The Axiom of ChoiceComprehensive in its selection of topics and results, this self-contained text examines the relative strengths and consequences of the axiom of choice. Each chapter contains several problems, graded according to difficulty, and concludes with some historical remarks. An introduction to the use of the axiom of choice is followed by explorations of consistency, permutation models, and independence. Subsequent chapters examine embedding theorems, models with finite supports, weaker versions of the axiom, and nontransferable statements. The final sections consider mathematics without choice, cardinal numbers in set theory without choice, and properties that contradict the axiom of choice, including the axiom of determinacy and related topics |
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a₁ aleph antichain assume atoms automorphisms Axiom of Choice Axiom of Determinateness B₁ basic Cohen model Boolean algebra Boolean-valued models cardinal number choice function closed unbounded compact complete Boolean algebra construct contradiction countable sets D-finite define Dependent Choices disjoint E₁ E₂ elements Embedding Theorem equivalent exists finite sets finite subsets function f Hint holds isomorphic least support Lemma Let F Let G linear linearly ordered Math maximal measurable cardinal model of ZF Mostowski model nonempty sets normal filter one-to-one function one-to-one mapping Ordering Principle P₁ pairs partially ordered set partition permutation model Prime Ideal Theorem Principle of Dependent Problem properties prove Ramsey's Theorem real numbers regular cardinal RO(P satisfies Section Selection Principle sequence set of reals set theory symmetric model transitive model ultrafilter w₁ well-orderable well-ordered X₁ ZF+AC л Є