Some Applications of the method of forcing

Todorchevich S., Farah I.

During the Fall Semester of 1991 Stevo Todorchevich (Todorcevic) gave a course on applications of the method of forcing at the Mathematical Institute in Belgrade. This text contains material presented in the course, as well as some additional closely related results included for completeness. The method of forcing, i.e. the method of adding a generic object to a given structure, is frequently used to get independent results, that is, results showing that certain statements cannot be proved (or disproved) in ZFC or some other similar theory. The main purpose of these notes is to present to a general mathematical audience a number of applications of the method of forcing to other branches of mathematics, such as general topology and measure theory. Most of the presented results do not require any additional axioms of set theory, but use standard set-theoretical and forcing constructions, such as Suslin tree, generic models^ Cohen and random reals, etc. Among topics included are Borel Equivalence Relations, Halpern-Laiichli Theorem, the Open Coloring Axiom and the Proper Forcing Axiom. For more ambitious readers there are a few exercises scattered throughout the text, and a list of yet unsolved problems is included.