Schedule:
 Lectures: Monday, Wednesday, Friday, 11–12, MR13.
Example Classes:
#1: Monday 3 February 2020, 3:30–5, MR14.
Example Sheet #1.
#2:
Monday 17 February 2020, 5–6:30, MR14.
Example Sheet #2.
#3:
Monday 2 March 2020, 3:30–5, MR14.
#4:
Monday 16 March 2020, 3:30–5, MR14.
Revision Session:
TBA
Friday, 17 January 2020
 First Lecture. Infinite length, two player, zerosum, perfect
information and perfect recall games. Discussion of the types of games
that will not be covered in the lecture course: more than two players,
cooperative games, imperfect information. A brief historical overview:
Zermelo (1913), the Polish school and the Scottish book (Banach, Mazur,
Ulam), Gale & Stewart (1953), Blackwell and the Californian
stochasticians, Mycielski's Axiom of Determinateness, Solovay.
Games, positions, plays, strategies, the result of two strategies
playing against each other. The notion of a winning strategy.

Monday, 20 January 2020
 Second Lecture. Guest Lecture Imre Leader
(nonexaminable). Positional games. Examples. Proof of determinacy of
positional games with finite winning lines. Strategystealing. Examples
of player I not winning in bounded time. Open problem: could 5inarow
be a win but not in bounded time? Open problems concerning Ramsey games.

Wednesday, 22 January 2020
 Third Lecture. Notation and concepts: sequences, trees,
branches of a tree, concatenation, the Ipart of a sequence and the
IIpart of a sequence. Strategic trees and reformulation of being a
winning strategy in terms of strategic trees. Necessary criterion for
being a win for player I or player II in terms of the cardinality of the
set.

Friday, 24 January 2020
 Fourth Lecture. Sufficient criterion for being a win for
player I or player II: countability. Finite games. Zermelo's Theorem.
Backwards induction and construction of labellings. Finitary games. The
GaleStewart Theorem: finitary games are determined. First half of the
proof: transfinite recursive definition of the partial labelling.

Monday, 27 January 2020
 Fifth Lecture. Wellfounded trees and their height function.
Continuation of the GaleStewart proof: proof that the transfinite
recursion terminates; proof that the labelling gives winning strategies.
The axiom of choice implies the existence of nondetermined sets.

Wednesday, 29 January 2020
 Sixth Lecture. Baire space and its topology:
zerodimensional, totally disconnected, metric. The GaleStewart theorem
in its topological formulation: all open sets are determined. Intuitive
understanding of convergence. Tree representation of closed sets.
Cardinality of the set of open and closed sets. The Borel
\(\sigma\)algebra. The Borel hierarchy.

Friday, 31 January 2020
 Seventh Lecture. The Borel hierarchy is a semilinearly
ordered wellfounded hierarchy. On countable topological spaces, it has
height at most two; in general, it has height at most \(\aleph_1\). A
\(\boldsymbol{\Sigma}^0_2\) set and its determinacy:
in general, \(\boldsymbol{\Sigma}^0_2\) sets do not admit constructive
labellings (cf. Löwe & Semmes 2007). A brief overview of determinacy
in the Borel hierarchy (without proofs):
\(\boldsymbol{\Sigma}^0_2\) determinacy (Wolfe 1955),
\(\boldsymbol{\Sigma}^0_3\) determinacy (Davis 1963),
\(\boldsymbol{\Sigma}^0_4\) determinacy (Paris 1972),
Borel determinacy (Martin 1975).
Universal sets.
Further literature: B. Löwe, B. Semmes, The
extent of constructive labellings, J. Log. Comput. 17(2), 2007,
285–298.

Monday, 3 February 2020
 Eighth Lecture. Pointclasses: boldface, coherent, closure
properties. Excursion on the use of the Axiom of Choice: closure of
\(\boldsymbol{\Sigma}^0_2\) under countable unions uses countable
choice; in the FefermanLévy model, every set is
\(\boldsymbol{\Sigma}^0_4\). Universal sets. The Universal Set Lemma.
The Universal Set Theorem: first half of the proof (construction of a
universal open set).
Further literature:
A. W. Miller, On the length of Borel hierarchies, Ann. Math. Log. 16, 1979, 233–267.
A. W. Miller, Long Borel hierarchies, Math. Log. Q. 54(3), 2008, 307–322.
Example Class #1.
Example Sheet #1.

Wednesday, 5 February 2020
 Ninth Lecture. Proof of the Universal Set Theorem (including
coding infinite sequences of elements of Baire space in an element of
Baire space). Lebesgue's error: the claim that the Borel sets are closed
under continuous images. Suslin's counterexample (without proof).
Intuitive definition of the projective hierarchy. In the early 1970s, it
was known that determinacy in the projective hierarchy needed to rely on
strong axioms of infinity (cannot be proved in \(\mathsf{ZFC}\) alone).

Friday, 7
February 2020
 Tenth Lecture. Finite products of Baire space.
Characterisation of continuity in terms of coherent functions (no
proof). Discussion of game representations of continuity. Analytic sets
and equivalent characterisations. Closure properties of analytic sets:
countable unions and intersections, continuous images. The projective
hierarchy. Discussion of firstorder definability and the projective
hierarchy. Suslin's Theorem: the projective hierarchy does not collapse
(start of proof).

Monday, 10
February 2020
 Eleventh Lecture. Proof of Suslin's theorem finished.
Regularity properties: Lebesgue measurability on Baire space; the Baire
property (Baire Category Theorem mentioned); the perfect set property
(PSP). CantorBendixson Theorem (proof idea), perfect sets and perfect
trees, cardinality of nonempty perfect sets. CantorBendixson as a
definable version of the Continuum Hypothesis. PSP and the Continuum
Hypothesis. Sketch of the proof that the Axiom of Choice implies that
there is a set without the perfect set property.

Wednesday, 12
February 2020
 Twelfth Lecture. Definability of the Axiom of Choice:
wellorderings of the Baire space as subsets of the
\(\omega^\omega\times\omega^\omega\); projective wellorderings and
their consequence for regularity of projective sets. Coding of countable
ordinals as elements of Baire space: \(\mathrm{WO}_\alpha\) and
\(\mathrm{WO}\). Choice functions for \(\{\mathrm{WO}_\alpha\,;\,\alpha<\aleph_1\}\) and the regularity of \(\aleph_1\). Choice functions for
families of closed sets: the leftmost branch in a tree. The set
\(\mathrm{WO}\) is \(\boldsymbol{\Pi}^1_1\).

Friday, 14
February 2020
 Thirteenth Lecture. Tree representation of
\(\boldsymbol{\Pi}^1_1\) sets. Every \(\boldsymbol{\Pi}^1_1\) set is a
union of \(\aleph_1\) many Borel sets. Boundedness theorem for
\(\mathrm{WF}\). Construction of a subset of \(\mathrm{WF}\) that does
not have the perfect set property. If there is a project wellordering of
the reals, then there is a projective set without the perfect set
property. Goal: if every projective set is determined, then every
projective set has the perfect set property. Perfect game: coding the
perfect game \(\mathrm{G}^*(A)\) as a game \(\mathrm{G}(A^*)\); if
\(\boldsymbol{\Gamma}\) is a boldface pointclass, then
\(\boldsymbol{\Gamma}\)determinacy implies
\(\boldsymbol{\Gamma}\)perfect set property. Game characterisation of
the perfect set property (no proof yet).

Monday, 17
February 2020
 Fourteenth Lecture. Proof of the game characterisation of the
perfect set property. Relationship between projective regularity
properties and large cardinals. Regular and singular cardinals. Weakly
and strongly inaccessible cardinals. Basic model theory of set theory:
transitive submodels and absoluteness of some properties (without any
proofs).
Further literature:
Kenneth Kunen, Set Theory, An Introduction to Independence Proofs. (Elsevier, 1980).
Studies in Logic and the Foundations of Mathematics Vol. 102. Section IV.3.
Example Class #2.
Example Sheet #2.
[In Example (18), assume that \(X\) is a Hausdorff space.]

