Math 281A (Set Theory) Fall 2013

Optional textbooks

Outline of lectures by week (subject to change)

  1. Extensionality and Separation axioms, Russell's paradox, classes, Pairing and Union axioms, Axiom of Infinity, natural numbers, ordinary induction, ordinals (parts of Jech, Ch. 1 and 2) Exercises. Solutions.
  2. Ordinals, transfinite induction, relations and functions, Replacement, transfinite recursion, normal functions, ordinal arithmetic (the rest of Jech, Ch. 1 and 2) Exercises. Solutions.
  3. Injections and surjections, finiteness, countability, cardinals, the power set axiom, uncountable sets, Cantor's theorem, Hartogs numbers (parts of Jech, Ch. 3) Exercises. Solutions.
  4. The aleph sequence, products and exponentiation, cardinal arithmetic, cofinality, regular and singular cardinals (the rest of Jech, Ch. 3) Exercises. Solutions.
  5. The Axiom of Choice, uncountable cofinalities, infinite sums and products, Koenig's theorem (parts of Jech, Ch. 5) Exercises. Solutions.
  6. The reals, the Baire Category Theorem, perfect sets, the Cantor–Bendixson Theorem (Jech, Ch. 4) Exercises. Solutions.
  7. The Axiom of Foundation, $\in$-induction and recursion, ranks of sets, the sets $V_\alpha$ and $H_\kappa$, the collection principle (Jech, Ch. 6) Exercises.
  8. General well-founded relations, Mostowski collapse, models of set theory, relative consistency (Jech, Ch. 12) Exercises.
  9. More models of set theory and relative consistency. (Jech, Ch. 12) Exercises.
  10. Model theory in $\mathsf{ZFC}$, soundness and completeness, Gödel's incompleteness theorems, compactness, the Löwenheim–Skolem theorem, the undefinability of truth (Jech, Ch. 12) No exercises assigned.

Homework policy

The homework assigned during a given week will usually be due at the beginning of the following Wednesday's lecture. If you prefer, you can e-mail it to me as a DVI or PDF file.

You may get help on the homework assignment from any source you choose, including other students and the Internet. Make sure to credit your collaborators and cite your sources (citation format is not important as long as you include enough information for me to find your source.)

Grading policy

The grade will be based entirely on homework. There will be no exams.