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V.B.Schechtman

Axiomatic Set Theory (Spring 1997)

• Axiomatic theories.
• Theory ZF.
• Well-ordered sets. Ordinals. Transifinite induction.
• Comparison of cardinalities. Cantor-Bernstein theorem. Cantor theorem.
• Hartogs' Theorem. Cardinals. Squares of cardinals.
• Confinality. Singular and regular cardinals.
• The sets Z, Q, R. Continuum hypothesis.
• Finite and D-finite sets.
• Axiom of choice, Zermelo theorem, Zorn lemma.
• Cardinal arithmetic. Powers of cardinals. Generalized Continuum hypothesis (GCH).
• Models of first order theories.
• Notations of formulas in ZF. Provability predicate.
• Truth in a model. Consistency and model existence axioms.
• Transitive classes. Extensionality.
• Absolute formulas. Operations defined by formulas.
• Examples of absolute formulas and operations. Goedel operations.
• Almost universal classes. Sufficient conditions to get a model of ZF.
• Regularity axiom (AR). Von Neumann's universum. Relative consistency of (AR).
• Goedel operations and models of ZF.
• Constructive sets and constructivity axiom (V=L).
• Relative consistency of V=L.
• Consistency of the Axiom of Choice.
• Isomorphism theorem. Reflexivity principle.
• Consistency of GCH.
• Complete Boolean algebras.
• Generic sets and Boolean valued models of ZF.
• Independency of V=L.
• Independency of GCH.