Handbook of Set Theory

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Matthew Foreman, Akihiro Kanamori
Springer Science & Business Media, Dec 10, 2009 - Mathematics - 2230 pages
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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.
 

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Contents

Stationary Sets
93
Partition Relations
129
Coherent Sequences
214
Borel Equivalence Relations
297
Proper Forcing
333
Combinatorial Cardinal Characteristics of the Continuum
395
Invariants of Measure and Category
491
Constructibility and Class Forcing
556
Cardinal Arithmetic
1148
Successors of Singular Cardinals
1229
PrikryType Forcings
1351
Beginning Inner Model Theory
1448
The Covering Lemma
1497
An Outline of Inner Model Theory
1595
A Core Model Toolbox and Guide
1685
Structural Consequences of AD
1752

Fine Structure
605
Sigma Fine Structure
657
Elementary Embeddings and Algebra
737
Iterated Forcing and Elementary Embeddings
775
Ideals and Generic Elementary Embeddings
885
Determinacy in LR
1877
Large Cardinals from Determinacy
1951
Forcing over Models of Determinacy
2121
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