Set theory (NAIL063), 2025/2026

Jan Kynčl, KAM (kyncl - at - kam.mff.cuni.cz)

The lecture takes place on Thursdays at 9:00 in S4.

We can arrange consultations in person or by email.

Recommended literature

[BS] B. Balcar, P. Štěpánek, Teorie množin, Academia, Praha, 2001 (or earlier editions). In Czech. Available in the library in Troja.

Other resources:

[HJ] Karel Hrbacek, Thomas Jech: Introduction to Set Theory, 3. edition, Marcel Dekker, 1999 - an English alternative to [BS], slightly different approach, but a great overlap of topics
Thomas Jech, Set theory, Springer, 2003 (Part I)
Richard Evan Schwartz, Gallery of the infinite, American Mathematical Society, Providence, RI, 2016 (link)
Mirek Olšák, Essence of Set Theory - very nice animated videos (slightly different terminology - axiom of existence, axiom of infinity, ordering, ...)
Joel David Hamkins: Infinity, Gödel Incompleteness, and the Paradoxes that Broke Mathematics | Lex Fridman Podcast #488

The lecture will be in English, the topics very similar to the last year.

Introductory information
Brief history of set theory, paradoxes
Language of set theory (and logic): symbols, formulas, extension by additional symbols
Examples of axioms of propositional and predicate logic
Examples of axioms of equality, examples of deduction rules (we will use logic intuitively)

Zermelo–Fraenkel theory (ZF)
Axiom of existence
Axiom of extensionality
Axiom schema of separation (also comprehension), intersection, set difference, empty set
Axiom of pairing (also pair), unordered set {a,b}, one-element set {a}, ordered pair (a,b), (x,y)=(u,v) implies x=u and y=v, ordered n-tuple
Axiom of union, union of a set a as a unary operation, union of two sets a, b, unordered n-tuple
Axiom of power set, power set of a set a
Axiom Schema of replacement

Axiom of foundation (regularity)
Classes
Class term, class determined by a formula ("definable collection of sets"), every set is a class, other classes are proper classes
Extension of the language of set theory by class terms and class variables, elimination of class terms from atomic formulas
Binary class operations: intersection, union, difference
Universal class V, (absolute) complement of a class
Predicates "to be a subclass", "to be a proper subclass"
Unary class operations: union, intersection, power class
The universal class is not a set (exercise)
The intersection of a set and a class is a set
Cartesian product of two classes, the Cartesian product of two sets is a set
Cartesian power X^n as a class of all ordered n-tuples

Relations
(Binary) relation, n-ary relation
Relations of membership and identity
Domain of (a class) X, range of X, image of a class Y by a class X, restriction of a class X to a class Y
If x is a set, then the following classes are also sets: the domain of x, the range of x, the image of a class Y by the set x, the restriction of the set x to a class Y
Inverse relation, composition of relations
Shortcut for quantification over a class (relativization of quantifiers)
Mapping (of a class X into a class Y, onto a class Y, injective), class ^aA of all mappings of a set a into a class A; if A is a set, then ^aA is also a set; if a is non-empty and A is a proper class, then ^aA is also a proper class
Orders
Reminder of basic properties of relations (reflexive etc.) on a class A, hereditability

Order (ordering) on a class A, comparable elements
Linear ordering, strict ordering, notation
Majorant (upper bound), maximal element, maximum or largest element, supremum; maximum element is always maximal, in a linear ordering there is at most one maximal element (which is then also maximum), maximum element and supremum are unique and can be denoted as max(X), sup(X)
A set bounded from above, lower set, lower set determined by an element x (principal ideal determined by x), x≤y implies an inclusion of the lower sets determined by x,y
Remark: Dedekind cuts on the set of rational numbers as a definition of real numbers
Well-ordering, it is a hereditary property, every well-ordering is linear
Reminder of basic information about equivalences (definition, equivalence classes, equivalence classes form a partition)