Prof Dr Benedikt Loewe: Forcing & the Continuum Hypothesis (L25)

Forcing & the Continuum Hypothesis
Part III of the Mathematical Tripos
Lent Term 2025

Lectures. Tuesday, Saturday 12-13.

Examples Classes. Lyra Gardiner.

Overview. The method of forcing is one of the most important model constructions in set theory and its versatility is the reason for the plethora of independence results in set theory. It was developed to solve one of the most important foundational problems of the 20th century: determining the cardinality of the set of real numbers. Cantor's Continuum Hypothesis asserts that the cardinality of the set of real numbers has the smallest possible value:

\(\mathsf{CH}\): "Every infinite set of reals is either equinumerous with the set of natural numbers or equinumerous with the set of all real numbers. Equivalently, \(2^{\aleph_0} = \aleph_1\)."

When he presented his list of twenty-three problems for the 20th century at the International Congress of Mathematicians in Paris in 1900, David Hilbert listed the the question whether the Continuum Hypothesis is true as the very first problem on his list. It turned out that this question cannot be solved on the basis of \(\mathsf{ZFC}\): in 1938, Kurt Gödel showed that \(\mathsf{CH}\) cannot be disproved in \(\mathsf{ZFC}\); in 1963, Paul Cohen invented the method of forcing to show that \(\mathsf{CH}\) cannot be proved in \(\mathsf{ZFC}\).

In this course we shall study the basics of the method of forcing in order to present Cohen's proof. The course will discuss:

models of set theory (in particular, transitive models, the Lévy reflection theorem, and the theory of absoluteness);
the construction of the generic extension of a countable transitive model (in particular, the syntactic and semantic forcing relation, the forcing theorem, and the generic model theorem);
showing relative consistency proofs with the method of forcing;
applications to cardinal arithmetic (in particular, preservation results and the consistency of \(\neg\mathsf{CH}\) and related results).

Prerequisites The Part II course Logic & Set Theory or an equivalent course is essential. This course uses some content from the Part III course Model Theory (Lent 2025). It is recommended (but not strictly necessary) to take Model Theory to supplement the discussion of models of set theory in this course.

Literature Kenneth Kunen. Set Theory. An Introduction to Independence Proofs.
Elsevier 1980 [Studies in Logic and the Foundations of Mathematics, Vol. 102].

L E C T U R E S.
Week 1.	Lecture I. Saturday, 25 January 2025. Hilbert's problems: the continuum problem. Gödel and Cohen: unsolvability of the continuum problem in \(\mathsf{ZFC}\). Relative consistency proofs. Analogy: relative consistency proofs in algebra; consistency of the existence of the square root of \(2\). Absoluteness, upwards absoluteness, downwards absoluteness. Absoluteness of atomic formulae for substructures (without proof). The language of set theory and its lack of atomic formulae. Non-absoluteness of the formula "\(x\) is empty" for structures of the language of set theory. Transitive models of set theory. Lecture Notes.	Lecture II. Tuesday, 28 January 2025. Transitive sets recognise the empty set, satisfy extensionality and foundation. Bounded quantifiers. Closure under bounded quantification. \(\Delta_0\) formulas and \(\Delta_0^T\) formulas. \(\Delta_0\) formulas are absolute for transitive models. \(\Sigma_1\) and \(\Pi_1\) formulas. \(\Sigma_1\) formulas are upwards absolute for transitive models and \(\Pi_1\) formulas are downwards absolute for transitive models. Examples of \(\Delta_0\) formulas. Absolute operations. Closure of absoluteness under composition. More examples of \(\Delta_0\) formulas. Lecture Notes.
Week 2.	Lecture III. Saturday, 30 January 2025. Absoluteness of ordinals. The set of ordinals in a transitive set model is an ordinal. Being a cardinal is \(\Pi_1\). Non-absoluteness of being a cardinal: countable transitive models. Existence of transitive models of set theory: \(\mathsf{ZFC}+\mathrm{Con}(\mathsf{ZFC})\) does not prove the existence of transitive models of set theory. The Lévy Reflection Theorem (statement). The Tarski-Vaught test. Generalisation of the Tarski-Vaught test to sets of formulas closed under subformulas. Lecture Notes.	Lecture IV. Tuesday, 3 February 2025. Warm-up: proof of the existence of countable elementary substructures using the Tarski-Vaught test. That construction does not in general produce transitive models. Proof of the Lévy Reflection Theorem using the Tarski-Vaught test. Every finite fragment of \(\mathsf{ZFC}\) is true in some transitive model. Mostowski collapse. Construction of a countable transitive model for every finite fragment of \(\mathsf{ZFC}\). Absoluteness is preserved under transfinite recursion: recapitulation of the proof of the reflection theorem. Lecture Notes.
Week 3.	Lecture V. Saturday, 7 February 2025. Proof of preservation of absoluteness under transfinite recursion. Absoluteness of \(\Delta_1\) concepts and closure under quantification bounded by absolute operations. Applications: set of formulae and definition of semantics are absolute. Definability over a structure. The definable power set \(\mathcal{D}(X)\). Basic properties. The constructible hierarchy. Basic properties and cardinality of the levels of the constructible hierarchy. Differences between the von Neumann and the constructible hierarchy. The axiom of constructibility. Lecture Notes.	Lecture VI. Tuesday, 11 February 2025. Axioms of \(\mathsf{ZF}\): structural axioms and functional axioms. The structural axioms hold in transitive models containing \(\omega\). Pairing and union hold in \(\mathbf{L}\). Powerset holds in \(\mathbf{L}\): the rank of the powerset of \(x\) is difficult to determine. Separation holds in \(\mathbf{L}\). Remarks on the axiom of choice: sketch of the construction of the global choice ordering on \(\mathbf{L}\). Lecture Notes.
Week 4.	Lecture VII. Saturday, 15 February 2025. Relevance of bounding the constructible rank of relevant objects. Combination of Löwenheim-Skolem and Mostowski: Löwenheim-Skolem-Mostowski. The condensation sentence. The condensation characterisation. Gödel's condensation lemma. The constructible universe satisfies \(\mathsf{GCH}\). Extending countable transitive models by witnesses: possibility of accidentally adding other objects that affect the cardinal structure. Lecture Notes.	Lecture VIII. Tuesday, 18 February 2025. Construction of a countable transitive model extending \(M\) and containing a given countable set using the Löwenheim-Skolem-Mostowski method. Partial orders, incompatibility, antichains, dense sets, filters, genericity. Jerusalem notation. Existence of generic filters for countably many dense sets. Example 0. \(\mathrm{Fn}(X,Y)\): generic filter produces a function from \(X\) to \(Y\). Example 1. Cohen forcing \(\mathrm{Fn}(\omega,2)\): generic filter produces a new Cohen real. Example 2. Collapse forcing \(\mathrm{Fn}(\omega,\alpha)\): generic filter produces a surjection from \(\omega\) to \(\alpha\) and makes \(\alpha\) countable. Example 3. \(\mathrm{Fn}(X\times Y,2)\): generic filter produces an injection from \(Y\) into \(\wp(X)\). Discussion of the existence of generic filters over countable transitive models \(M\). Lecture Notes.
Week 5.	Lecture IX. Saturday, 22 February 2025. \(\mathbb{P}\)-generics over a countable transitive model \(M\). \(\mathbb{P}\)-names. Absoluteness of \(\mathbb{P}\)-names. Examples. Interpretation of \(\mathbb{P}\)-names. Absoluteness of the interpretation of \(\mathbb{P}\)-names. Examples. The generic extension \(M[F]\). Transitivity of \(M[F]\). Canonical names. A name for the generic filter. Minimality of the generic extension. Name for the unordered pair. Pairing in \(M[F]\). Name for the union. Union in \(M[F]\). Lecture Notes.	Lecture X. Tuesday, 25 February 2025. Height of the generic extension. Technical problems with proving powerset directly. Proof of the axiom of choice in the generic extension. The forcing language. The semantic forcing predicate. The forcing theorem and the definability theorem (without proofs). Preview of the proof of the forcing theorem. Consequences of the forcing theorem. Proof of Separation in the generic extension. Lecture Notes.
Week 6.	Lecture XI. Saturday, 1 March 2025. Proof of Replacement in the generic extension. Density below \(p\). Lemma about density below \(p\) (Example Sheet #3). Definition of the syntactic forcing relation. Lemma about the syntactic forcing relation. Proof strategy: deriving the definability theorem and the forcing theorem from the syntactic forcing theorem. Proof of the syntactic forcing theorem: induction on formula complexity (proof for \(\neg\); the other two cases on Examples Sheet #3). Lecture Notes.	Lecture XII. Tuesday, 4 March 2025. Proof of the syntactic forcing theorem: \(\in\) and \(=\). Summary of results obtained so far. Lecture Notes.
Week 7.	Lecture XIII. Saturday, 8 March 2025. Meta-mathematical discussion: version of the generic model theorem for finite fragments of \(\mathsf{ZFC}\). Preserving cardinals. The countable chain condition (c.c.c.). Forcings with the c.c.c. preserve cardinals. Main lemma: for every function, there is a countable set of potential values. Proof of the fact that c.c.c. forcings preserve cardinals. If \(Y\) is countable, \(\mathrm{Fn}(X,Y)\) has the c.c.c. Consequences: consistency of \(2^{\aleph_0} \geq \aleph_2\). Question: what is the precise value of the continuum in the generic extension? Lecture Notes.	Lecture XIV. Tuesday, 11 March 2025. Using \(\mathrm{Fn}(\omega\times\aleph_\alpha^M,2)\) to obtain better lower bounds for \(2^{\aleph_0}\). Lower bounds depend on the value of \(2^{\aleph_0}\) in \(M\). Which cardinals can be \(2^{\aleph_0}\)? Kőnig's lemma: \(2^{\aleph_0} \neq \aleph_\omega\). Cofinal sets and cofinality. The cofinality of \(2^{\aleph_0}\) cannot be \(\aleph_0\). Nice names. Counting nice names. Upper bounds for \(2^{\aleph_0}\). Consistency of \(2^{\aleph_0} = \aleph_2\). Consistency of \(2^{\aleph_0} = \aleph_n\) for \(n\geq 1\). The value of \(2^{\aleph_0}\) after forcing with \(\mathrm{Fn}(\omega\times\aleph_\omega^M,2)\). Consistency of \(2^{\aleph_0} = \aleph_{\omega_1}\). Lecture Notes.
Week 8.	Lecture XV. Saturday, 15 March 2025. Generalised nice names technique: \(\lambda\)-names and upper bound for \(2^\lambda\). Calculation of \(2^{\aleph_1}\) in the model of adding \(\aleph_2\) Cohen reals. The power set axiom: discussion of its naturality. Changing \(2^{\aleph_1}\) with \(\mathrm{Fn}(\aleph_1\times\aleph_3,2)\) always also adds \(\aleph_3\) many reals. Forcing with infinite conditions: \(\mathrm{Fn}(X,Y,\lambda)\). Chain condition. \(\mathsf{CH}\) implies that \(\mathrm{Fn}(\aleph_1\times\aleph_3,2,\aleph_1)\) has the \(\aleph_2\)-c.c. Closure and the closure lemma (no proof yet). \(\mathrm{Fn}(\aleph_1\times\aleph_3,2,\aleph_1)\) is \(\aleph_1\)-closed and therefore does not add reals and preserves \(\aleph_1\).´Consistency of \(\mathsf{CH}+2^{\aleph_1}=\aleph_3\). Lecture Notes.	Lecture XVI. Tuesday, 18 March 2025. Proof of the closure lemma. \(\mathrm{Fn}(\lambda^+\times\kappa,2,\lambda^+)\) always adds a surjection from \(\lambda^+\) to \(2^\lambda\cap M\). If \(2^{\aleph_0} = \aleph_2\), then forcing with \(\mathrm{Fn}(\aleph_1\times\aleph_3,2,\aleph_1)\) collapses \(\aleph_2\). Iterations: two-step iteration. Consistency of \(2^{\aleph_0} = \aleph_2+2^{\aleph_1}=\aleph_3\). The order of iteration matters: if the forcings are interchanged, then the resulting model is a model of \(\mathsf{CH}\). Remarks on forcing \(\mathsf{CH}\) by collapsing \(\aleph_1\) in a model of \(2^{\aleph_0} = \aleph_2\). Lecture Notes.
Example Sheets & Examples Classes.	Example Sheet #1. pdf file. Solutions provided by Ioannis Eleftheriadis: pdf file. Example Sheet #2. pdf file. Solutions provided by Ioannis Eleftheriadis: pdf file. Example Sheet #3. pdf file. Solutions provided by Ioannis Eleftheriadis: pdf file.