC: Primary 03; Secondary 54
This work proposes a major new extension of “non”standard mathematics. Addressed to a general mathematical audience, the book is intended to be philosophically provocative. The model theory on which “non”standard mathematics has been based is first reformulated within point set topology, which facilitates proofs and adds perspective. These topological techniques are then used to give new, uniform conservativity proofs for the various versions of “non”standard mathematics proposed by Nelson, Hrbáček, and Kawai. The proofs allow for sharp comparison. Addressing broader issues, Ballard then argues that what is novel in these forms of “non”standard mathematics is the introduction, however tentative, of relativity in one's mathematical environment. This hints at the possibility of a mathematical environment which is radically relativistic. The work's major and final feature is to present and prove conservative a version of “non”standard mathematics which, for the first time, illustrates this full radical relativism. The book is entirely selfcontained, with all necessary background in point set topology, model theory, “non”standard analysis, and set theory provided in full.
ReadershipGeneral mathematical audiences.

Table of Contents

Chapters

Introduction

Part 1: Preliminaries

Chapter 1. Point Set Topology

Chapter 2. Model Theory

Chapter 3. "Non" standard Analysis

Part 2: Topological Aspects

Chapter 4. Introduction

Chapter 5. Theory of CL Spaces

Chapter 6. Topological Determinacy of Local Internal Domains

Chapter 7. Topological Determinacy of Internal Domains

Part 3: Set Theoretic Aspects

Chapter 8. Introduction

Chapter 9. Standard Set Theory

Chapter 10. Current "Non" standard Set Theories

Chapter 11. Proofs Of Conservativity

Chapter 12. Critical Review With Proposal: EST

Chapter 13. Conservativity of EST

Chapter 14. Concluding Remarks

References

Index

Symbols


Reviews

The author contributes a number of sweeping ideas toward a new settheoretic foundation for nonstandard analysis ... places a number of important topics in a new light.
Mathematical Reviews 
In this interesting research monograph the author shows that the theory of nonstandard mathematics (Robinson, Nelson, etc.) can be developed from a topological viewpoint, and that, by this approach, new insight is gained. By adding ample background material and comments, the author has tried to make his exposition as accessible to a broad audience of mathematicians as possible.
Monatshefte für Mathematik


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This work proposes a major new extension of “non”standard mathematics. Addressed to a general mathematical audience, the book is intended to be philosophically provocative. The model theory on which “non”standard mathematics has been based is first reformulated within point set topology, which facilitates proofs and adds perspective. These topological techniques are then used to give new, uniform conservativity proofs for the various versions of “non”standard mathematics proposed by Nelson, Hrbáček, and Kawai. The proofs allow for sharp comparison. Addressing broader issues, Ballard then argues that what is novel in these forms of “non”standard mathematics is the introduction, however tentative, of relativity in one's mathematical environment. This hints at the possibility of a mathematical environment which is radically relativistic. The work's major and final feature is to present and prove conservative a version of “non”standard mathematics which, for the first time, illustrates this full radical relativism. The book is entirely selfcontained, with all necessary background in point set topology, model theory, “non”standard analysis, and set theory provided in full.
General mathematical audiences.

Chapters

Introduction

Part 1: Preliminaries

Chapter 1. Point Set Topology

Chapter 2. Model Theory

Chapter 3. "Non" standard Analysis

Part 2: Topological Aspects

Chapter 4. Introduction

Chapter 5. Theory of CL Spaces

Chapter 6. Topological Determinacy of Local Internal Domains

Chapter 7. Topological Determinacy of Internal Domains

Part 3: Set Theoretic Aspects

Chapter 8. Introduction

Chapter 9. Standard Set Theory

Chapter 10. Current "Non" standard Set Theories

Chapter 11. Proofs Of Conservativity

Chapter 12. Critical Review With Proposal: EST

Chapter 13. Conservativity of EST

Chapter 14. Concluding Remarks

References

Index

Symbols

The author contributes a number of sweeping ideas toward a new settheoretic foundation for nonstandard analysis ... places a number of important topics in a new light.
Mathematical Reviews 
In this interesting research monograph the author shows that the theory of nonstandard mathematics (Robinson, Nelson, etc.) can be developed from a topological viewpoint, and that, by this approach, new insight is gained. By adding ample background material and comments, the author has tried to make his exposition as accessible to a broad audience of mathematicians as possible.
Monatshefte für Mathematik