Math 5020 Spring 2017

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Course Information: Syllabus

In this course we give a development of modern set theory. We begin with a develop of the \(\mathsf{ZFC}\) axioms of set theory, and develop the basic properties of models of set theory. We give a rigorous presentation of ordinals and cardinals, and fundamental notions such as absoluteness and definability. We discuss independence and the incompleteness theorem as it pertains to set theory. We discuss inner models such as \(L\) and HOD, and show, for example, that GCH holds in \(L\). We develop forcing and us it to get various consistency results as well as a tool for proving theorems. We discuss combinatorial principles such as \(\diamond\) and MA. Other topics as time permits.
We will not follow a single text exclusively, but a good reference is Set Theory, an Introduction to Independence Proofs by K. Kunen. Another good reference is Set Theory by T. Jech. We will also use class notes which I will post on the website.

Here are two notes to get started. I will be modifying these notes as the class progresses.

Axioms of set theory

Ordinals, Ordinal arithmetic,

Cardinals, Absoluteness

Forcing, Independence of CH

HW #1.
1.) Exercise 21 in the notes: show that for all sets \( x_1,\dots,x_n\), the set \(y= \{ x_1,\dots,x_n\} \) exists.
2.) Exercise 24 in the notes: show that for any relation \( R\) that \( \text{dom}(R) \) and \( \text{ran}(R) \) are sets.
3.) Show that \( \mathsf{ZF} \vdash \neg \exists x \, \forall y\, (y \in x) \). Follow the informal argument of Russell's paradox, but give the argument rigorously from \( \mathsf{ZF} \).
4.) Exercise 6 from the ordinal notes.

HW #2. Due Thursday, March 1.
1.) Write out a formula in the language of set theory \( \varphi(x,y) \) which holds in a model of ZF iff \( y\) is the transitive closure of \( x\). Show that \( \mathsf{ZF} \vdash \forall x\ \exists y\ \varphi(x,y) \).
2.) Show that if \( \alpha, \beta\) are ordinals and each can be order-embedded into the other, then \( \alpha =\beta\).
3.) View the exponential function \( f(n)=2^n\) as being defined by recursion: \( f(0)=1\) and \( f(n+1)= 2\cdot f(n)\). Write a formula \( \varphi(m,n)\) in the languge of set theory
which holds in a model of \( \mathsf{ZF} \) iff \(m, n\) are integers and \( n=2^m\). You don't have to go down to the level of a formal wff just using \( \epsilon\), but it should be clear that you could continue to do so.

HW #3.
1.) Let \( \alpha= \omega^3\cdot 2 +\omega^2\cdot 5 +3\) and \( \beta= \omega^4\cdot 3 +\omega\cdot 6+7\). Compute \( \alpha+\beta\), \( \beta+\alpha\), \( \alpha \cdot \beta\), and \( \alpha^\beta\).

2.) Let \( \kappa\) be a strong limit cardinal. Show that \( \text{cof}(\gimel(\kappa)) >\kappa\).

3.) Exercise 13 from the notes: Suppose \( \kappa\) is a singular limit cardinal. Suppose there is a \( \lambda <\kappa\) such that \( \lambda^{\text{cof}(\kappa)} \geq \kappa\), and let \( \lambda\) be the least such. Show that \( \text{cof}(\lambda) \leq \text{cof}(\kappa)\). [hint: suppose \( \text{cof}(\kappa) < \text{cof}(\lambda)\), and show that \( \lambda^{\text{cof}(\kappa)} <\kappa\). You will need to show that \( \text{sup} \{ \mu^{\text{cof}(\kappa)} \colon \mu <\lambda\} <\kappa \)]

HW #4.
1.) Exercise 1 of the forcing notes: show every generic \( G\) meets every predense set and every maximal antichain.

2.) Exercise 3 of the forcing notes: show that \( p \Vdash \neg \neg \varphi\) iff \( p \Vdash \varphi\).

3.) Exercise 2 of the continuum notes: show that \( \prod_{\alpha \in I} {\mathbb{R}}_{\text{std}}\) is c.c.c.