# Forcing and Independence in Set Theory

**Part of the UCLA Logic Center 2009 Summer School for Undergraduates**

*When:* Monday through Friday, 3PM-5PM. The first day is July 13 and the last is July 31.

*Where:* The lectures will be held in MS 5217. Problem solving sessions will be held daily in my office.

*Instructor:* Justin Palumbo

*Office: *MS 6617F

*Office hours:* I will be available by appointment. Just let me know.

Here is the original abstract for the course.

Set-theoretic forcing is a technique originally introduced by Paul Cohen to prove that the axiom of choice and the continuum hypothesis are independent of the classical axioms of mathematics. Since then it has led to an explosive growth in research, and has been used by set theorists to prove literally hundreds of independence results. In this course we will introduce students to the forcing technique by first studying some of the combinatorial consequences of Martin's Axiom, an axiom which when added to ZFC makes it possible to prove consistency results by means very similar to those used in forcing. We will then introduce forcing itself and show how it can be applied to settle the independence of the continuum hypothesis.

Here is an interesting youtube video of Paul Cohen reminiscing on his mathematical work leading up to his solution to the continuum hypothesis, and on his interactions with Kurt Gödel shortly thereafter.

The summer school has now ended. Below are the problem sets and portions of the notes covered each day. And here are the complete notes for the course put together in a single pdf.

A couple of first day problems, with solutions.

Day 1 (pdf): A (succinct) review. Ordinals, cardinals. Cantor-Bernstein Theorem, Cantor's Theorem, Konig's Lemma.

Relevant problems.

Day 2 (pdf): Some motivation. Cardinal
characteristics of the continuum, the bounding number, almost disjoint number. Solomon's theorem.

Relevant problems.

Day 3 (pdf): Technology and terminology. Posets,
dense sets, antichains, and filters. MA(\kappa), and the provable extent thereof.

Relevant problems.

Day 4 (pdf): Applications of MA. MA implies the bounding and almost disjoint numbers are the continuum, and that
the continuum is regular.

Relevant

Day 5 (pdf): MA implies that the union of less than continuum many Lebesgue measure zero sets has Lebesgue measure zero.

Relevant problems.

Day 6 (pdf): Ultrafilters and MA. Generating filters. ultrafilters are the same as maximal filters. Ramsey's
theorem and an application. MA implies the existence of a Ramsey ultrafilter.

Relevant problems.

Challenging additional MA problems (pdf).

Day 7 (pdf): Models of Set Theory. Transitive models,
absoluteness, $\Delta_0$ sets, review of the ZFC axioms.

Relevant problems.

Day 8 (pdf): The cumulative hierarchy. Neither Infinity
nor Replacement are provable from the other ZFC axioms. Mostowski collapse.

Relevant problems.

Day 9 (pdf): The reflection principles. Countable transitive
models of finite fragments of ZFC. Introduction to forcing - P-names, their evaluations.

Relevant problems.

Day 10 (pdf): M[G] satisfies pairing, union, foundation, extensionality. The
forcing relation and an example.

Relevant problems.

Day 11 (pdf): Statement of the forcing theorems. M[G] is the
minimal transitive model of ZFC including both M and G.

Relevant problems.

Day 12 (pdf): Proofs of the Forcing Theorems.
Countably closed posets.

Relevant problems.

Day 13 (pdf): Consistency of CH.
The Delta-System Lemma.

Relevant problems.

Day 14 (pdf): Consistency of the failure of CH.
The value of the continuum in the Cohen model.