Weekly outline

  • General

    Set Theory and Forcing, Fall 2014, 7.5 hp

    The course treats modern set theory and independence results. Many statements of mathematics are neither provable nor disprovable from the basic axioms of set theory, they are independent of the axioms. The most famous one is Cantor's Continuum Hypothesis. The course gives an introduction to methods for proving independence results, in particular Cohen's method of forcing.

    Contents

    Classical set theory: Axioms of Zermelo-Fraenkel (ZF) set theory. Ordinal numbers, well-orderings and cardinal number theory in ZF. The Continuum Hypotheses. Independence results for ZF: permutation models and independence of the axiom of choice. Forcing and the independence of the Continuum. Boolean-valued models. Consequences of independence results for mathematics. A selection of the following subjects: Infinitary combinatorics, Gödel's constructible sets and the V=L axiom. Alternative axioms: Projective Determinacy and Martin's axiom. Large cardinals. Constructive set theory: CZF and IZF.

     

    Course literature

    John L. Bell. Boolean-Valued Models and Independence Proofs. Third Edition. Oxford 2005. (E-book)

    Peter Aczel and  Michael Rathjen. Notes on Constructive Set Theory. Institut Mittag-Leffler 2001. www.mittag-leffler.se/preprints/files/IML-0001-40.pdf

    Yiannis N. Moschovakis. Notes on Set Theory.  2nd ed. Springer 2006.  (E-book)

    Thomas Jech, Set Theory, Third edition. Springer 2000. (E-book)

    Reference literature

    Kenneth Kunen, Set Theory – an Introduction to Independence Proofs. North-Holland 1980.

    Note: The "E-books" above can accessed through the Stockholm University Library (www.sub.su.se) provided you have library card (which all students registered at SU can get).

    Teacher

    Palmgren, Erik

    Language: The course will be given in English if there is interest in this.

    Examination: Home work problems through out the course (75% of total score) and a short written exam (25% of total score) at the end of the course. See grading criteria below. The home work problems will be posted below, marked Home Work.

    Written exam: Tuesday 16 December, 15:15 - 17:15 , Room 22, Building 5, Kräftriket (Department of Mathematics).

    The written exam is marked. The last set of home work problems will be marked before January 7.

    Schedule:

    Time and place

  • Lecture 1: September 1

    Some history of foundational systems for mathematics: set theories (classical, intuitionistic and constructive) and type theories (classical and constructive). Introduction to consistency and independence questions. The continuum hypothesis.

    Axioms of Zermelo-Fraenkel set theory. First-order classical and intutionistic logic. Examples of formal proofs in set theory. See Handout 1. First half of section 2 of Aczel and Rathjen 2001 covered. Study rest of section 2 and section 3 until next time.

  • Lecture 2: September 8

    Axiom of replacement. Class notation. Proving that a class is not a set. Power set axiom versus the Fullness axiom. Axioms of CZF and IZF. CZF+LEM is equivalent to ZF.

    Study chapters 1 and 2 of Moschovakis 2006 until next time.

  • Lecture 3: September 15

    Encoding pairs and cartesian products. Cantor's theorem. Equinumerosity (equally numerous, equal cardinality). Schröder-Bernstein theorem and its use in cardinal computations. Some important cardinality identities. Equivalents of the axiom of choice: existence of sections of surjective functions, existence of selection functions for multivalued relations, existence of choice function, Zorn's Lemma, cardinal comparison principle.

  • Lecture 4: September 22

    Linear orders and wellorders, and order preserving maps. The well ordering principle (WO). WO implies AC. Ordinals as sets, and their basic properties. Order types. Successor and limit ordinals. Ordinal induction. (Jech 2000, pp 17-21)

  • Lecture 5: September 29

    Transfinite recursion. Ordinal arithmetic. Addition and mulitplication of ordinals in terms of operations on wellorders. Cantor's normal form. Cardinal numbers. Hartog's function - find the cardinal. Aleph numbers. Proof of the wellordering principle from AC using transfinite recursion. Cardinal arithmetic operations. Wellordering of Ord x Ord. (Jech 2000,  pp 21-24, pp 27-30)

    • Lecture 6: October 6

      Wellordering of Ord x Ord (cont.). Proof of k x k = k for cardinals.  Cofinality of ordinals. Regular and singular cardinals. Characterization of singular cardinals. Computing cardinality of general unions, disjoint unions and general products. König's theorem generalizing several cardinal inequalities. (Jech 2000,  pp 30 - 35, p 49, pp 51 - 55).

      • Lecture 7 = Guest lecture on October 15

        15 October: Guest lecture (SMC Colloquium) 15.15-16.15, Oskar Klein Auditorium, AlbaNova University Center. See:

        Michael Rathjen: Is Cantor’s continuum problem still open?

        Reference: M.Rathjen. Indefiniteness in semi-intuitionistic set theories: On a conjecture of Feferman, Preprint 2014

        http://www1.maths.leeds.ac.uk/~rathjen/Indefinite.pdf

        • Lecture 8: October 20

          König's Theorem on infinite sums and products of cardinals. The Gimel function. Computation of cardinal exponentiation with and without GCH. Strong limit cardinals. Inaccessible cardinals. Formalized soundness. ZFC cannot prove inaccessible cardinals exist. The cumulative hierarchy. Rank of a set. (Jech  2000, pp 54-58, p. 64, pp 155-157.)

          • Lecture 9: November 3

            Applications of ranks of sets. Transitive closures. Transitive classes. Well-founded induction. A comparison to CZF. Mostowski's collapsing theorem. More on models of sets. Skolem's paradox. Relative consistency of the regularity axiom. Transitive models. Absoluteness. Definition of the constructible hierarchy.

            • Lecture 10: November 10

              Properties of Gödel's L (cont.). Consequences of V=L: AC and GCH.

              • NO LECTURE November 17

                • Lecture 11: November 24

                  Properties of Gödel's L (cont.). The Levy hierarchy. Delta_1 absolute notions. Consequences of V=L: AC and GCH.

                • Lecture 12: December 1

                  Main theorems of forcing. Generic sets. 

                  • Lecture 13: December 8

                    Boolean valued models of the language of set theory. Soundness and consistency proofs using such models..Complete Boolean algebras and Heyting algebras. Homomorphisms of Boolean algebras. Filters and ideals on Boolean algebras. Ultrafilters and prime ideals. Embedding a (separative) partial order in a complete Boolean algebra.

                  • Lecture 14: December 15

                    • Written exam

                      December 16, 15:15 - 17:15, Room 22, Building 5, Kräftriket (Department of Mathematics).