(in Polish) Nierozstrzygalność i forcing iterowany 1000-1M18NFI
This course is a continuation of the course "The method of forcing" (1000-1M09MEF). The purpose of the course is an introduction to more advanced proofs of undecidability and independence of solutions of natural problems which appear in mathematical practice and require complex combinatorial and forcing methods, in particular the iterated forcing.
We will cover iterations with finite supports, countable supports, proper forcings, adding generic structures etc. Beginner skills will be developed based on simple classical examples of undecidability of properties of the Lebesgue measure, the Baire category and of Boolean algebras. The complexity of topological and functional analytic examples will be adapted to the background and interests of the students. Examples of concrete topics which may be covered are:
1) The methods related to the Cohen and the random models: properties of the measure and the category, Ulam's rectangle problem, versions of the Fubini theorem, the problem of the existence of universal compact spaces and universal Banach spaces in various classes. Properties of the Cech-Stone reminders in the Cohen model.
2) Finite support iterations and countable support iterations: adding dominating functions, Sacks forcing, Prikry-Silver forcing,
Mathias forcing; Axiom A; special types of ultrafilters, cardinal invariants, Open Coloring Axiom (OCA), the Borel conjecture, Banach spaces with the Grothendieck property, Efimov compact spaces.
3) Adding generic structures: finite and countable approximations, historical forcing, large Lindelof spaces with points G_delta, large structures with few endomorphisms, anti-ramsey colorings, generic gaps.
4) Forcing and large cardinals: the Levy collapse, iterations of inaccessible length: applications in trees and infinitary combinatorics.
5) Proper forcings.
6) Useful combinatorial principles: Jensen's diamond, OCA, PFA: consistent constructions of C*-algebras and gaps in P(N)/Fin.
Type of course
Mode
Prerequisites
Prerequisites (description)
Learning outcomes
A student upon completing the course will:
1. understand consistency proofs employing models obtained using iterated forcing.
2. be able to decide whether given facts concerning the Lebegue measure, the Baire category or cardinal invariants are true or false
in the Cohen model, the random model or in the Sacks model etc.
3. know how to use iterated forcing (with finite or countable supports, including iterations of the length which is a large cardinal) to construct models of ZFC where given combinatorial, topological or functional analytic facts are true or false.
4. know how to use the Proper Forcing Axiom and Jensen's diamond.
5. be able to invent a notion of forcing which adds a generic structure (a Boolean algebra, a Banach space, a topological space)
with prescribed properties.
Assessment criteria
Quizzes, problem sets, written final exam.
Bibliography
Bibliography:
U. Abraham, Proper forcing. Handbook of set theory. Vols. 1, 2, 3, 333--394, Springer, Dordrecht, 2010.
T. Bartoszyński, H. Judah, Set theory. On the structure of the real line. A K Peters, Ltd., Wellesley, MA, 1995.
J. Baumgartner, Iterated forcing. Surveys in set theory, 1--59, London Math. Soc. Lecture Note Ser., 87, Cambridge Univ. Press, Cambridge, 1983.
J. Baumgartner, Applications of the proper forcing axiom. Handbook of set-theoretic topology, 913--959, North-Holland, Amsterdam, 1984.
M. Goldstern, Tools for your forcing construction. Set theory of the reals (Ramat Gan, 1991), 305--360, Israel Math. Conf. Proc., 6, Bar-Ilan Univ., Ramat Gan, 1993.
T. Jech, Set theory. The third millennium edition, revised and expanded. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003.
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:
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