# Book:Raymond M. Smullyan/Set Theory and the Continuum Problem/Revised Edition

## Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (Revised Edition)

Published $\text {2010}$, Dover

ISBN 0-486-47484-4

### Contents

Preface to the Revised Edition
Preface

$\text I$ AXIOMATIC SET THEORY
CHAPTER $1$ GENERAL BACKGROUND
$\S 1$ What is infinity?
$\S 2$ Countable or uncountable?
$\S 3$ A non-denumerable set
$\S 4$ Larger and smaller
$\S 5$ The continuum problem
$\S 6$ Significance of the results
$\S 7$ Frege set theory
$\S 8$ Russell's paradox
$\S 9$ Zermelo set theory
$\S 10$ Sets and classes

CHAPTER $2$ SOME BASICS OF CLASS-SET THEORY
$\S 1$ Extensionality and separation
$\S 2$ Transitivity and supercompleteness
$\S 3$ Axiom of the empty set
$\S 4$ The pairing axiom
$\S 5$ The union axiom
$\S 6$ The power axiom
$\S 7$ Cartesian products
$\S 8$ Relations
$\S 9$ Functions
$\S 10$ Some useful facts about transitivity
$\S 11$ Basic universes

CHAPTER $3$ THE NATURAL NUMBERS
$\S 1$ Preliminaries
$\S 2$ Definition of the natural numbers
$\S 3$ Derivation of the Peano postulates and other results
$\S 4$ A double induction principle and its applications
$\S 5$ Applications to natural numbers
$\S 6$ Finite sets
$\S 7$ Denumerable classes
$\S 8$ Definition by finite recursion
$\S 9$ Supplement -- optional

CHAPTER $4$ SUPERINDUCTION, WELL ORDERING AND CHOICE
$\S 1$ Introduction to well ordering
$\S 2$ Superinduction and double superinduction
$\S 3$ The well ordering of $g$-towers
$\S 4$ Well ordering and choice
$\S 5$ Maximal principles
$\S 6$ Another approach to maximal principles
$\S 7$ Cowen's theorem
$\S 8$ Another characterization of $g$-sets

CHAPTER $5$ ORDINAL NUMBERS
$\S 1$ Ordinal numbers
$\S 2$ Ordinals and transitivity
$\S 3$ Some ordinals

CHAPTER $6$ ORDER ISOMORPHISM AND TRANSFINITE RECURSION
$\S 1$ A few preliminaries
$\S 2$ Isomorphisms of well orderings
$\S 3$ The axiom of substitution
$\S 4$ The counting theorem
$\S 5$ Transfinite recursion theorems
$\S 6$ Ordinal arithmetic

CHAPTER $7$ RANK
$\S 1$ The notion of rank
$\S 2$ Ordinal hierarchies
$\S 3$ Application to the $R_\alpha$ sequence
$\S 4$ Zermelo universes

CHAPTER $8$ FOUNDATION, $\epsilon$-INDUCTION, AND RANK
$\S 1$ The notion of well-foundedness
$\S 2$ Descending $\epsilon$-chains
$\S 3$ $\epsilon$-induction and rank
$\S 4$ Axiom $E$ and Von Neumann's principle
$\S 5$ Some other characterizations of ordinals
$\S 6$ More on the axiom of substitution

CHAPTER $9$ CARDINALS
$\S 1$ Some simple facts
$\S 2$ The Bernstein-Schröder theorem
$\S 3$ Denumerable sets
$\S 4$ Infinite sets and choice functions
$\S 5$ Hartog's theorem
$\S 6$ A fundamental theorem
$\S 7$ Preliminaries
$\S 8$ Cardinal arithmetic
$\S 9$ Sierpinski's theorem

$\text {II}$ CONSISTENCY OF THE CONTINUUM HYPOTHESIS
CHAPTER $10$ MOSTOWSKI-SHEPHERDSON MAPPINGS
$\S 1$ Relational systems
$\S 2$ Generalized induction and $\Gamma$-rank
$\S 3$ Generalized transfinite recursion
$\S 4$ Mostowski-Shepherdson maps
$\S 5$ More on Mostowski-Shepherdson mappings
$\S 6$ Isomorphisms, Mostowski-Shepherdson, well-orderings

CHAPTER $11$ REFLECTION PRINCIPLES
$\S 0$ Preliminaries
$\S 1$ The Tarski-Vaught theorem
$\S 2$ We add extensionality considerations
$\S 3$ The class version of the Tarski-Vaught theorems
$\S 4$ Mostowski, Shepherdson, Tarski, and Vaught
$\S 5$ The Montague-Levy reflection theorem

CHAPTER $12$ CONSTRUCTIBLE SETS
$\S 0$ More on first-order definability
$\S 1$ The class $L$ of constructible sets
$\S 2$ Absoluteness
$\S 3$ Constructible classes

CHAPTER $13$ $L$ IS A WELL-FOUNDED FIRST-ORDER UNIVERSE
$\S 1$ First-order universes
$\S 2$ Some preliminary theorems about first-order universes
$\S 3$ More on first-order universes
$\S 4$ Another result

CHAPTER $14$ CONSTRUCTIBILITY IS ABSOLUTE OVER $L$
$\S 1$ $\Sigma$-formulas and upward absoluteness
$\S 2$ More on $\Sigma$-definability
$\S 3$ The relation $y = \map \FF x$
$\S 4$ Constructibility is absolute over $L$
$\S 5$ Further results
$\S 6$ A proof that $L$ can be well-ordered

CHAPTER $15$ CONSTRUCTIBILITY AND THE CONTINUUM HYPOTHESIS
$\S 0$ What we will do
$\S 1$ The key result
$\S 2$ Gödel's isomorphism theorem (optional)
$\S 3$ Some consequences of Theorem $\text G$
$\S 4$ Metamathematical consequences of Theorem $\text G$
$\S 5$ Relative consistency of the axiom of choice
$\S 6$ Relative consistency of $\text {GCH}$ and $\text {AC}$ in class-set theory

$\text {III}$ FORCING AND INDEPENDENCE RESULTS
CHAPTER $16$ FORCING, THE VERY IDEA
$\S 1$ What is forcing?
$\S 2$ About equality
$\S 3$ What is modal logic?
$\S 4$ What is $S 4$ and why do we care?
$\S 5$ A classical embedding
$\S 6$ The basic idea

CHAPTER $17$ THE CONSTRUCTION OF $S 4$ MODELS FOR $ZF$
$\S 1$ What are the models?
$\S 2$ About equality
$\S 3$ The well founded sets are present
$\S 4$ Four more axioms
$\S 5$ The definability of forcing
$\S 6$ The substitution axiom system
$\S 7$ The axiom of choice
$\S 8$ Where we stand now

CHAPTER $18$ THE AXIOM OF CONSTRUCTIBILITY IS INDEPENDENT
$\S 1$ Introduction
$\S 2$ Ordinals are well-behaved
$\S 3$ Constructible sets are well-behaved too
$\S 4$ A real $S 4$ model, at last
$\S 5$ Cardinals are sometimes well-behaved
$\S 6$ The status of the generalized continuum hypothesis

CHAPTER $19$ INDEPENDENCE OF THE CONTINUUM HYPOTHESIS
$\S 1$ Power politics
$\S 2$ The model
$\S 3$ Cardinals stay cardinals
$\S 4$ $\text {CH}$ is independent
$\S 5$ Cleaning it up
$\S 6$ Wrapping it up

CHAPTER $20$ INDEPENDENCE OF THE AXIOM OF CHOICE
$\S 1$ A little history
$\S 2$ Automorphism groups
$\S 3$ Automorphisms preserve truth
$\S 4$ Model and submodel
$\S 5$ Verifying the axioms
$\S 6$ $\text {AC}$ is independent

CHAPTER $21$ CONSTRUCTING CLASSICAL MODELS
$\S 1$ On countable models
$\S 2$ Cohen's way
$\S 3$ Dense sets, filters, and generic sets
$\S 4$ When generic sets exist
$\S 5$ Generic extensions
$\S 6$ The truth lemma
$\S 7$ Conclusion

CHAPTER $22$ FORCING BACKGROUND
$\S 1$ Introduction
$\S 2$ Cohen's version(s)
$\S 3$ Boolean valued models
$\S 4$ Unramified forcing
$\S 5$ Extensions

BIBLIOGRAPHY
INDEX
LIST OF NOTATION

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