Squares, Scales and Lines in Set Theory - Todorcevic's Theorem and PCF Theory

Squares, Scales and Lines in Set Theory - Todorcevic's Theorem and PCF Theory

Explore a mathematical lecture on the interplay between square principles, PCF theory, and linear orders. Delve into Todorcevic's result showing that for a singular cardinal kappa of cofinality omega, if square_kappa holds, there exists a linear order of cardinality kappa^+ that is not sigma-scattered, while all its small suborders are sigma-wellordered. Examine how similar results can be derived from PCF-theoretic principles, which follow from both weaker forms of square and certain failures of SCH. Gain insights into related problems in this field, as presented by James Cummings during the Workshop on "Determinacy, Inner Models and Forcing Axioms" at the Erwin Schrödinger International Institute for Mathematics and Physics.