Kardinalna aritmetika

1. Axioms and structures
   • The axioms and their consequences
   • Product structures
   • Natural number structures
   • The (parameterised) recursion theorem
   • Von Neumann numbers
2. Basic cardinal arithmetic
   • Equality and the order relations on cardinals
   • The Schröder-Bernstein theorem
   • Sums and products of cardinals
   • Cardinals that are their own squares.
3. Ordinals and well-orders
   • Ordinal structures
   • Von Neumann ordinals
   • Formulations of transfinite induction
   • Trensfinite recursion
   • Ordinal arithmetic
   • Well-founded relations and well-orders
   • The classification theorem
   • Sums and products of well-orders
4. The cardinal arithmetic of alephs (without choice)
   • Hartog's lemma
   • Initial ordinals
   • Definition of the omena and aleph herarchies
   • Addition and multiplication of alephs
   • Alephs are their own squares
5. The axiom of choice
   • X well-orderable iff P(X) has a choice function
   • Formulastion of AC
   • Proof that Zorn's lemma is equivalent to AC
   • Other equivalent statements to AC
   • Countable and dependent choice
   • Vitali's theorem
   • The statement (but not proof) of the Banach-Tarski theorem
6. Cardinal arithmetic with choice
   • Indexed sums and products of cardinals
   • Laws involving sums, products and exponentiation
   • König's theorem and its proof
   • Provable and unprovable properties of 2 to the power of aleph_alpha
   • Cofinality and the cofinality bound for 2 to the power of aleph_alpha
7. Singular, regular and inaccessible cardinals
   • Cofinality and its basic properties
   • Singular and regular cardinals
   • Successor cardinals are regular
   • Inaccessible cardinals
   • Grothendieck universes
8. Sets of real numbers
   • Cardinalities of open and closed sets
   • The cardinality of the set of all open sets
   • Perfect sets and the Cantor set as an example
   • Every uncountable closed set contains a perfect subset
   • The Borel hierarchy
   • The cardinality of the set of all Borel sets

Notes: The following are specifically not examinable:

• The proof of the Banach-Tarski theorem
• Gale-Stewart games
• Lecture 14