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Preface | |
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The Elements | |
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Introduction | |
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Extensionality | |
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Sets | |
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Mathematical Proof | |
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Relations | |
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Some Binary Relations | |
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Equivalence Relations | |
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Orderings | |
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Functions | |
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Numbers | |
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Propositional Logic | |
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Introduction | |
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Induction and Formal Languages | |
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Syntax | |
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Semantics | |
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Unique Readability | |
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Recursive Definitions | |
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Expressive Completeness | |
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Expressively Complete Languages | |
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First-Order Logic: Syntax and Semantics | |
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Introduction | |
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Properties, Relations and Individuals | |
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Truth-Functional Composition | |
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Functions | |
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Identity | |
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Quantification | |
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Syntax | |
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Semantics | |
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Truth | |
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Logical Consequence and Other Logical Notions | |
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Models | |
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Equivalence Relations | |
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Orderings | |
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Structures for the First-Order Language with No Extralogical Symbols | |
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Groups | |
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Arithmetic | |
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Deduction | |
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Introduction | |
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The Basic Rule and the Connective Rules | |
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Propositional Deduction | |
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Substitution | |
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Quantifier and Identity Rules | |
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Establishing (First-Order) Deducibility Claims | |
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The Definition of Deducibility | |
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Deducibility in Different Languages | |
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Soundness and Completeness | |
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Soundness | |
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Completeness, Consistency and Model Existence | |
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Canonical Structures and Well-Rounded Sets, I: First-Order Languages without Identity | |
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Canonical Structures and Well-Rounded Sets, II: First-Order Languages with Identity | |
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A Well-Rounded Set of Sentences Is the Theory of its Canonical Structure | |
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Negation Completeness | |
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Henkin Constants and Henkin Axioms | |
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Consistent, Negation-Complete Henkin Sets Are Well Rounded | |
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Compactness | |
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Cardinality | |
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The Size of Infinity | |
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Criteria for Size Claims | |
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Counting | |
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Functional Criteria | |
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Same Size | |
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Big and Small | |
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Addition | |
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Multiplication | |
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Exponentiation | |
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Cardinality | |
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Contextual Definition | |
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Continuities | |
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Denumerable Sets | |
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Diagonals | |
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ZF | |
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Choice | |
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Zorn's Lemma | |
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Completeness for Uncountable Languages | |
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Expressive Limitations | |
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Representation and Indiscernibility | |
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Isomorphic Structures | |
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Isomorphic Structures and Contextual Definitions | |
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Representing Finite Structures | |
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The Downward Lowenheim-Skolem Theorem | |
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Skolem's Paradox | |
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The Upward Lowenheim-Skolem Theorem | |
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Finite Structures Again | |
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Nonstandard Models of Arithmetic | |
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Strict Dense Linear Orderings with no Minimal or Maximal Elements | |
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Categoricity in Power | |
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Decidability | |
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Decidable Sets of Sentences and Indiscernible Structures | |
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Vaught's Test | |
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Strict Dense Linear Orderings with No Minimal or Maximal Elements | |
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Successor Arithmetic | |
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Quantifier Elimination | |
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Strict Dense Linear Orderings with No Minimal or Maximal Elements | |
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Successor Arithmetic | |
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Ordered Successor Arithmetic | |
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The Decision Problem | |
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Decidable Structures | |
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Godel's First Incompleteness Theorem | |
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Set Theory Revisited | |
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Symbols and Notation | |
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Greek Letters | |
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Index | |