Bounded Forcing Axioms and Inner Models

• Applicant: Dr. Andreas Lietz
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 542745507

The main objective of this project is to further the understanding of the connection between Bounded Forcing Axioms and canonical inner models. How strong, in terms of consistency strength, is Bounded Martin's Maximum exactly? How many large cardinals do we need to force Woodin's axiom (*)? What impact does Bounded Martin´s Maximum have on the structure of the canonical ideal on the first uncountable cardinal, the nonstationary ideal? All of these are important long standing open questions. With this project, I propose a strategy which can ultimately lead to a solution to these questions. Both Bounded Martin's Maximum and Woodin's axiom (*) assert, in some sense, maximality of the set theoretical universe. Bounded Martin's Maximum does this more directly by demanding maximality of the set of sets of hereditary size at most the first uncountable cardinal with respect to existential formulas and stationary set preserving forcings. Woodin's axiom (*) does this through asserting that a big chunk of the universe is a forcing extension of a canonical inner model of determinacy by the forcing Pmax. That this leads to a natural maximality axiom is justified through Woodin's remarkable Pmax theory. In a recent breakthrough, Asperó-Schindler show that under the existence of a proper class of Woodin cardinals, Woodin's axiom (*) is equivalent to a version of Bounded Martin's Maximum. In my dissertation, I greatly expanded this connection to many different variations of Bounded Martin's Maximum on the one side and variations of Woodin's axiom (*) on the other side. The most important case is given by a new forcing axiom BQM and the version of (*) related to the famous sibling Qmax of Pmax. Additionally, I developed novel methods in the area of iterated forcing. My method allows for the first time to construct iterated forcings with a version of countable support which preserve the first uncountable cardinal while destroying many stationary sets. I could combine these results to solve a question of Woodin which had been open for a quarter of a century. Namely, I could show that assuming the existence of an inaccessible limit of supercompact cardinals, there is a stationary set preserving forcing extension in which the nonstationary ideal is dense. This project aims to develop related forcing techniques and methods in Inner Model Theory which can shed light on the open questions above.

• DFG Programme: WBP Fellowship
• International Connection: Austria
• Host: Professorin Dr. Sandra Müller