内容説明
Numbers imitate space, which is of such a di?erent nature -Blaise Pascal It is fair to date the study of the foundation of mathematics back to the ancient Greeks. The urge to understand and systematize the mathematics of the time led Euclid to postulate axioms in an early attempt to put geometry on a ?rm footing. With roots in the Elements, the distinctive methodology of mathematics has become proof. Inevitably two questions arise: What are proofs? and What assumptions are proofs based on? The ?rst question, traditionally an internal question of the ?eld of logic, was also wrestled with in antiquity. Aristotle gave his famous syllogistic s- tems, and the Stoics had a nascent propositional logic. This study continued with ?ts and starts, through Boethius, the Arabs and the medieval logicians in Paris and London. The early germs of logic emerged in the context of philosophy and theology. The development of analytic geometry, as exempli?ed by Descartes, ill- tratedoneofthedi?cultiesinherentinfoundingmathematics. Itisclassically phrased as the question ofhow one reconciles the arithmetic with the geom- ric. Arenumbers onetypeofthingand geometricobjectsanother? Whatare the relationships between these two types of objects? How can they interact? Discovery of new types of mathematical objects, such as imaginary numbers and, much later, formal objects such as free groups and formal power series make the problem of ?nding a common playing ?eld for all of mathematics importunate. Several pressures made foundational issues urgent in the 19th century.
目次
Handbook of Set Theory,
Volume I,
Akihiro Kanamori, 0. Introduction
Thomas Jech, 1. Stationary Sets
Andras Hajnal and Jean Larson, 2. Partition Relations
Stevo Todorcevic, 3. Coherent Sequences
Greg Hjorth, 4. Borel Equivalence Relations
Uri Abraham, 5. Proper Forcing
Andreas Blass, 6. Combinatorial Cardinal Characteristics of the Continuum
Tomek Bartoszynski, 7. Invariants of Measure and Category
Sy Friedman, 8. Constructibility and Class Forcing 48
Ralf-Dieter Schindler and Martin Zeman, 9. Fine Structure 52
Philip Welch, 10. S* Fine Structure 80
Volume II,
Patrick Dehornoy, 11. Elementary Embeddings and Algebra
James Cummings, 12. Iterated Forcing and Elementary Embeddings
Matthew Foreman, 13. Ideals and Generic Elementary Embeddings
Uri Abraham and Menachem Magidor, 14. Cardinal Arithmetic
Todd Eisworth, 15. Successors of Singular Cardinals
Moti Gitik, 16. Prikry-Type Forcings
Volume III,
William Mitchell, 17. Beginning Inner Model Theory
William Mitchell, 18. The Covering Lemma
John Steel, 19. An Outline of Inner Model Theory
Ernest Schimmerling, 20. A Core Model Tool Box and Guide
Steve Jackson, 21. Structural Consequences of AD
Itay Neeman, 22. Determinacy in L(R)
Peter Koellner and Hugh Woodin, 23. Large Cardinals from Determinacy
Paul Larson, 24. Forcing over Models of Determinacy
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