Suslin's hypothesis, the diamond principle and the proper forcing axiom

• Grant number: 22/07650-0
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: March 01, 2023
• End date: June 30, 2024
• Field of knowledge: Physical Sciences and Mathematics - Mathematics
• Principal Investigator: Christina Brech
• Grantee: Maurício Rossetto Corrêa
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil

Abstract

The following topics will be studied in this project:* The Suslin hypothesis (SH): a statement first studied by the mathematician Mikhail Yakovlevich Suslin in the 1910s. It consists in the characterization of the poset of the real numbers (R) with the usual order from three properties. That is X is order isomorphic to R if, and only if, the linearly ordered set X satisfies the three properties delimited by Suslin.* The diamond principle: an additional axiom of set theory that implies the continuum hypothesis and guarantees the existence of Suslin lines, linearly ordered sets that have the aforementioned properties and are not isomorphic (with respect to order) to the line real.* The proper forcing axiom (PFA): a stronger version of Martin's axiom where the assumption of forcings that satisfy the countable chain condition is replaced by proper forcings. This statement implies equality between the cardinality of the continuum and the cardinal aleph 2, in the isomorphism between any subsets of R of cardinality aleph 1, etc. It serves as a gateway to the study of several other forcing axioms that have gained importance in contemporary mathematics.