The Constructible Universe
Sommersemester 2024
Universität Hamburg
Fachbereich Mathematik
Lecture I: 8 April 2024. Geom H1.
§1. Historical background and motivation.
Hilbert's problems: Hilbert's first problem, the continuum hypothesis \(\mathsf{CH}\).
Meaning of \(2^{\aleph_0} = \aleph_1\) and the role of the axiom of choice.
Proof that there is a surjection from \(\wp(\mathbb{N})\) onto \(\aleph_1\) without
use of the axiom of choice. Emil du Bois Raymond: Grenzen des Naturerkennens and
the ignorabimus.
Hilbert's "Wir müssen wissen, wir werden wissen".
Independence of \(\mathsf{CH}\): Gödel's 1938 result and Cohen's 1962 result.
§2. Substructures. Definitions. Absoluteness. Upwards and downwards absoluteness.
Atomic formulas, propositional formulas, \(\exists\), \(\forall\), \(\exists\forall\),
and \(\forall\exists\)-formulas. Absoluteness of propositional formulas.
Preservation of upwards absoluteness by \(\exists\) and downwards absoluteness by \(\forall\).
Elementary substructures. Reminder: the isomorphism lemma. Elementary embeddings.
§3. The language of set theory.
Formulas describing 0 and 1 are not propositional and not absolute for substructures.
Example. The formula describing the power set is not absolute.
Transitive substructures. The formula describing 0 is absolute for transitive
substructures.
Lecture Notes.
Lecture II: 15 April 2024. Geom H1.
§4. Transitive substructures.
Bounded quantification and \(\Delta_0\) formulas. Absoluteness of \(\Delta_0\) formulas
for transitive models. \(\Delta_0^T\) formulas for a theory \(T\). Examples of
\(\Delta_0^T\) formulas: subset, pair, singleton, empty set, union, intersection.
Extensionality and Foundation hold in transitive submodels of models of Extensionality
or Foundation, respectively. Conditions for having Pairing, Union, and Power set
in transitive submodels. Definable operations. Absoluteness of definable operations.
Concatenations of absolute formulas and operations are absolute. Examples of further
absolute formulas.
§5. Relativisation. Definable substructures. Definition of the relativisation
of a formula. Proof of relative consistency via proving relativised formulas.
§6. The von Neumann hierarchy.
Definition and basic properties. Remark about the consistency proof of Foundation.
Lecture Notes.
Lecture III: 22 April 2024. Geom H1.
§6. The von Neumann hierarchy.
The von Neumann hierarchy is defined by a formula, i.e., an inner model. Closure properties
of the von Neumann hierarchy. Proof of \(\mathsf{ZF}\) in the von Neumann hierarchy. Conditions under which
axioms hold in the levels of the von Neumann hierarchy. All axioms of Zermelo set theory hold in
limit ranks, not replacement does not hold in \(\mathbf{V}_{\omega+\omega}\) and
\(\mathbf{V}_{\omega_1}\).
§7. More absoluteness and wellfoundedness.
\(\Sigma_1\), \(\Pi_1\), and \(\Delta_1^T\) formulas. Wellfoundedness is \(\Pi_1\) and so
not necessarily absolute. Wellfoundedness is not expressible in first-order logic: proof
that there are illfounded models of \(\mathsf{ZF}\) using the compactness theorem.
Being an ordinal is equivalent to a \(\Delta_0\) formula in set theory with foundation, so
it is absolute between transitive models of \(\mathsf{ZF}\). Being a cardinal is \(\Pi_1\).
§8. Replacement.
Regularity and strong limits. Inaccessible cardinals. Zermelo's Theorem: if \(\kappa\) is
inaccessible, then \(\mathbf{V}_\kappa\) is a model of replacement. The proof proves a
stronger statement; discussion of the consequences for the provability of the existence of
inaccessible cardinals. Large cardinals.
Lecture Notes.
Lecture IV: 29 April 2024. online.
§8. Replacement.
The difference between Replacement and Second-Order Replacement. Shepherdson's Theorem (without proof).
§9. Countable transitive submodels.
Löwenheim-Skolem and countable models. The Tarski-Vaught Lemma. Construction of a countable
elementary submodel. Mostowski's Collapsing Theorem. Construction of a countable transitive
submodel that elementarily embeds. Height of the model. Transitive submodels that contain
all of the subsets of \(\omega\): preservation of \(\aleph_1\) and upwards preservation of
\(\mathsf{CH}\). Transitive cardinal-preserving submodels: downwards preservation of \(\mathsf{CH}\).
Lecture Notes.
Lecture V: 6 May 2024. Online.
§10. Defining definability.
Definition of definability in the meta-language. Definability cannot be expressed
in the object language: the least non-definable ordinal. Coding of the language
in a model of set theory. Recursive definition of formulas.
Transfinite recursion preserves absoluteness. The notion of formula is absolute
between transitive models of set theory. Recursive definition of truth in a structure.
The notion of definability in a (set) structure is absolute.
§11. Definition of the constructible universe.
The definable power set operation \(\mathcal{D}\) and its absoluteness.
The constructible hierarchy. Transitivity of the \(\mathrm{L}_\alpha\).
Constructibility. Absoluteness of constructibility. Every transitive model of \(\mathsf{ZF}\) contains all constructible sets.
Lecture Notes.
Lecture VI: 13 May 2024. Online.
§11. Definition of the constructible universe.
\(\mathrm{L}_\alpha\cap\mathrm{Ord} = \alpha\); the constructive universe is a proper class; the constructible rank function; differences between constructible rank and Mirimanoff rank;
\(\mathrm{L}_{\omega+1} \neq \mathrm{V}_{\omega+1}\).
§12. Simple axioms of \(\mathsf{ZF}\) in the constructive universe.
Extensionality and Foundation hold in all transitive classes.
Infinity holds since \(\omega\in\mathrm{L}_{\omega+1}\).
Pairing and Union hold since the classical (von Neumann) proof provides the witnesses
in \(\mathrm{L}_{\alpha+1}\). Power set uses Replacement in \(M\) to find
a bound on the ranks of all constructible subsets. (We do not know anything about this bound.)
A failed attempt to prove Separation in the constructible universe.
§13. The Lévy Reflection Theorem. Hierarchies. The Lévy Reflection
Theorem. An improved version of the Tarski-Vaught lemma for collections of formulas closed
under subformulas. Proof of the Lévy Reflection Theorem.
Lecture Notes.
Pentecost Monday: 20 May 2024. No lecture.
Lecture VII: 27 May 2024. Online.