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 constructible 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. §14. \(\mathsf{ZF}\) in the constructible universe. Separation. Replacement. The Axiom of Constructibility \(\mathbf{V}{=}\mathbf{L}\). The constructible universe is the minimal model of \(\mathsf{ZF}\).
§15. \(\mathsf{AC}\) in the constructible universe. Global choice. Inductive construction of a well-ordering of \(\mathbf{L}_\alpha\) in the constructible universe. Sizes of constructible ranks.
§16. The condensation sentence. Absoluteness for transitive models of \(\mathsf{ZF}\) shows that there are finite sets of axioms that imply absoluteness. The condensation sentence \(\sigma^*\). A transitive set is a model of \(\sigma^*\) if and only if it is a constructible rank (no proof yet).
Lecture Notes (corrected version, 7 June 2024).

Lecture VIII: 3 June 2024. Online. §16. The condensation sentence. Proof of the finite axiomatisability of "being an \(\mathbf{L}_\alpha\)" for transitive sets. (Correction of the originally given condensation sentence \(\sigma\) to the correct condensation sentence \(\sigma^*\).)
§17. Constructible ranks and inaccessibility. Sets of hereditary size \(<\kappa\); \(\mathbf{H}_\kappa\). Second order replacement and power set in \(\mathbf{H}_\kappa\). Inaccessibility and \(\mathbf{V}_\kappa = \mathbf{H}_\kappa\). Constructible ranks and \(\mathbf{H}_\kappa\) under \(\mathbf{V}{=}\mathbf{L}\). Existence of countable ordinal \(\alpha\) such that \(\mathbf{L}_\alpha\models\mathsf{ZFC}\). Brief discussion of the analogous construction for the von Neumann hierarchy. Inaccesible cardinals do not say whether \(\mathbf{V}{=}\mathbf{L}\) holds.
Lecture Notes.

Lecture IX: 10 June 2024. Online. §17. Constructible ranks and inaccessibility. Finitely axiomatisable hierarchies. Models of \(\mathsf{ZFC}\) in finitely axiomatisable hierarchies. If \(\mathbf{V}_\alpha\models\mathsf{ZFC}\), then \(\alpha\) is a cardinal. The von Neumann hierarchy is not finitely axiomatisable.
§18. \(\mathsf{CH}\) in the constructible universe. The condensation lemma (countable case). \(\mathbf{L}\models\mathsf{CH}\).
§19. Cardinal arithmetic in \(\mathbf{L}\). The generalised continuum hypothesis. The general case of the condensation lemma. \(\mathbf{L}\models\mathsf{GCH}\). Cardinal arithmetic under \(\mathsf{GCH}\).
Lecture Notes.

Lecture X: 17 June 2024. Online. §20. Suslin's Problem. Total orders: density, unboundedness, completeness, separability. Cantor's theorem on countable dense orders (no proof). Characterisation of the real line as a total order. Suslin's problem. Suslin's hypothesis.
§21. Diamond. Closed and unbounded sets; stationary sets. Stationary sets are unbounded. Diamond sequences. The diamond principle \(\diamondsuit\). The diamond principle implies \(\mathsf{CH}\).
§22. \(\diamondsuit\) in \(\mathbf{L}\). Construction of the diamond sequence in \(\mathbf{L}\). Properties of countable elementary submodels of \(\mathbf{H}_{\omega_2}\).
Lecture Notes.

Lecture XI: 24 June 2024. Online. §22. \(\diamondsuit\) in \(\mathbf{L}\). The countable submodel \(H\) and its Mostowski collapse \(T\). Particular values of the Mostowski isomorphism. \(\aleph_1^T\) is a counterexample to the claim that the sequence is not a diamond sequence.
§23. Cantor's theorem on dense linear orders. All countable dense and unbounded total orders are isomorphic to the rationals. All countable total orders embed into the rationals. Back-and-forth method.
§24. Effective versions of \(\mathsf{CH}\). The Cantor sets. Perfect sets (no formal definition yet). The perfect (sub)set property. \(\mathsf{PSP}\) implies \(\mathsf{CH}\). The Cantor-Bendixson theorem (no proof). Hausdorff's theorem (no proof). \(\mathsf{PSP}\) contradicts the axiom of choice. Lebesgue's error. Analytic sets. Suslin's theorem: analytic sets have the perfect set property (no proof). The projective hierarchy. Goal: in \(\mathbf{L}\) there is a projection of a co-analytic set that does not have the perfect set property.
Lecture Notes.

Lecture XII: 1 July 2024. Geom 414 & online. §25. Perfect sets & Bernstein's theorem. Cantor space. Distance function. Bijection with the power set of the natural numbers. Relation to binary expansions of real numbers. Closed sets in Cantor space. Countable sets are \(\mathrm{F}_\sigma\). Isolated points and perfect sets. Trees. The tree defined by a set in Cantor space. The set of branches through a tree. Characterisation of closed sets by trees. Splitting nodes and perfect trees. Characterisation of perfect sets by trees. Cantor's lemma about the size of perfect sets. The perfect set property. Theorems of Cantor-Bendixson and Hausdorff repeated (still without proof). Cardinality of the sets of trees and perfect trees. Bernstein sets. Bernstein's theorem.
§26. Descriptive complexity. Borel sets and \(\sigma\)-algebras: complement and countable union. Lebesgue's error repeated. Interlacing and homeomorphism of Cantor space with its finite products. Zero-dimensionality of Cantor space. Projections. Pointclasses. The dual pointclass; the pointclass formed by projections. The analytic sets. The projective hierarchy. Notations \(\boldsymbol{\Sigma}^1_n\) and \(\boldsymbol{\Pi}^1_n\).
Lecture Notes.

Lecture XIII: 8 July 2024. Geom 414 & online. §26. Descriptive complexity. Remarks on the proof and its converse: There is a \(\boldsymbol{\Pi}^1_1\) set without the perfect set property in \(\mathbf{L}\): no proof. The technique of forcing can construct models where every \(\boldsymbol{\Pi}^1_1\) set has the perfect set property (Solovay); this proof requires the assumption of an inaccessible cardinal: no proofs. Remark on quantifiers and descriptive complexity classes.
§27. Coding. Coding of countable models as elements of Cantor space. Coding of functions from \(\mathbb{N}\) to \(\mathbb{N}\) as elements of Cantor space. Wellfoundedness of a countable model is a \(\Pi^1_1\) property. Coding of validity of a formula in a countable model is Borel. Evaluating the Mostowski collapse for natural numbers and elements of Cantor space is Borel.
§28. Descriptive complexity in \(\mathbf{L}\). Warm-up: the complexity of the wellorder on Cantor space in \(\mathbf{L}\) is \(\Sigma^1_2\). Remark on the Fubini proof that a wellorder of order type \(\omega_1\) cannot be Lebesgue measurable. Bernstein sequences. Reconstruction of Bernstein's proof: the Bernstein set is \(\Sigma^1_2\).
Lecture Notes.