2022: Constructibility
Suggested Readings
- Keith J. Devlin,
Constructibility.
The first three chapters cover what we covered in the last four lectures.
The lectures
The lectures are being recorded and are stored in this
Vimeo showcase
(easy password: 71RT).
- 2022-09-12: History and Axioms
- A short description of the beginnings of Set Theory and a discussion
of Zermelo's Axioms.
Material
- 2022-09-19: The Axioms as they are now.
- The first few axioms, as they appear in Kunen's book.
With Skolem's formulation of what Zermelo called a `definite property':
describable using a well-formed formula in the language of Set Theory.
- 2022-09-26: More Axioms
- How various familiar notions, especially relations and functions
(or maps), work in Set Theory.
Well-orders and the Axiom of Choice \(\mathsf{AC}\).
- 2022-10-03: Ordinals
- Zermelo's proof of the Well-Ordering Theorem.
Ordinals and their role as standard well-ordered sets.
- 2022-10-10: More ordinals
- Ordinal arithmetic.
Induction and Recursion.
- 2022-10-17: Cardinalities and Cardinals
- Cardinalities: comparison by way of injective and bijective maps.
Cardinals: special ordinal numbers that assign a `number of elements'
to well-orderable sets.
Definition of finiteness.
- 2022-10-24: Arithmetic and Foundation
- Cardinal arithmetic and the continuum function.
The Axiom of Foundation (or Regularity).
The cumulative hierarchy:
\(\langle V_\alpha:\alpha\in\mathbf{On}\rangle\)
and its union \(\mathbf{V}\).
- 2022-10-31: More cardinal arithmetic
- How to compute \(\kappa^\lambda\) for cardinals \(\kappa\)
and \(\lambda\), and its dependence on
\(\gimel(\kappa)=\kappa^{\mathrm{cf}\kappa}\).
- 2022-11-07: Cub and stationary sets and applications
- The Fodor's Pressing-Down Lemma; the \(\Delta\)-system lemma;
Hajnal's Free Set Lemma.
- 2022-11-14: 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.
- 2022-11-21: Trees
- Kőnig's Infinity Lemma (\(\aleph_0\) has the tree property)
and its relation to Ramsey's Theorem.
Construction of an Aronszajn tree (\(\aleph_1\) does not have the tree
property).
A discussion of Souslin's Problem and its relation to trees.
- 2022-11-28: The Constructible Universe \(\mathbf{L}\), I
- The informal definition of \(\mathbf{L}\): iterating the operation
of taking the definable subsets to get
\(\langle L_\alpha:\alpha\in\mathrm{On}\rangle\) and
\(\mathbf{L}=\bigcup_{\alpha\in\mathrm{On}}L_\alpha\)
- 2022-12-05: The Constructible Universe \(\mathbf{L}\), II
- The formal definition of \(\mathbf{L}\): redefining definability using
the Gödel operations and \(\Delta_0\)-formulas.
Absoluteness of \(\langle L_\alpha:\alpha\in\mathrm{On}\rangle\) and
\(\mathbf{L}\), as well as of the notion of constructibility itself.
- 2022-12-12: The Constructible Universe \(\mathbf{L}\), III
- The Axiom of Choice and the Generalized Continuum Hypothesis in
\(\mathbf{L}\), elementary substructures, and the Condensation Lemma.
- 2022-12-19: The Constructible Universe \(\mathbf{L}\), IV
- Trees in \(\mathbf{L}\): a Souslin tree and the
Diamond-principle \(\lozenge\).
Also, elementary substructures again: minimality.
A short description of Kurepa's Hypothesis, its status
in \(\mathbf{L}\), and \(\lozenge^+\).
- 2023-01-16: The Exam
- You can have a go at
K_dot_P_dot_Hart_at_TUDelft_dot_nl
Last modified: dinsdag 14-02-2023 at 12:12:13 (CET)