Foundations of constructive mathematics : metamathematical studies  U.S. ~ Germany

Michael J. Beeson

[目次]

  • One. Practice and Philosophy of Constructive Mathematics.- I. Examples of Constructive Mathematics.- 1. The Real Numbers.- 2. Constructive Reasoning.- 3. Order in the Reals.- 4. Subfields of R with Decidable Order.- 5. Functions from Reals to Reals.- 6. Theorem of the Maximum.- 7. Intermediate Value Theorem.- 8. Sets and Metric Spaces.- 9. Compactness.- 10. Ordinary Differential Equations.- 11. Potential Theory.- 12. The Wave Equation.- 13. Measure Theory.- 14. Calculus of Variations.- 15. Plateau's Problem.- 16. Rings, Groups, and Fields.- 17. Linear Algebra.- 18. Approximation Theory.- 19. Algebraic Topology.- 20. Standard Representations of Metric Spaces.- 21. Some Assorted Problems.- II. Informal Foundations of Constructive Mathematics.- 1. Numbers.- 2. Operations or Rules.- 3. Sets and Presets.- 4. Constructive Proofs.- 5. Witnesses and Evidence.- 6. Logic.- 7. Functions.- 8. Axioms of Choice.- 9. Ways of Constructing Sets.- 10. Definite Presets.- III. Some Different Philosophies of Constructive Mathematics.- 1. The Russian Constructivists.- 2. Recursive Analysis.- 3. Bishop's Constructivism.- 4. Objective Intuitionism.- 5. Sets in Intuitionism.- 6. Brouwerian Intuitionism.- 7. Martin-Lof s Philosophy.- 8. Church's Thesis.- IV. Recursive Mathematics: Living with Church's Thesis.- 1. Constructive Recursion Theory.- 2. Diagonalization and "Weak Counterexamples".- 3. Continuity of Effective Operations.- 4. Specker Sequences.- 5. Failure of Konig's Lemma: Kleene's Singular Tree.- 6. Singular Coverings.- 7. Non-Uniformly Continuous Functions.- 8. The Infimum of a Positive Function.- 9. Theorem of the Maximum Revisited.- 10. The Topology of the Disk in Recursive Mathematics.- 11. Pointwise Convergence Versus Uniform Convergence.- 12. Connectivity of Intervals.- 13. Another Surprise in Recursive Topology.- 14. A Counterexample in Descriptive Set Theory.- 15. Differential Equations with no Computable Solutions.- V. The Role of Formal Systems in Foundational Studies.- 1. The Axiomatic Method.- 2. Informal Versus Formal Axiomatics.- 3. Adequacy and Fidelity: Criteria for Formalization.- 4. Constructive and Classical Mathematics Compared.- 5. Arithmetic of Finite Types.- 6. Formalizing Constructive Mathematics in HA?.- Two. Formal Systems of the Seventies.- VI. Theories of Rules.- 1. The Logic of Partial Terms.- 2. Combinatory Algebras.- 3. Axiomatizing Recursive Mathematics.- 4. Term Reduction and the Church-Rosser Theorem Ill.- 5. Combinatory Logic and ?-Calculus.- 6. Term Models.- 7. Continuous Models.- 8. Finite Type Structures and Continuity in Combinatory Algebras.- 9. Set-Theoretic and Topological Models.- 10. Discussion: Adequacy and Fidelity of EON?.- VII. Readability.- 1. Definition and Soundness of Realizability.- 2. Realizability and Models.- 3. Some Simple Applications.- 4. Existence Properties.- 5. q-Realizability.- 6. Rules of Choice.- 7. Discussion: Numerical Meaning.- VIII. Constructive Set Theories.- 1. Intuitionistic Zermelo-Fraenkel Set Theory, IZF.- 2. Non-Extensional Set Theory.- 3. The Double-Negation Interpretation for IZF.- 4. Realizability for Set Theory Without Extensionality.- 5. Realizability and Models.- 6. Realizability for IZF.- 7. Connection with Realizability for Arithmetic.- 8. Consistency of Church's Thesis with IZF.- 9. The Numerical Existence Property for IZF.- 10. Discussion.- 11. The Theory B.- 12. More Discussion.- 13. Intermediate Constructive Set Theories.- IX. The Existence Property in Constructive Set Theory.- 1. Introduction.- 2. The Set Existence Property for HAS.- 3. The Existence Property in Set Theories.- X. Theories of Rules, Sets, and Classes.- 1. The Theory FML.- 2. The Theories EM0 and EM0 +J.- 3. Models of Feferman's Theories.- 4. Realizability.- 5. The Axiom of Choice.- 6. q-Realizability.- 7. Term Existence Property.- 8. Evaluation of Numerical Terms.- 9. Numerical Existence Property.- 10. Decidable Equality.- 11. Extensionality in Feferman's Theories.- 12. Some Remarks on Formalizing Mathematics in FML + DC.- 13. Functions, Operations, and Axioms of Choice.- 14. The Theory SOF (Sets, Operations, Functions).- 15. Some Theorems of SOF.- 16. Realizability for SOF.- 17. Discussion.- XI. Constructive Type Theories.- 1. Forms of Judgment.- 2. Philosophical Remarks on Sets, Categories, and Canonical Elements.- 3. Hypothetical Judgments.- 4. Families of Sets.- 5. Disjoint Union and Existential Quantification.- 6. Abstraction.- 7. Cartesian Product and Conjunction.- 8. The Constant E and Projection Functions.- 9. Product and Universal Quantification.- 10. Implication.- 11. N, Nk, and R.- 12. Disjoint Union.- 13. The I-Rules.- 14. How Martin-Lof's Rules Actually Define a Formal System.- 15. List of Rule Schemata of Martin-Lof's System ML0.- 16. Other Formulations of ML0.- 17. Interpretation of HA? + AC + EXT in ML0.- 18. Formalizing Mathematics in ML0.- 19. Is the Theory HA? + EXT + AC Constructive?.- 20. The Realizability Model of ML0.- 21. Martin-Lof's Universes.- 22. The Arithmetic Theorems of Martin-Lof's Systems, and a New Realizability for Arithmetic.- 23. Discussion.- Three. Metamathematical Studies.- XII. Constructive Models of Set Theory.- 1. Aczel's "Iterative Sets".- 2. Interpreting Subtheories of Intuitionistic ZF in Feferman's Theories.- XIII. Proof-Theoretic Strength.- 1. Definitions About Proof-Theoretic Strength.- 2. ?1/1-AC and ID1-.- 3. The Strength of Feferman's Theory EM0 + J and Related Theories.- 4. The Strength of Constructive Set Theory T2.- 5. The Strength of Martin-Lof's Theories.- 6. Theories with the Strength of Arithmetic.- 7. Conservative Extension Theorems.- XIV. Some Formalized Metamathematics and Church's Rule.- 1. The Theories Tb and Ta.- 2. Formalization of Normal-Term Arguments.- 3. Formalized Models.- 4. Church's Rule.- 5. Truth Definitions and Reflection Principles.- 6. Formalized Realizability and Existence Properties.- 7. Discussion.- XV. Forcing.- 1. Ordinary Forcing.- 2. Conservative Extension Results.- 3. Uniform Forcing.- XVI. Continuity.- 1. Continuity Principles.- 2. Continuity and Church's Thesis.- 3. Consistency of Brouwer's Principle and Uniform Continuity.- 4. Derived Rules Related to Continuity.- Four. Metaphilosophical Studies.- XVII. Theories of Rules and Proofs.- 1. Review of the Relevant Literature.- 2. The Main Issues.- 3. A Theory of Rules and Proofs.- 4. Undecidability of the Proof-Predicate in C.- 5. Consistency of C.- 6. Discussion.- 7. Frege Structures.- 8. Existence of Frege Structures.- 9. Set Theory and Frege Structures.- 10. A Theory of Rules, Proofs, and Sets.- Historical Appendix.- 1. From Gauss to Zermelo: The Origins of Non-Constructive Mathematics.- 2. From Kant to Hilbert: Logic and Philosophy.- 3. Brouwer and the Dutch Intuitionists.- 4. Early Formal Systems for Intuitionism.- 5. Kleene: The Marriage of Recursion Theory and Intuitionism.- 6. The Russian Constructivists and Recursive Analysis.- 7. Model Theory of Intuitionistic Systems.- 8. Logical Studies of Intuitionistic Systems.- 9. Bishop and his Followers.- 10. The Latest Decade.- References.- Index of Axioms, Abbreviations, and Theories.- Index of Names.- Index of Symbols.

「Nielsen BookData」より

この本の情報

書名 Foundations of constructive mathematics : metamathematical studies
著作者等 Beeson, Michael J.
Beeson M.J.
シリーズ名 Ergebnisse der Mathematik und ihrer Grenzgebiete
巻冊次 U.S.
Germany
出版元 Springer-Verlag
刊行年月 c1985
版表示 Softcover reprint of the original 1st ed. 1985
ページ数 xxiii, 466 p.
大きさ 25 cm
ISBN 3540121730
0387121730
9783642689543
NCID BA00496985
※クリックでCiNii Booksを表示
言語 英語
出版国 ドイツ
この本を: 
このエントリーをはてなブックマークに追加

このページを印刷

外部サイトで検索

この本と繋がる本を検索

ウィキペディアから連想