Syllabus for T. Jech's lectures at the
Fall school(Sept.'01)

Impossibilities in set theory

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Lecture 1. Impossibility of a consistency proof.

We give a proof of Goedel's Second Incompleteness Theorem stating that there is no proof in set theory of the consistency of set theory.

lecture 2. Inaccessibility of measurable cardinals.

We introduce the measure problem and show that the assumption of the existence of a nontrivial sigma additive measure implies the existence of inaccessible cardinals, and is therefore unprovable in set theory.

Lecture 3. Measurable cardinals and constructible sets.

We introduce elementary embeddings of transitive models and prove Scott's theorem stating that in the constructible universe L, measurable cardinals do not exist.

Lecture 4. Impossibility of a nontrivial embedding of the universe.

We prove Kunen's theorem that shows that there exists no nontrivial elementary embedding of the universe V.

Lecture 5. The Singular Cardinal Problem.

We introduce the problem and present Silver's theorem on the exponentiation of singular cardinals of uncountable cofinality.

Lecture 6. Jensen's Covering Theorem.

We state the Covering Theorem, prove its consequences for cardinal arithmetic, and outline its proof and state some generalizations.

Lecture 7. Projective Determinacy.

We introduce infinite games and discuss their determinacy, with emphasis on its relation to Lebesgue measurability, the Baire property and the perfect set property.

Lecture 8. Independence of the Axiom of Choice.

We discuss the role of the Axiom of Choice, prove its independence from a weak version of set theory and discuss the method of forcing.