The method of forcing 1000-1M09MEF
0. Martin's Axiom and its consequences. Basic concepts in the theory of forcing: forcing notion, dense set, antichain, generic filter.
1. Axiomatic Set Theory ZFC: the axioms, models of ZFC, reflection principle, Mostowski collapse, absoluteness.
2. Generic extensions, forcing relation and Truth Lemma, ZFC in the extension.
3. Appliactions of forcing in cardinal arithmetic and infinite combinatorics: independence of CH, consistency of diamond principle, consistency of given values of 2^kappa for a few regular cardinals kappa simultaneously.
4. Applciations of forcing in the study of the structure of the real line, measure theory and topology, Cohen and Solovay models.
5*. Products and iterations, finite support iterations of c.c.c. forcings, consistency of Martin's Axiom.
6*. Schoenfield's absoluteness theorem, proving ZFC theorems using forcing.
*) parts marked with a star will be presented, should time and the interest of the audience permit that."Set Theory" or "Introduction to Set Theory" courses are required before
taking the course of forcing and a course of Mathematical Logic is advised.
Type of course
Learning outcomes
A student upon completing the course will:
1. know applications of Martin's Axiom in set theory and topology,
2. understand the concepts of relative consistency and independence of a given sentence from ZFC theory,
3. learn the basics of the forcing method and understand Cohen's proof of independence of CH,
4. see examples of applications of the method of forcing in set theory, topology and measure theory.
Assessment criteria
the final grade will be based on students' activity during the semester and an oral exam.
Bibliography
1. K. Kunen - Set Theory. An Introduction to independence proofs.
2. T. Jech - Set Theory.
3. W. Kubiś - Notes on forcing theory,
4. T. Bartoszyński, H. Judah - Set Theory. The structure of the real line.
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:
Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system: