Mastermath: Set Theory (spring 2011)

Form of the exam

• 25 multiple choice questions designed to test your knowledge and insight of basic facts and major results; here are some sample questions (more may follow).
• a number of open questions that will test your ability to reproduce/recreate/create proofs of theorems given in class as well as your ability to come up with arguments for results that were not necessarily included in the lectures; here are some sample problems.

Program

Week 1: Axioms

An introduction to Set Theory with an explanation of the need for axioms. The first three axioms: Extensionality, Existence of empty set, Pairing.

Week 2: More Axioms

The axioms of Union, power set, separation, infinity, replacement and regularity. A discussion of classes.
Homework: problems 1.4, 1.5, 1.14 and 1.15. Hand in in week 3.

Week 3: Well-orderings and ordinal numbers

Introduction to well-ordered sets and ordinal numbers. Main results: any two well-ordered sets are comparable and every well-ordered set is isomorphic to an ordinal number.

Week 4: Well-orderings and ordinal numbers and Cardinal numbers

• Induction and recursion;
• ordinal arithmetic with Cantor normal form (here is a description of a set that represents the ordinal power and a link to Goodstein's paper can be found on the Leiden set theory page, see below);
• well-founded relations
• Definition of `the same cardinality' and a proof that a the cardinality of a set is strictly smaller than that of its power set.

Homework: problems 2.7, 2.8, 2.11 and 2.13. Hand in in week 5.

Week 5: Cardinal numbers and Cardinal Arithmetic.

• Cantor-Bernstein Theorem;
• elementary cardinal arithmetic for cardinalities;
• definition of cardinal numbers and Hessenberg's theorem; the original proof can be found here (on page 109) and yet another proof, by Jourdain, is here;
• cofinality and regular versus singular; a model in which $\omega_1$ is singular can be found in Chapter 15 of the book (Example 15.57).

Week 6: Axiom of Choice.

Some proofs that use the axiom of choice from ordinary mathematics, recapitulation of Hartogs' Theorem, definition of choice functions, definition of arbitrary sums and products of sets, various formulations of the axiom of choice, fragments of the axiom of choice (e.g., countable choice), some mathematical equivalents and consequences of the axiom of choice (without proofs), equivalence of AC, the wellordering theorem, and the comparability of cardinals. Definition of Zorn's Lemma.
Homework 5.1, 5.3 and 5.4. Hand in in week 7. (Theorem 4.5 of Chapter 4 may be useful).

Week 7: Axiom of Choice and Cardinal Arithmetic

• equivalence of AC and Zorn'sLemma
• cardinal arithmetic under the Axiom of Choice

Week 8: Axiom of Choice and Cardinal Arithmetic and The Axiom of Regularity

Inaccessibility, SCH, Axiom of Regularity vs Axiom Scheme of Regularity (proof that the set version implies the class version), Transitive Closure of a set, von Neumann (cumulative) hierarchy, proof (with regularity) that $V$ is the universe, Scott's Trick (cardinals without the AC), Induction and Recursion on sets, Axioms in the ranks $V_\alpha$: $V_\lambda$ for $\lambda > \omega$ limit is a model of ZC, $V_{\omega+\omega}$ does not satisfy Replacement.
Homework: Exercises 5.8, 5.15, 5.27, and 6.5. Hand in in week 9.
As to the construction of $\mathbb{Z}$, $\mathbb{Q}$ and $\mathbb{R}$, these are quite algebraic:

• $\mathbb{Z}$ is constructed from $\mathbb{N}$ using an equivalence relation on the pairs: $\langle k,l\rangle$ is equivalent to $\langle m,n\rangle$ if $k+n=l+m$
• $\mathbb{Q}$ is constructed from $\mathbb{Z}$ as its field of quotients, via an equivalence relation on $\mathbb{Z}\times\mathbb{N}_{>0}$: $\langle k,l\rangle$ is equivalent to $\langle m,n\rangle$ if $k\times n=l\times m$.
• $\mathbb{R}$ is constructed from $\mathbb{Q}$ as the set of Dedekind cuts (see page 39 of the book)

Week 9: Models of Set Theory, Inaccessible cardinals and Measurable cardinals

$V_\alpha$ models Replacement implies $\alpha$ is a limit cardinal and an $\aleph$-fixed point. $\kappa$ inaccessible implies $V_\kappa$ models Replacement. A hierarchy of large cardinal axioms:

• IC ("there is an inaccessible"),
• 2IC ("there are two inaccessibles"),
• nIC ("there are n inaccessibles"),
• OrdIC ("there is a proper class of inaccessibles"),
• "there is an inaccessible limit of inaccessibles".

Proof that this is a sequence strictly increasing in logical strength. Measurable cardinals. Proof that a measurable cardinal is inaccessible.

Week 10: Measurable cardinals and Combinatorial set theory

Three partition theorems: Ramsey, Erdos-Rado, Dushnik-Miller.
Homework: 10.4, 12.10, 9.5 and the exercise in this file (exercises 9.8 and 9.9 were about material that was not yet covered in class). This exercise needs a stronger property of measurable cardinals: namely that they carry a normal ultrafilter, see page 134 and following. Using this property one can show that the homogeneous set can be found in the ultrafilter.

Week 11: Combinatorial set theory and Stationary sets

More on weakly compact cardinals. The order topology on ordinals. Closed unbounded and stationary sets. The club filter. Mahlo cardinals.
Correction: The proof of the statement done in the last few minutes of class (``If $\kappa$ is the least weakly inaccessible cardinal, then the set of regular cardinals below $\kappa$ is not stationary in $\kappa$.'') was incorrect.
Here is a correction: As argued in class, the set of limit cardinals less than $\kappa$ is club in $\kappa$. But since $\kappa$ is the least weakly inaccessible, all limit cardinals below $\kappa$ are singular, so the set $S$ of singular cardinals is club in $\kappa$. The set Reg of regular cardinals below $\kappa$ is disjoint from $S$ and hence not stationary.

Week 12: Stationary sets and Combinatorial set theory

Mahlo cardinals, properties of the club filter, diagonal intersection, Fodor's lemma, the tree property, every weakly compact cardinal has the tree property.
Homework: 8.5, 8.6 and 9.9.

Week 13: Weakly Compact Cardinals and Powers of singular cardinals

Finished the discussion of weakly compact cardinals with compactness for infinitary logics. Silver's theorem on the GCH at singular cardinals of uncountable cofinality.

Week 14: Powers of singular cardinals

The Galvin-Hajnal theorem. Ordinal functions and scales.
Homework: 17.17, 17.19, 24.2, 24.6.
Two other sources: Burke and Magidor and Abraham and Magidor.

Week 15: Powers of singular cardinals

We finished the proof of 24.8.
After a discussion of the reasons for studying the powers of singular cardinals we talked about elementary substructures of $H(\theta)$ and we looked ahead at what pcf can tell us about, specifically, $2^{\aleph_\omega}$.
We embarked upon the proof of 24.18. Thus far we did 24.10, 24.19 and 24.20.
Here is a summary of all we learned about cardinal exponentation in Chapter 8.
Here is the paper by Kojman and Shelah on the normal space whose product with the interval $[0,1]$ is not normal.

Week 15: Powers of singular cardinals

We finished the proof of Theorem 24.18. We ended the course with a sketch/overview of the proof of Theorem 24.33; the theorem on Club-guessing sequences is in Chapter 23. The chapter on cardinal arithmetic by Abraham and Magidor in the Handbook of Set Theory is highly recommended reading; it contains forms of Theorems 24.18 and 24.33 that do not need the strong limit assumptions on $\aleph_\omega$.

Literature

We also recommend the following books

• K. Kunen, Set Theory. An introduction to Independence proofs
• Y. N. Moschovakis, Notes on Set Theory
• K. J. Devlin, The Joy of Sets.

Course in Leiden

Other material