(in Polish) Canonical Models of Set Theory

elective monographs
general courses

The course begins with a review of the fundamentals of set theory from Logic II, and then further develops the theory of ordinals and cardinals. The highlights of the course consist of the close study of canonical models of set theory, especially the von Neumann universe V, the inner model HOD (the class of hereditarily ordinal definable sets), Gödel's constructible universe L, and the verification of ZFC and the continuum hypothesis in L. The technical development of the material will be interwoven with the discussion of historical and philosophical aspects of the ambitious enterprise of formulating precise principles governing the set-theoretic universe.

Lecture offered within the University of Warsaw Integrated Development Programme (ZIP), co-financed by the European Social Fund under POWER, track 3.5.

Cantor's development of set theory in the late of 19th century, a grand and intricate theory of infinite objects of various sizes, together with the accompanying system of transfinite ordinals and cardinals, was carried out in a rigorous yet non-axiomatic framework. The paradoxes in set theory that arose in the early part of the twentieth century, and the bitter controversies surrounding the axiom of choice, prompted Zermelo to axiomatize set theory; a process that was eventually enhanced (independently) by Fraenkel and Skolem into a first order axiomatic system known as Zermelo-Fraenkel set theory ZF, which is often supplemented with the axiom of choice under the guise of ZFC. By the 1950s, ZFC came to be unanimously viewed in the eyes of set theorists and philosophers of mathematics as the most compelling foundational system for the subject, and thereby for the entirety of mathematics. Thus ZFC has come to be seen as superior to the venerable Russell-Whitehead foundational system of Principia Mathematicae and its descendants (e.g., Quine's New Foundations system). A number of pivotal and highly technical developments brought this transformation about; the most important of which was the construction of the von Neumann hierarchy and the so-called Gödel hierarchy of sets. The former provided a dynamically structured picture of the universe of sets, while the latter revealed the “thinnest” possible universe of sets (known as the constructible universe) that not only satisfies ZFC, but also validates the notorious Continuum Hypothesis, a hypothesis that had befuddled Cantor and many other luminaries (including Hilbert and his School). The course will flesh out, technically, historically and philosophically, the aforementioned turning points that have brought about our modern, highly nuanced understanding of the familiar yet elusive notion of sets.

The specific topics covered by the course are as follows:

(a) Review of relevant material from Logic II

(b) Axioms of Zermelo-Fraenkel set theory

(c) Ordinal arithmetic

(d) Cardinal arithmetic

(e) The axiom of choice

(f) The von Neumann hierarchy of sets

(g) The inner model HOD and the consistency of the axiom of choice

(h) The inner model L and the consistency of the generalized continuum hypothesis

(i) Historical and Philosophical interludes

(a) D. Goldrei, Classic Set Theory, Chapman & Hall Mathematics, 1996.

(b) T. Jech and K. Hrbáček, Introduction to Set Theory (3rd ed.), Marcel Dekker Inc., 1999.

(c) K. Kunen, Set Theory, North Holland/College Publications, 1980/2011.

(d) Mary Tiles, The Philosophy of Set Theory, Basil Blackwell/Dover, 1989/2004.

(e) Philosophy of Mathematics (2nd ed., edited by P. Benacerraf and H. Putnam, Cambridge University Press, 1984.

(f) G. Moore, Zermelo's Axiom of Choice: Its Origins, Development, and Influence, Springer-Verlag, 1984.

KNOWLEDGE: The central concepts and axioms of modern set theory; the logical relationship between them exemplified by the structure of various models of set theory; and the understanding of the philosophical and historical development of set theory.

SKILLS: Fluency in set theoretical concepts, especially those pertaining to ordinal and cardinal arithmetic and the axiom of choice. Logical translation of mathematical statements into set theory, and the determination of their veracity by rigorous analysis and argumentation.

SOCIAL COMPETENCE: Improving in the communication of scientific claims by employing an axiomatic framework for their evaluation.

A final test and short quizzes throughout the term.

Number of absences: 2