**Instructor:** Itay Neeman.

Office: MS 6334.

Email:

Phone: 794-5317.

Office hours: Mondays and Wednesdays, 10-11am.

The class will cover the basics of forcing, a technique introduced by Cohen to prove the independence of the Continuum Hypothesis from the axioms of mathematics.

Recall that two sets $A$ and $B$ have the same cardinality if there is a bijective function $f \colon A\rightarrow B$. The Continuum Hypothesis states that every infinite subset of ${\mathbb R}$ has either the same cardinality as ${\mathbb R}$, or the same cardinality as ${\mathbb N}$; there are no cardinalities in between.

It turns out that the Continuum Hypothesis is {\em independent}, meaning neither provable nor refutable, from the axioms of mathematics. That it cannot be refuted was shown by G\"odel (1940's) and that it cannot be proved was shown by Cohen (1960's). Cohen's technique has since been used in proofs of many other independence results. For example Solovay used it to show that the existence of a non-measurable set of reals cannot be proved without the axiom of choice.

The class will cover the basics of the forcing technique, preservation of cardinals in forcing extensions, applications to cardinal arithmetic (including the independence of the CH), iterated forcing, Martin's axiom, and applications of Martin's axiom.

**Time and Place:** MWF 12-12:50pm, in MS 6118.

**Text:** Set Theory, an Introduction to Independence Proofs, by Kenneth Kunen.

**Grading and assignments**: Students will be asked
to solve assigned questions from Kunen's book
and present the solutions in class.
Grading will be based on the presentations.

List of assigned questions (updated as the term progresses):

Given in class:

Question A, claimed by Darren Creutz.

Question B, claimed by Konstantinos Palamourdas.

From Chapter 2 of Kunen.

Question 8, claimed by Ioannis Souldatos.

Question 9, claimed by Justin Palumbo.

Question 10, claimed by Anush Tserunyan.

Question 20, claimed by Justin Palumbo.

Question 21, claimed by Ioannis Souldatos.

Question 25, claimed by Darren Creutz.

Question 26, claimed by Konstantinos Palamourdas.

Question 27, claimed by Anush Tserunyan.

Question 30, claimed by Ngoc Chi Le.

From Chapter 7 of Kunen.

Question E1, claimed by Konstantinos Palamourdas.

Question E2, claimed by Justin Palumbo.

Question G8, claimed by Anush Tserunyan.

Question H1, claimed by Ioannis Souldatous.

Question H2, claimed by Darren Creutz.