Set Theory and its Philosophy A Critical Introduction

Name: Set Theory and its Philosophy A Critical Introduction | A Critical Introduction
Brand: Oxford University Press
Price: 57.89 EUR
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A Critical Introduction

POTTER MICHAEL(Autor*in)

Oxford University Press

1. Auflage

Erschienen am 15. Januar 2004

360 Seiten

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Inhalt

Intro
Contents
I: Sets
Introduction to Part I
1 Logic
1.1 The axiomatic method
1.2 The background logic
1.3 Schemes
1.4 The choice of logic
1.5 Definite descriptions
Notes
2 Collections
2.1 Collections and fusions
2.2 Membership
2.3 Russell's paradox
2.4 Is it a paradox?
2.5 Indefinite extensibility
2.6 Collections
Notes
3 The hierarchy
3.1 Two strategies
3.2 Construction
3.3 Metaphysical dependence
3.4 Levels and histories
3.5 The axiom scheme of separation
3.6 The theory of levels
3.7 Sets
3.8 Purity
3.9 Well-foundedness
Notes
4 The theory of sets
4.1 How far can you go?
4.2 The initial level
4.3 The empty set
4.4 Cutting things down to size
4.5 The axiom of creation
4.6 Ordered pairs
4.7 Relations
4.8 Functions
4.9 The axiom of infinity
4.10 Structures
Notes
Conclusion to Part I
II: Numbers
Introduction to Part II
5 Arithmetic
5.1 Closure
5.2 Definition of natural numbers
5.3 Recursion
5.4 Arithmetic
5.5 Peano arithmetic
Notes
6 Counting
6.1 Order relations
6.2 The ancestral
6.3 The ordering of the natural numbers
6.4 Counting finite sets
6.5 Counting infinite sets
6.6 Skolem's paradox
Notes
7 Lines
7.1 The rational line
7.2 Completeness
7.3 The real line
7.4 Souslin lines
7.5 The Baire line
Notes
8 Real numbers
8.1 Equivalence relations
8.2 Integral numbers
8.3 Rational numbers
8.4 Real numbers
8.5 The uncountability of the real numbers
8.6 Algebraic real numbers
8.7 Archimedean ordered fields
8.8 Non-standard ordered fields
Notes
Conclusion to Part II
III: Cardinals and Ordinals
Introduction to Part III
9 Cardinals
9.1 Definition of cardinals
9.2 The partial ordering
9.3 Finite and infinite
9.4 The axiom of countable choice
Notes
10 Basic cardinal arithmetic
10.1 Finite cardinals
10.2 Cardinal arithmetic
10.3 Infinite cardinals
10.4 The power of the continuum
Notes
11 Ordinals
11.1 Well-ordering
11.2 Ordinals
11.3 Transfinite induction and recursion
11.4 Cardinality
11.5 Rank
Notes
12 Ordinal arithmetic
12.1 Normal functions
12.2 Ordinal addition
12.3 Ordinal multiplication
12.4 Ordinal exponentiation
12.5 Normal form
Notes
Conclusion to Part III
IV: Further Axioms
Introduction to Part IV
13 Orders of infinity
13.1 Goodstein's theorem
13.2 The axiom of ordinals
13.3 Reflection
13.4 Replacement
13.5 Limitation of size
13.6 Back to dependency?
13.7 Higher still
13.8 Speed-up theorems
Notes
14 The axiom of choice
14.1 The axiom of countable dependent choice
14.2 Skolem's paradox again
14.3 The axiom of choice
14.4 The well-ordering principle
14.5 Maximal principles
14.6 Regressive arguments
14.7 The axiom of constructibility
14.8 Intuitive arguments
Notes
15 Further cardinal arithmetic
15.1 Alephs
15.2 The arithmetic of alephs
15.3 Counting well-orderable sets
15.4 Cardinal arithmetic and the axiom of choice
15.5 The continuum hypothesis
15.6 Is the continuum hypothesis decidable?
15.7 The axiom of determinacy
15.8 The generalized continuum hypothesis
Notes
Conclusion to Part IV
Appendices
A: Traditional axiomatizations
A.1 Zermelo's axioms
A.2 Cardinals and ordinals
A.3 Replacement
Notes
B: Classes
B.1 Virtual classes
B.2 Classes as new entities
B.3 Classes and quantification
B.4 Classes quantified
B.5 Impredicative classes
B.6 Impredicativity
B.7 Using classes to enrich the original theory
C: Sets and classes
C.1 Adding classes to set theory
C.2 The difference between sets and classes
C.3 The metalinguistic perspective
Notes
References
List of symbols
Index of definitions
A
B
C
D
E
F
G
H
I
K
L
M
N
O
P
Q
R
S
T
U
V
W
Z
Index of names
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
Z

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