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Forcing & the Continuum Hypothesis |
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Forcing & the Continuum Hypothesis |
Lecturer. Benedikt Löwe
Lectures. Tuesday, Thursday 10-11.
Examples Classes. Lyra Gardiner.
| L E C T U R E S. | |||
| Week 1. | Lecture I.
Tuesday, 14 October 2025. Hilbert's problems. The continuum
problem. Cantor's theorem and the \(\beth\) hierarchy. Hartogs's theorem
and the \(\aleph\) hierarchy. The continuum hypothesis and the
generalised continuum hypothesis. Gödel's 1938 result and Cohen's 1962
result. Relative consistency proofs. Analogy from algebra: adding square
roots to fields. Substructures, absoluteness, substructure lemma. The
language of set theory: being empty is not quantifier-free. Being empty
is not absolute for substructures. Lecture Notes. |
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| Week 2. | Lecture
II. Thursday, 16 October 2025. The importance of
non-absoluteness for independence proofs. Transitive models. Transitive
sets. Absoluteness; upwards absoluteness; downwards absoluteness.
Transitive sets satisfy extensionality and foundation. Closure under
bounded quantification. \(\Delta_0\), \(\Sigma_1\), and \(\Pi_1\), also
relative to a theory \(T\). \(\Delta_0^T\) formulas are absolute for
transitive models of \(T\). Lecture Notes. |
Lecture
III. Saturday, 18 October 2025. \(\Sigma_1\) and
\(\Pi_1\) formulas are upwards and downwards absolute, respectively.
\(\Delta_1\) formulas and their absoluteness. Absolute operations:
concatenation and quantifiers bounded by absolute operations.
Preservation of absoluteness under transfinite recursion. Examples of
absolute formulas and operations: basic set theoretic operations,
functions, ordinals, \(\omega\), and arithmetical facts. Lecture Notes. |
Lecture
IV. Tuesday, 21 October 2025. Arithmetical formulas
and their absoluteness. Consistency statements are absolute for
transitive models. \(T^* := T+\mathrm{Cons}(T)\). The existence of a
transitive model of \(\mathsf{ZFC}\) implies \(\mathsf{ZFC}^{(n)}\) for
all natural numbers \(n\). Existence of transitive models of finite
fragments of \(\mathsf{ZFC}\). Hierarchies. Lévy Reflection Theorem. The
Tarski-Vaught-Test. Löwenheim-Skolem in general does not give transitive
models. Lecture Notes. |
| Week 3. | Lecture V.
Thursday, 23 October 2025. Proof of Löwenheim-Skolem via
Tarski-Vaught-Test and analysis of the resulting model. Mostowski
Collapsing Theorem. Every finite \(T \subseteq \mathrm{ZFC}\) has a
countable transitive model. Non-absoluteness of countability and being a
cardinal for transitive models. Gödel's method of inner models:
construction of the minimal inner model; issues with guaranteeing the
truth of axioms via adding witnesses. Lecture Notes. |
Lecture
VI. Tuesday, 28 October 2025. Sufficiently strong
theories. Absoluteness of the truth in a model relation. (Locally)
definable subsets. The definable power set. The constructible hierarchy.
Properties and absoluteness. Minimality properties of the constructible
hierarchy. Axioms of \(\mathsf{ZFC}\). The structural axioms in the
constructible universe. Lecture Notes. |
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| Week 4. | Lecture
VII. Thursday, 30 October 2025. Proof of
\(\mathsf{ZF}\) in \(\mathbf{L}\): Pairing, Union, Powerset, Separation.
(Cf. Example Sheet #1 for Replacement.) Remarks on the Axiom of Choice.
The Axiom of Constructibility. Gödel's Condensation Lemma. Lecture Notes. |
Lecture
VIII. Tuesday, 4 November 2025. Size of
\(\mathbf{L}_\alpha\). Proof of \(\mathsf{CH}\) in \(\mathbf{L}\).
Remark about \(\mathsf{GCH}\) in \(\mathbf{L}\). Cohen's result for
countable transitive models of finite fragments of \(\mathsf{ZFC}\).
Deriving the relative consistency result from Cohen's theorem. Adding a
set to a countable transitive model of \(\mathsf{ZFC}\). Lecture Notes. |
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| Week 5. | Lecture
IX. Tuesday, 11 November 2025. Forcing partial
orders. Conditions. Interpretation: Jerusalem convention. Compatibility.
Antichains. Dense sets. Filters. Genericity. All countable collections
of dense sets have a generic filter. Example: finite partial functions.
Names. Examples. Interpretations of names. Examples. The extension of a
model. Lecture Notes. |
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| Week 6. | Lecture X.
Thursday, 13 November 2025. Convention on countable transitive
models (ctm). Generic filters over a ctm: existence and objects defined
from them, e.g., collapsing \(\aleph_1\). The (generic) extension of a
model: countability, transitivity. Canonical names: \(M\subseteq M[F]\)
and \(F\in M[F]\). Structural \(\mathsf{ZFC}\) axioms. Pairing and
Union. Discussion of Separation. Lecture Notes. |
Lecture
XI. Saturday, 15 November 2025. The forcing
language. The Forcing Theorem (statement). The Forcing Theorem implies
Separation, Powerset, Replacement, and Choice in \(M[G]\). Being dense
below \(p\). Properties of being dense below \(p\). Lecture Notes. |
Lecture
XII. Tuesday, 18 November 2025. Proof of the Forcing
Theorem: definition of the forcing relation; roadmap of the proof; proof
for the \(\neg\) case; proof for the \(\in\) case; first half of the
proof of the \(\subseteq\) case. Lecture Notes. |
| Week 7. | Lecture
XIII. Thursday, 20 November 2025. Proof of the
Forcing Theorem: second half of the \(\subseteq\) case. The main example
\(\mathrm{Fn}(X,Y)\). Applications: collapsing cardinals;
\(\neg\mathsf{CH}\), and adding \(\aleph_2^M\) many subsets of
\(\omega\). Lecture Notes. |
Lecture
XIV. Tuesday, 25 November 2025. Warm-up:
\(\omega_1\) is preserved by countable forcing. Chain conditions.
\(\kappa\) is preserved by \(\kappa\)-chain condition forcing. Potential
values for functions generated by \(\kappa\)-c.c. forcing. Preservation
of cardinals by \(\kappa\)-c.c. forcing. \(\Delta\) systems. The
\(\Delta\) System Lemma. \(\mathrm{Fn}(X,2)\) has the c.c.c. Lecture Notes. |
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| Week 8. | Lecture
XV. Thursday, 27 November 2025. Summary: proof of
Cohen's theorem proving the consistency of
\(\mathsf{ZFC}+\neg\mathsf{CH}\). Possible values for the continuum. The
method of nice names: nice names, the Nice Name Theorem, counting nice
names. Consistency of \(2^{\aleph_0} = \aleph_2\) and in general
\(2^{\aleph_0} = \aleph_n\) for \(n\neq 0\). Lecture Notes. |
Lecture
XVI. Tuesday, 2 December 2025. Consistency of
\(\mathsf{CH}+\neg\mathsf{GCH}\). Forcing with
\(\mathrm{Fn}(\omega_3^M\times\omega_1^M,2)\) adds lots of subsets of
\(\omega\). \(\mathrm{Fn}(X,Y,\lambda)\) and its chain condition.
Closure of forcings. \(\mathrm{Fn}(X,2,\aleph_1)\) is
\(\aleph_1\)-closed. Closed forcings do not add new functions.
\(\mathrm{Fn}(X,2,\aleph_1)\) preserves \(\mathsf{CH}\) and
\(\aleph_1\). Remark about controlling \(2^{\aleph_0}\) and
\(2^{\aleph_1}\) independently by forcing iterations. Lecture Notes. |
Example Sheets & Examples Classes. |
Examples Class #1. Monday 3 November 2025, 3:30–5:30pm,
MR3.
Example Sheet #1: pdf file.
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