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
- 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
- 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
- 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
- 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
- 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
- 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
- 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)
- 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.
K_dot_P_dot_Hart_at_TUDelft_dot_nl
Last modified: Tuesday 21-12-2021 at 15:10:39 (CET)