Tukey relations between the null ideal and classical tree forcing ideals

• Applicant: Professor Dr. Otmar Spinas
• Subject Area: Mathematics
• Term: from 2015 to 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 276055566

A map from an ideal I into another ideal J is called Tukey reduction if every subset of I whose union is not in I gets mapped onto a subset of J with the same property. If I is in this sense Tukey reducible to J, then add(I), its additivity coefficient, is at least as big as add(J). Motivation for the main open problem of this proposal is my latest result which says that the meager ideal is Tukey reducible to the Mycielski ideal. This is the tree ideal associated with Silver forcing. By a famous result of Pawlikowski, the meager ideal is also Tukey reducible to the Lebesgue null ideal. Putting the two together, one would like to know whether my result can be strengthened in the sense that even the null ideal is Tukey reducible to the Mycielski ideal.The Mycielski ideal is one of the classical tree forcing ideals. For most other ones of theses like the Laver, Miller, and the Sacks ideal it has been known for a long time that the meager ideal is not Tukey reducible to any of them. Core element of the proof in all cases has been the construction of an amoeba forcing that does not add Cohen reals. By iterating this it was possible to construct a model in which the additivity of the meager ideal is smaller than the one of the respective tree ideal. If it turns out that we are unable to give a positive answer to our main question, we shall try to construct a Silver amoeba forcing that does not add random reals. Then in a similar way it should be possible to show that consistently the additivity of the null ideal is smaller than the one of the Mycielski ideal. This would imply a negative answer to our problem.Another goal of our project is to prove that the additivities of different such tree ideals are independent. A more modest goal would be to show that there are no provable Tukey reductions between any two of them.Looking at the very definition of these tree ideals suggests that one cannot really understand there behaviour without understanding the antichain structure of the underlying tree forcing. This intuition has been proved to be right by the proof of my latest result mentioned above, where a major ingredient is a result about Silver antichains from my recent work with my student Marek Wyszkowski.I apply that this project be worked out in cooperation with Prof. Saharon Shelah, one of the most prominent logicians of our time. He has done great and deep work in the area of the problems mentioned, such as null ideal, tree forcings and antichain structures. Over many years we have been having a very fruitful cooperation.

• DFG Programme: Research Grants
• International Connection: Israel
• Cooperation Partner: Professor Dr. Saharon Shelah