Instructor: Prof. Dr. Benedikt
Löwe
Teaching Assistants: Takanori Hida,
Zhenhao Li.
Vakcode: WI305096
ECTS: 6
Time & Place:
February & March 2013: Tuesday 1517 (SP C1.112),
Friday 1315 (SP C1.112)
April & May 2013: Tuesday 1315 (SP C1.112), Wednesday
1315 (SP F1.02)
Course language: English
Intended audience: B.Sc. students of Mathematics,
M.Sc. students of Logic.
Goal of this course: Understanding of the connections between
logic and set theory, in particular the axiomatic approach. Skillful
handling of ordinals and cardinals, in particular the methods of
transfinite induction and recursion.
Content of the course: Axioms of Set Theory, Set Theory as a
Foundations of Mathematics, Ordinal Numbers, Cardinal Numbers, Axiom of
Choice. Possibly basics of some additional topics such as set theory of
the reals, descriptive set theory, and large cardinals.
Prerequisites:
Mathematical maturity, decent understanding of firstorder logic.
Evaluation:
 Homework.
 There will be 12 to 13 homework sheets.

You are allowed to either work alone or in a group of at most two people
for the homework. It is not necessary to stay in the same group for
every homework set.
 Homework is handed in either before class on the day of the
deadline mentioned on the homework set or by email to
T.Hida(at)uva.nl.
Late homework is not accepted.
 Exam. The exam will be on 28 May 2013,
1316, USC Sporthal 2.
 Final grade.
The final grade is the average of the grade of the
Homework component and the Exam component calculated
according to the OER regulations (Part A, Article 23).
If you do not pass the course in the first attempt,
there will be a hertentamen. In those cases where
it becomes necessary to redo (parts of) the homework
component, we shall discuss individual solutions.
Literature.
 Keith Devlin, The Joy of Sets,
amazon.co.uk
 Yiannis N. Moschovakis, Notes on Set Theory, amazon.co.uk
 Herbert B. Enderton, Elements of Set Theory,
amazon.co.uk
 Heinz D. Ebbinghaus, Einführung in die Mengenlehre.
 Thomas S. Jech, Set Theory, amazon.co.uk
 Kenneth Kunen, Set Theory,
amazon.co.uk
Course syllabus.
5 February 2013 
Hoorcollege 1517 C1.112 
Motivation (1): Set theory as a field of mathematics,
paradoxes of the infinite, definition of "of equal size",
Cantor's theorem without proof, existence of
transcendental numbers, transfinite procedures.
Motivation (2): Set theory as a foundations of mathematics.


8 February 2013 
Hoorcollege 1315 C1.112 
Models of set theory as graphs. The language of set theory. The existence axiom, the extensionality axiom, the Aussonderungsaxiom. Proof that the empty set exists and is unique.
Homework Set #1


12 February 2013 
Hoorcollege 1517 C1.112 
Consistency proofs for very weak systems of set theory. The pairing
axiom. Proof that no model of Ex+Ext+Aus+Pair can be finite. Desired set
operations from our ordinary mathematical practice: the Cartesian
product.


15 February 2013  Hoorcollege
1315 C1.112  Operations defined by formulas; notational
abbreviations in set theory. Proving that an operation is total. The
binary union axiom and the power set axiom. Construction of the
Cartesian product with binary union and power set.
Homework Set #2

19 February 2013 
Hoorcollege 1517 C1.112  Models of
(Ex)+(Ext)+(Aus)+(Pow) are infinite. The union axiom and arbitrary unions
and intersections. The axiom system FST of finite set theory.
Relations and functions in FST.


22 February 2013  Hoorcollege
1315 C1.112 
Cantor's theorem.
The model HF of the hereditarily finite sets.
Dedekindfiniteness. Inductive sets. The axiom of infinity. (Brief discussion of HW set
#1.)
Homework Set #3


26 February 2013 
Hoorcollege 1517 C1.112 
The definition of the natural numbers and the set of natural numbers. Properties of the
natural numbers. Constructing the integers and rational numbers as settheoretic
quotients.


1 March 2013  Hoorcollege
1315 C1.112 
Construction of the real numbers: Dedekind cuts. Size of infinite sets: the notion of
equinumerosity. Zermelo set theory proves that there is an uncountable set. Skolem paradox.
Homework Set #4


5 March 2013 
Hoorcollege 1517 C1.112
 Classes and proper classes. Examples of proper classes: there is no set of all singletons,
no set of all groups, etc. For every nonempty set, the class of sets equinumerous to it is a
proper class. The axiom scheme of Replacement. Sketch that if you iterate the HF
construction, you get a model of Z in which Replacement is false. The axiom scheme of
Foundation and the axiom of Foundation.


8 March 2013  Werkcollege
1315 C1.112 
Homework Set #5


12 March 2013 
Hoorcollege 1517 C1.112 
Wellfoundedness. Principle of Induction, principle of weak induction, and least number
principle on the natural numbers. Proof of the recursion theorem for functions on the natural
numbers.


15 March 2013  Werkcollege
1315 C1.112 
Homework Set #6


19 March 2013 
Hoorcollege 1517 C1.112 
Grassmann recursion equations for addition and multiplication on the
natural numbers. Induction on wellorders. The recursion theorem for
wellorders. Ordinals: the natural numbers are ordinals. Closure properties
of ordinals. Statement of the fundamental theorem on wellorders (without
proof).


22 March 2013  Werkcollege
1315 C1.112 
Homework Set #7


2 April 2013 
Hoorcollege 1315 C1.112 
Properties of ordinals: irreflexivity,
linear order, wellfoundedness of the
class of ordinals, transitive sets of
ordinals are ordinals. BuraliForti
paradox: there is no set of all ordinals.
Class version of the recursion theorem. Application: ordinal addition. Some
properties of ordinal addition: failure of commutativity.


3 April 2013  Werkcollege
1315 F1.02 
Homework Set #8


9 April 2013 
Hoorcollege 1315 C1.112 
Limit ordinals. Ordinal multiplication and exponentiation. Gamma, delta and epsilonnumbers.
Normal ordinal operations and the fixed point theorem. The existence of uncountable ordinals via
the coding method of coding countable ordinals as subsets of the natural numbers.


10 April 2013  Werkcollege
1315 F1.02 
Homework Set #9

16 April 2013 
Hoorcollege 1315 C1.112 
Hartogs' Theorem. The aleph function and its properties. Fixed points of the aleph function and
their size. Infinite cardinals. Proof that every wellordered set is in bijection with a
cardinal. Zermelo's wellordering theorem (without proof).


17 April 2013  Werkcollege
1315 F1.02 
Homework Set #10

23 April 2013 
Hoorcollege 1315 C1.112 
Proof of Zermelo's wellordering theorem. Choice functions. The axiom of choice. Cardinality of sets in ZFC. Fragments of the axiom of
choice. Some history of the axiom of choice and some consequences (without proof). Countable unions of countable sets are countable (using
the countable axiom of choice).


24 April 2013  Werkcollege
1315 F1.02 
Homework Set #11

7 May 2013 
Hoorcollege 1315 C1.112 
The wellordering theorem implies the Axiom of Choice. Zorn's lemma and its equivalence to the Axiom of
Choice. Cardinal addition and multiplication. Proof that κ+κ = κ. Hessenberg's theorem
(without proof).


8 May 2013  Werkcollege
1315 F1.02 
Homework Set #12

14 May 2013 
Hoorcollege 1315 C1.112 
Proof of Hessenberg's Theorem. Cardinal exponentiation. The continuum function. Hausdorff's Formula.


15 May 2013  Werkcollege
1315 F1.02 

21 May 2013 
Hoorcollege 1315 C1.112 
Cofinality. The gimel function. Fundamental theorem of cardinal arithmetic
(recursive determination of κ^{λ} in terms of the continuum
function and the gimel function. Von Neumann hierarchy. Mirimanoff rank.
Characterization of the axiom
of foundation in terms of the von Neumann hierarchy.


22 May 2013  Werkcollege
1315 F1.02 
Training Exam

