Generalized Baire spaces - definability and paradoxical sets

• Applicant: Professor Dr. Ralf Schindler
• Subject Area: Mathematics
• Term: from 2019 to 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 428519768

• Final Report Year: 2025

Final Report Abstract

The proof by David Asperó together with the leader of the current project, according to which Martin’s Maximum++ implies Woodin’s Pmax axiom (*) introduced a forcing technique which opened up the door for many other applications. The key new result of this project is that in the presence of Martin’s Maximum the nonstationary ideal on ω1 cannot be definable over the structure H(ω2) by a universal (i.e., Π1) formula with a parameter from H(ω2). Further investigations discuss extensions of this as well as the meaning of Π11 over H(ω2) in the presence of forcing axioms.