Arbeitsgemeinschaft zur Mathematischen Logik

Organizers

Time and Place

Wednesday 9:00-11:00 SRF

Program

In this research seminar we will treat two-cardinal problems. Shelah's historic forcing, Todorcevic's rho-functions, and morasses have been applied successfully in this area, but there are still many open problems. Our starting point is the first result in this area solved by Shelah using historic forcing. Then we look at the other methods which have been applied in this field and we look at some open problems.

  1. Introduction

    October 18th 2006
    Bernhard Irrgang

    Abstract An introduction to two-cardinal problems: A sketch of the forcing construction of [1].

  2. Historic Forcing I

    October 25th 2006
    Ioanna Dimitriou

    Abstract How to use Shelah's historic forcing to get a function with property delta

  3. Historic Forcing II

    November 8th 2006
    Ioanna Dimitriou

    Abstract Continuation of Historic forcing I.

  4. Historic Forcing III

    November 15th 2006
    Ioanna Dimitriou

    Abstract Continuation of Historic forcing I.

  5. Getting a superatomic Boolean Algebra

    November 22th 2006
    Bernhard Irrgang

    Abstract Using historic forcing we can get a function with property delta. In this session we see how to get a forcing extension with a superatomic algebra from this assumption.

  6. rho-functions

    November 29th 2006
    Jip Veldman

    7. Abstract Using coherent sequences and Todorcevic's minimal walks we can define the function Rho. In this session we define coherent sequences, minimal walks, and the function Rho. Then we show some of its basic properties, in particular its subadditivity.

  7. rho-functions II

    December 6th 2006
    Jip Veldman

    8. Abstract Continuation of the previous talk.

  8. The function rho and Aronszajn trees

    December 13th 2006
    Bernhard Irrgang

    9. Abstract We interpret the function rho in terms of Aronszajn trees.

  9. The unboundedness of rho

    December 20th 2006
    Dominik Klein

    10. Abstract We prove the unboundedness property of rho. This is used to construct a forcing that adds a function f: omega_2 x omega_2 --> omega that is not constant on any rectangle with infinite sides.

  10. The unboundedness of rho II

    January 10th 2007
    Dominik Klein

    11. Abstract Continuation of the previous talk.

  11. The function D

    January 17th 2007
    Alex Rothkegel

    12. Abstract Using the rho-function, we define a function D and prove that it has property Delta.

  12. The function D II

    January 31st 2007
    Alex Rothkegel

    13. Abstract Continuation of the previous talk.

  13. A ccc forcing that adds a Kurepa tree

    February 7th 2007
    Merlin Carl

    14. Abstract Using a function with property Delta, we construct a ccc forcing that adds a Kurepa tree.

Literature

  1. J. Baumgartner, S. Shelah - Remarks on superatomic Boolean algebras, Annals of Pure and Applied Logic, vol. 33, S. 109 - 129.
  2. P. Koszmider - Applications of rho-Functions, S. 83 - 98 in
  3. C.A. Di Prisco et al. (eds.) - Set Theory
  4. S. Todorcevic - Coherent Sequences (http://www.math.toronto.edu/stevo)
    5. Last changed January 17th 2007