IPAC · Independence Proofs and Combinatorics

Independence Proofs and Combinatorics

The goal of the project has been to advance our knowledge in the field of set theory, a subfield of mathematical logic. Mildenberger worked together with Shelah on some combinatorial questions in an area where independence of the axioms of mathematics, i.e. the Zermelo Fraenkel axiom system with together with the axiom of choice, short ZFC, is very likely. Therefore the main part of the proposed work was to develop forcing techniques. We also wanted to emphasise combinatorial methods in the analysis of existing notions of forcing with respect to new properties. Mildenberger investigated near coherence classes of ultrafilters, combinatorics of Milliken-Taylor ultrafilters. Mildenberger and Shelah investigated possible cofinalities of ultrapowers, the cofinality of the symmetric group, and preservation of Souslin trees. Here are the results achieved in the project: Paper 1: On Milliken-Taylor ultrafilters, by Heike Mildenberger We show that there may be a Milliken-Taylor ultrafilter with infinitely many near coherence classes of ultrafilters in its projection to omega, answering a question by Lopez-Abad. We show that k-coloured Milliken-Taylor ultrafilters have at least k+1 near coherence classes of ultrafilters in its projection to omega. We show that the Mathias forcing with a Milliken-Taylor ultrafilter destroys all Milliken-Taylor ultrafilters from the ground model. Paper 2: The minimal cofinality of an ultrapower of omega and the cofinality of the symmetric group can be larger than b+, by Heike Mildenberger and Saharon Shelah We developped a notion of forcing that allowed us to prove the statement in the title. Paper 3: Many countable support iterations of proper forcings preserve Souslin trees, by Heike Mildenberger and Saharon Shelah We show that many countable support iterations of proper forcings preserve Souslin trees. We establish sufficient conditions in terms of games and we draw connections to other preservation properties. We present a proof of preservation properties in countable support iterations in the so-called case A that does not need a division into forcings that add reals and those who do not. Section 1 give some conditions on a forcing in terms of games that imply that the forcing is Souslin preserving. A special case of Souslin preserving is preserving the Souslinity of an omega-1-tree. In Section 2, we show that some tree-creature forcings developped by Roslanowski and Shelah satisfy the sufficient condition for one of the strongest games. Without the games, in Section 3 we show that some linear creature forcings are Souslin preserving. There are non-Cohen preserving examples. For the wider class of non-elementary proper forcings we show in Section 4 that omega-Cohen preserving for certain candidates implies Souslin preserving. In Section 5, we give more easily readable presentation of a result from Shelah's book on proper and improper forcing: If all iterands in a countable support iteration are proper and Souslin preserving, then also the iteration is Souslin preserving. This is a presentation of the so-called case A in which a division in forcings that add reals and those who do not is not needed. All papers can be downloaded in their most recent version or in their published version from http://home.mathematik.uni-freiburg.de/mildenberger/publist.html The project was carried out for 13 months. By 1 October 2010, Mildenberger became a full professor at the University of Freiburg im Breisgau.

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