Oris teme

  • Kardinalna Aritmetika (Cardinal arithmetic)

    Skrči vse Razširi vse



  • IZPITI

    Izpitni roki:

    1. 07. 06. 2022, 10:00-12:00 v predavalnici 3.07.
    2. 28. 06. 2022, 10:00-12:00 v predavalnici 3.07.

  • PREDAVANJA (LECTURES)

    Lectures are from 14h to 17h on Wednesdays in lecture room 3.07 (Jadranska 21).

    There is no lecture on 9th March (strike), nor on 27th April (public holiday).

    On Thursday 28th April (17.00-19.00, room 2.02) there is a nonstandard lecture on mathematical consequences of the axiom of choice (instead of the exercise class).

    From 23. 02. 2022 lectures will also broadcast on Zoom (meeting id: 987 2628 2122, passcode: 535897, direct link).

    • Predavanje 1 (16. 02. 2022): Course assessment and logistics. Background Cantor's continuum hypothesis (CH) and the development of cardinal arithmetic. Reasons to be interested in set theory. Classes, sets and Russell's paradox. Axioms of  set theory. Cartesian products and exponentials.

    • Ikona Datoteka
      Lecture 1: LaTeX notes Datoteka


    • Predavanje 2 (23. 02. 2022): Natural number structures. The (parameterised) recursion theorem and applications. Construction of a natural number structure using von Neumann numbers. Real number structures. Construction of the real numbers as Dedekind cuts. Proof of the recursion theorem. 

    • Ikona Datoteka
      Lecture 2: rough board notes. Datoteka
    • Ikona URL
      Lecture 2: video URL


    • Predavanje 3 (02. 03. 2022): Comparison of size for classes and sets: bijective correspondence (a.k.a. equipotency, equal cardinality); non-strict, strict and very strict order comparisons, including the Schröder-Bernstein theorem. The natural numbers are the smallest infinite set. Classification of countable sets. Cantor's theorem that powerset incteases cardinality. The cardinality of the continuum (the real numbers).

    • Ikona Datoteka
      Lecture 3, rough board notes Datoteka
    • Ikona URL
      Lecture 3: video URL


    • No lecture (09. 03. 2022): Strike.


    • Predavanje 4 (16. 03. 2022): Sum (coproduct) of classes and sets. Sums, products and exponentials preserve cardinal equalities and (usually) inequalities The basic arithmetical laws for cardinal arithmetic. Failure of cancellation. Cardinals that are their own squares. Removing a (very) strictly smaller subset from an infinite set (that is its own square) does not change cardinality.

    • Ikona Datoteka
      Lecture 4: rough board notes Datoteka
    • Ikona URL
      `Lecture 4: video URL


    • Predavanje 5 (23. 03. 2022): Motivation for ordinals: counting into the transfinite. Definition of ordinal structure. Example proofs by transfinite induction. Successor and limit ordinals. Reformulations of transfinite induction. The (parameterised) transfinite recursion theorem. Definitions of ordinal addition, multiplication and exponentiation by transfinite recursion. Brief outline of how the von Neumann ordinals furnish an ordinal structure.

    • Ikona Datoteka
      Lecture 5: rough board notes Datoteka
    • Ikona URL
      Lecture 5: video URL

      Unfortunately I forgot to resume the recording for the last hour of the lecture. The video ends with the reformulations of transfinite induction.


    • Predavanje 6 (30. 03. 2022): The least-ordinal principle. Well-founded relations and their properties. Definition of well-order. Order embeddings between and initial segments of well-ordered sets. The classification  theorem for well-ordered sets. Order types and basic properties. Sum and product operations on well-orders. (The exponentiation operation is described in the notes.) The rigidity of well-orders. Proof of the classification theorem. 

    • Ikona Datoteka
      Lecture 6: rough board notes Datoteka
    • Ikona URL
      Lecture 6: video URL


    • Predavanje 7 (06. 04. 2022): Well-orderable sets. Hartog's lemma. Initial ordinals and their notation system. Alephs. Every aleph is its own square.

    • Ikona Datoteka
      Lecture 7: rough board notes Datoteka
    • Ikona URL
      Lecture 7: video URL


    • Predavanje 8 (13. 04. 2022): Choice functions. Relating well-orderability and choice functions. The axiom of choice (AC). Equivalents of AC: the well-ordering principle; Zorn's lemma; nonemptiness of cartesian products; surjections have sections. The axioms of countable choice (CC) and dependent choice (DC), and applications thereof. (AC) => (DC) => (CC).

    • Ikona Datoteka
      Lecture 8: rough board notes Datoteka
    • Ikona URL
      Lecture 8: video URL


    • Predavanje 9 (20. 04. 2022): AC simplifies the notion of cardinality. Indexed sums and products of cardinalities. The arithmetic of sums of cardinals. König's theorem relating sums and products of cardinals. The relation of cardinal exponentiation to sums and products. Three properties we can prove about cardinal exponentiation if we don't assume the generalised continuum hypothesis (GCH). The notion of cofinality. Expressing infinite cardinalities as sums of strictly smaller cardinalities. A strict lower bound on the cofinality of cardinal exponentials. 

    • Ikona Datoteka
      Lecture 9: rough board notes Datoteka
    • Ikona URL
      Lecture 9: video URL


    • Predavanje 10 (28. 04. 2022): Mathematical consequences of AC. Vitali's theorem. The Banach-Tarski theorem.

    • Ikona Datoteka
      Lecture 10: rough board notes Datoteka
    • Ikona URL
      Lecture 10: video (part 1) URL
    • Ikona URL
      Lecture 10: video (part 2) URL

      The video footage from the lecture on 28th April 2022 was unusable. The above videos, which contain a slightly more detailed presentation of the material, were pre-recorded in 2020 for a previous incarnation of the course.


    • Predavanje 11 (04. 05. 2022): Cofinality: definition (recap) and basic properties. Singular and regular cardinals. Every successor cardinal is regular. Weakly and strongly inaccessible cardinals. Grothendieck universes and their relationship to strongly inaccessible cardinals. The cumulative hierarchy.

    • Ikona Datoteka
      Lecture 11: rough board notes Datoteka
    • Ikona URL
      Lecture 11: video URL


    • Predavanje 12 (11. 05. 2022): Sets of real numbers. Open sets and their cardinalities. Closed sets and their cardinalities. The Cantor set as an example perfect set. Every perfect set has continuum cardinality. The perfect set property: every uncountable closed set contains a perfect subset.

    • Ikona Datoteka
      Lecture 12: rough board notes Datoteka
    • Ikona URL
      Lecture 12: video URL


    • Predavanje 13 (18. 05. 2022): The Borel hierarchy. The Borel hierarchy defines the Borel sets. Gale-Stewart games on Baire space. Example game: the perfect subset game. Martin's determinacy theorem (statement but not proof): every Borel game is determined.

    • Ikona Datoteka
      Lecture 13: rough board notes Datoteka
    • Ikona URL
      Lecture 13: video URL


    • Predavanje 14 (25. 05. 2022): Formalising our axioms in first-order logic. Axioms for ZF, and its relationship to our axioms. Formalised consistency statements. Two main independence results: AC is independent from ZF; CH is independent from ZFC. The independence of SIC and other large cardinal axioms requires `leaps of faith' (i.e., stronger consistency assumptions). Intrinsic and extrinsic reasons for accepting large cardinal axioms. Projective Determinacy (PD) and the Axiom of Determinacy (AD).

    • Ikona Datoteka
      Lecture 14: rough board notes Datoteka
    • Ikona URL
      Lecture 14: video (part 1) URL
    • Ikona URL
      Lecture 14: video (part 2) URL

  • VAJE