2021: Forcing

Here you will find, starting in November, the recorded lectures on cardinal arithmetic, some combinatorics, and forcing.

The lectures

Note: the course involves Group Interactions, where students work on some problems together. These problems are a bit more open-ended than the homework problems.

2021-11-01: Cardinal Arithmetic

Sum, products, and powers of cardinals and how to compute these recursively.
Material

• The recording
• The preparations
• The writings (what became of the preparations)
• The Homework
• The problems for Group Interaction

2021-11-08: Stationary sets and Δ-systems

Closed and unbounded subsets of regular uncountable cardinals; stationary subsets; Fodor's Pressing-Down Lemma; Solovay's theorem on partitioning stationary sets into many stationary sets; the Δ-system Lemma.
Material

• The recording
• The preparations
• The writings (what became of the preparations)
• The Homework
• The problems for Group Interaction

2021-11-15: The Free Set Lemma and some Partition Calculus

Proof of The Free Set Lemma, to show the difference between regular and singular cardinals. Ramsey's Theorem (infinite version) and the Erdős-Rado Theorem.
Material

• The recording
• The preparations
• The writings (what became of the preparations)
• The Homework
• The problems for Group Interaction

2021-11-22: Forcing I: models, absoluteness, and reflection

How to prove that the Continuum Hypothesis is not provable? Avoid Gödel's incompleteness theorems by building, given finitely many axioms, a countable structure in which these axioms are valid, and \(\mathsf{CH}\) is not. Today we see how to satisfy the axioms …
Material

• The recording
• The preparations
• The writings (what became of the preparations)
• The Homework
• The problems for Group Interaction

2021-11-29: Forcing II: how to extend a model to get a model of the negation of \(\mathsf{CH}\)

How to extend a countable transitive set that satisfies (a large portion of) \(\mathsf{ZFC}\) to a similar set that also satisfies \(\neg\mathsf{CH}\). Finite approximations to an injection of \(\omega_2\) into \(\mathcal{P}(\omega)\). An application of the Δ-system Lemma to show that \(\omega_1\) and \(\omega_2\) are preserved.
Material

• The recording
• The preparations
• The writings (what became of the preparations)
• The Homework
• The problems for Group Interaction

2021-12-06: Forcing III: the technical details

How to make \(M[G]\) out of \(M\) and \(G\). Generic extensions via arbitrary partial orders: \(M\)-generic filters, names, evaluations of names. The definability of forcing and the Truth Lemma. The proper definition of the relation that was used last week.
Material

• The recording
• The preparations
• The writings (what became of the preparations)
• The Homework
• The problems for Group Interaction

2021-12-13: Forcing IV: the technical details

Summary of what went before. \(\mathsf{ZFC}\) in \(M[G]\). Revisit Cohen's model for \(2^{\aleph_0}=\aleph_2\): cofinalities and cardinals are preserved; calculate \(2^{\aleph_1}\) in the extension. Show that \(2^{\aleph_0}\) can be anything not forbidden by what we discovered while doing cardinal arithmetic. Show how to get \(2^{\aleph_0}=\aleph_1\) plus \(2^{\aleph_1}>\aleph_2\). Problem (48) in the homework shows how to force \(\mathsf{CH}\).
Material

• The recording
• The preparations
• The writings (what became of the preparations)
• The Homework
• The problems for Group Interaction
• R. Solovay, \(2^{\aleph_0}\) can be anything it ought to be

2021-12-20: Forcing V: some examples

The power set axiom in \(M[G]\).

• Consistency of "every almost disjoint family of uncountable subsets of \(\omega_1\) has cardinality at most \(\aleph_2\) plus \(2^{\aleph_0}>\aleph_2\)" (a nice application of the Erdős-Rado theorem).
• Tennenbaum's construction of a Souslin tree using finite approximations.

Material

• The recording
• The preparations The preparations contain two more examples of forcing arguments:
  • Consistency, due to Kunen, of "the \(\sigma\)-algebra generated by the rectangles in \(\omega_2\times\omega_2\) is not equal to the whole power set (it need not contain the set \(\{\langle\alpha,\beta\rangle:\alpha < \beta\}\))".
  • If \(S\) is a stationary subset of \(\omega_1\) then there is a generic extension in which \(\omega_1\) is preserved and \(S\) contains a closed and unbounded subset (Devlin).
• The writings (what became of the preparations)
  • first page
  • the rest
• S. Tennenbaum, Souslin's problem

Material

Chapter 4 of the book Teorie Množin by Balcar and Štěpánek (translated by E. Copláková)

This chapter first investigates Boolean algebras in some depth. Then it treats generic extensions in a more-or-less axiomatic way and in the end applies the theory of complete Boolean algebras to show how generic extensions can be constructed.