Fast Track to Forcing 1st Edition Mirna 1108351964 9781108351966

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Product details:

• ISBN 10: 1108351964
• ISBN 13: 9781108351966
• Author: Mirna Džamonja

This quick yet detailed introduction to set theory and forcing builds the reader’s intuition about it as much as the mathematical detail. Intuition, rather absent from the existing literature on the subject, here plays a large role. The reader will not only learn the facts, but will understand why they are true and will be brought to ask: what else could be true? Having presented forcing in Part I, the second part of the book discusses contemporary issues in the theory of forcing. It includes known and some previously unpublished results as well as many open questions. This is ideal for those who want to start a research career in forcing but do not have a personal interlocutor. Obviously, not everything about forcing is in this book. Many references are included to help the reader further explore the vast amount of research literature available on the subject.

Table of contents:

1 Introduction
2 Axiomatic Systems
2.1 First Order Logic
2.2 Language of Set Theory
3 Zermelo–Fraenkel Axioms and the Axiom of Choice
3.1 Axioms of Set Theory
4 Well Orderings and Ordinals
4.1 Arithmetic Properties of Ordinals
4.2 More Countable Sets
4.3 Uncountable Ordinals
5 Cardinals
6 Models and Independence
7 Some Class Models of ZFC
7.1 The Cumulative Hierarchy V
7.2 Regularity and Singularity
7.3 The Constructible Universe L
7.4 Reflection Principle
7.5 ZFC in L
7.6 Facts About L
7.7 Mostowski Collapse
7.8 Trees and the Fine Structure of L
8 Forcing
8.1 Logic of Forcing
8.2 Combinatorics of Forcing
8.3 M[G] 8.4 The Forcing Relation
8.5 ZFC in M[G] 8.6 Ordinals in M[G] 8.7 What is New in M[G]?
9 Violating CH
9.1 Chain Conditions and Closure
9.2 Canonical Names, again
PART TWO WHAT IS NEW IN SET THEORY
10 Introduction to Part Two
11 Classical Extensions
11.1 Lévy Collapse
11.2 Solovay Model
12 Iterated Forcing and Martin’s Axiom
12.1 Martin’s Axiom
13 Some More Large Cardinals
13.1 Prikry Forcing
13.2 Supercompact Cardinals and Laver Preservation
14 Limitations of Martin’s Axiom and Countable Supports
14.1 Laver Reals and Countable Support Iterations
15 Proper Forcing and PFA
15.1 P[sub(κ)]λ-combinatorics
15.2 Definition and Properties of Proper Forcing
15.3 Axiom A and Proper Forcing
15.4 Iteration of Proper Forcing with Countable Supports
15.5 Proper Forcing Axiom
15.6 On ℵ[sub(1)]-dense Sets of Reals
15.7 Further on PFA
16 ℵ[sub(2)] and other Successors of Regulars
16.1 A version of MA for Successors of Regulars
16.2 A Different View of ccc for a new Forcing Axiom
16.3 Strong κ[sup(+)]-cc Forcing without Combinatorics
16.4 Method of Side Conditions
16.5 Side Conditions for ω[sub(2)] 16.6 Some Challenges
17 Singular Cardinal Hypothesis and Some PCF
17.1 Easton Forcing
17.2 Large Cardinals and Singular Cardinals
18 Forcing at Singular Cardinals and Their Successors
18.1 Unions of Ultrafilters
18.2 Tree Property at ℵ[sub(ω+1)]

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