The Forcing Method
Sommersemester 2026
Universität Hamburg
Fachbereich Mathematik

Vorlesung I: 9. April 2026. Sed 19 107.
§1. The incompleteness phenomenon. Science optimism vs limitations of science. Du Bois-Reymond: ignorabimus. Foundational crisis. The axiomatic method. Hilbert's programme. Gödel's completeness theorem. Gödel's incompleteness theorem.
§2. Incompleteness in set theory. The continuum problem. The continuum hypothesis. Cantor's theorem vs Hartogs's theorem. The Aleph and the Beth hierarchy. The generalised continuum hypothesis. Equivalent formulations of the continuum hypothesis. Gödel: method of inner models; Cohen: method of outer models. Problems proved independent with the method of forcing.
§3. Independence in algebra. Adding \(\sqrt{2}\) without adding \(\sqrt{3}\). Model-theoretic basics. Absoluteness. The substructure lemma.
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Vorlesung II: 16. April 2026. Sed 19 107.
§4. Absoluteness in set theory. The language of set theory. Upwards and downwards absoluteness. Formulas defining \(0\) and \(1\). The formula \(x=0\) is not absolute for substructures.
§5. Transitive models. Transitive models. Example: the formula \(x=0\) is absolute for transitive models.
§6. The extent of absoluteness for transitive models. The formula \(x=0\) is downwards absolute. The formula \(x\neq 0\) is upwards absolute. \(\Delta_0\) formulas. \(\Delta_0\)-formulas are absolute for transitive models. \(\Delta_0^T\) formulas. \(\Delta_0^T\)-formulas are absolute for transitive models of \(T\). \(\Sigma_1^T\)-formulas are upwards absolute for transitive models of \(T\). \(\Pi_1^T\)-formulas are downwards absolute for transitive models of \(T\). \(\Delta_1^T\)-formulas are absolute for transitive models of \(T\). The \(\Delta_1\) trick.
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Vorlesung III: 23. April 2026. Sed 19 107.
Absolute operations. Closure properties of absoluteness: quantification over absolute operations, concatenation, transfinite recursion. Examples of absolute formulas and operations: basic operations, cartesian products, functions, ordinals, \(\omega\), arithmetical functions.
§7. Existence of transitive models. Arithmetical formulas. Consistency statements are arithmetical. Arithmetical formulas are absolute. The existence of a transitive model of \(\mathsf{ZFC}\) implies iterated consistency statements.
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Vorlesung IV: 30. April 2026. Zoom.
§8. Proving relative consistency with transitive models. Cohen's outer model results for transitive models. Existence of transitive models of finite fragments of \(\mathsf{ZFC}\) (without proof). Proof of the main result from theorems 1 & 2.
§9. Model theory of set theory. Tarski-Vaught test. Application 1: improved downwards Löwenheim-Skolem theorem. Elementary substructures. Application 2: construction of countable transitive models from transitive models. Mostowski collapse.
§10. The Lévy reflection theorem. Hierarchies. Examples. Lévy reflection theorem. Proof of theorem 2 from the Lévy reflection theorem. Proof of the Lévy reflection theorem.
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Vorlesung V: 7. Mai 2026. Zoom. Note. Lecture starts later and is only 45 minutes: 17:15–18:00!
§11. Forcing partial orders. Forcing partial orders. Conditions. Jerusalem convention. Compatibility. Antichains. Dense sets. Filters. Generic filters. Existence of generic filters for countably many dense sets. Example: \(\mathrm{Fn}(X,Y)\). Defining a surjection from a generic filter.
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14. Mai 2026. Ascension Day: no lecture!

Vorlesung VI: 21. Mai 2026. Zoom.
Countable transitive models (ctm). Generic objects produce new functions. \(\mathbb{P}\)-generic filters over \(M\).
§12. Names and the generic extension. Names. Examples. Values of names. Examples. The generic extension: countability and transitivity.
§13. Basic properties of the generic extension. Extensionality and foundation hold in transitive models. Canonical names: the ground model is included in the generic extension. The generic filter is an element of the generic extension. The generic extension satisfies the pairing axiom.
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Vorlesung VII: 28. Mai 2026. Zoom. Note. Lecture starts later and is only 45 minutes: 17:15–18:00!
Overview of the axioms of \(\mathsf{ZFC}\). The generic extension satisfies the infinity axiom. The generic extension satisfies the union axiom. Difficulties in proving separation. Informal idea for separation.
§14. \(\mathsf{ZFC}\) in the generic extension under the assumption of the forcing theorem. Formulation of the forcing theorem.
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4. Juni 2026: no lecture!

Vorlesung VIII: 11. Juni 2026. Zoom.
The forcing language. Proof of Separation in \(M[G]\). Proof of Powerset in \(M[G]\). Proof of Replacement in \(M[G]\). Remark about Choice in \(M[G]\). Remark about proving the generic model theorem in finite fragments of \(\mathsf{ZFC}\).
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Vorlesung IX: 18. Juni 2026. Zoom.
§15. Proof of the Forcing Theorem. Density below \(p\). Lemma about density below \(p\). Definition of the forcing relation \(\Vdash\). Proof of the Forcing Theorem: Case 4 (\(\neg\)); Case 2 (\(\in\)); Case 1 (\(=\)).
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Vorlesung X: 25. Juni 2026. Zoom.
§16. Generic functions. Application A: collapses. Being a cardinal is not upwards absolute. Application B: adding subsets.
§17. General remarks about changing cardinalities. Notations \(\aleph_1^M\), \(\aleph_2^M\) etc. Possible situations of cardinals in two models. Changing the value of \(\aleph_1\) and subsets of the natural numbers.
§18. General remarks about generic subsets. Is every new subset part of the generic?
§19. Warm-up: how to avoid collapsing? If \(\mathbb{P}\) has size less than \(\kappa\), it cannot collapse \(\kappa\). Potential values of a function.
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Vorlesung XI: 2. Juli 2026. Zoom.
§20. Chain conditions & preservation theorems. The \(\kappa\)-chain condition. The countable chain condition. Size bounds on the set of potential values. Cardinal preservation from chain condtions.
§21. The Delta system lemma. Delta systems. The Delta system lemma and its proof.
§22. The proof of Cohen's theorem. Countable chain condition of \(\mathrm{Fn}(X,2)\). Detailed proof of the relative consistency of \(\mathsf{ZFC}+\neg\mathsf{CH}\).
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Vorlesung XII: 9. Juli 2026. Sed 19 107.

Vorlesung XIII: 16. Juli 2026. Zoom.