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

Subject Matter

 * Set Theory
 * Class Theory

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
 * $\S 10$ Serts 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
 * $\S 2$ Isomprphism 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
 * $\S 5$ Hatrog'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



Exercise $5.6$: $B - \paren {A - B} = \O$
Chapter $2$: Some Basics of Class-Set Theory:

Source work progress
* : Chapter $3$: The Natural Numbers: $\S 5$ Applications to natural numbers: Exercise $5.1$