Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 128 (Forcing and the Continuum Hypothesis)

The Lévy reflection theorem says that for every finite collection $\Phi$ of first-order formulas there are arbitrarily large ordinals $\alpha$ such that, for every $\phi \in \Phi$ and every tuple of parameters $a_1,\dots ,a_n\in V_{\alpha }$,Equivalently, the ordinals simultaneously reflecting all formulas in $\Phi$ form a closed unbounded class.

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Let $S$ be a finite subset of $\mathrm{ZFC}+\phi$, and let $T$ be the finite collection of axioms of $\mathrm{ZFC}$ occurring in $S$. Choose the finite fragment $T^{\ast }$ supplied by the hypothesis. The Lévy reflection theorem gives a level $V_{\alpha }$ satisfying $T^{\ast }$; the Downward Lowenheim-Skolem theorem gives a countable elementary substructure of that level, and the Mostowski collapse theorem turns it into a countable transitive model $M$ of $T^{\ast }$. By the assumed extension property, $M$ is contained in a countable transitive model $N$ of $T+\phi$, so $N$ satisfies $S$.

This argument is formalizable over $\mathrm{ZFC}$ for each finite $T$. Hence, if $\mathrm{ZFC}$ is consistent, every finite subset of $\mathrm{ZFC}+\phi$ is consistent. The compactness theorem now gives

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Take $\lambda =\omega \cdot 2$. This is a limit ordinal greater than $\omega$. There is a recursive well-order code $R$ for ${\omega }^2$: for example, use a recursive pairing of the natural numbers with $\omega \times \omega$ and the lexicographic order consisting of $\omega$ successive blocks of order type $\omega$. Since $R$ is definable over $L_{\omega }$, it belongs to $L_{\omega +1}$ and hence to $L_{\omega \cdot 2}$.

The representation of $R$ is ${\omega }^2$, butand the ordinals belonging to $L_{\lambda }$ are exactly those below $\lambda$. Thus $R\in L_{\lambda }$ while its representation is not in $L_{\lambda }$, violating the second requirement for a coding level of the constructible hierarchy.

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Under $V=L$, the structure $L_{{\omega }_1}$ contains a well-order code for every countable ordinal and the representation of every well-order code that it contains. Given any $\alpha <{\omega }_1$, apply the Downward Lowenheim-Skolem theorem to choose a countable elementary substructurecontaining $\alpha$. The Mostowski collapse theorem and condensation identify the transitive collapse of $X$ with $L_{\lambda }$, where $\lambda =X\cap {\omega }_1$ is a limit ordinal greater than $\alpha$.

Elementarity now verifies both coding properties. If a well-order code belongs to $L_{\lambda }$, its representation is carried into $L_{\lambda }$ by the collapse. Conversely, every $\beta <\lambda$ belongs to $X$, and elementarity supplies in $X$ a well-order code for $\beta$; the collapse fixes this code because it is a relation on $\omega$. Hence $L_{\lambda }$ is a coding level of the constructible hierarchy. Such $\lambda$ occur unboundedly below ${\omega }_1$, so $\Gamma$ has at least ${\aleph }_1$ elements; since $\Gamma \subseteq {\omega }_1$, it has exactly

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The two conventions reverse the order relation. In standard notation for forcing, $q\le p$ means that $q$ is stronger than $p$. Thus $p$ and $q$ are incompatible forcing conditions when there is no $r$ with $r\le p$ and $r\le q$, while $D\subseteq \mathbb{P}$ is a dense set whenIn Jerusalem notation for forcing, $p\le q$ means that $q$ is stronger than $p$. Incompatibility therefore means that there is no $r$ with $p\le r$ and $q\le r$, and density means

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Let $a,p\in M[G]$, and suppose a first-order formula $\psi (x,y,p)$ defines exactly one $y$ for every $x\in a$. Apply the Lévy reflection theorem to the formulas needed to express this assertion, choosing an ordinal $\alpha$ with $a,p\in V_{\alpha }^{M[G]}$ such thatfor every $x\in a$. Therefore every required value lies in the set $V_{\alpha }^{M[G]}$.

The already established axiom schema of separation forms the setFunctionality makes $b$ exactly the range of the definable function on $a$. This proves every instance of the Axiom schema of replacement in the generic extension $M[G]$.

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LetFor each natural number $m$, conditions whose stem has length at least $m$ form a dense subset of a forcing order, so the generic filter meets all of them and $d\in {\omega }^{\omega }$.

Fix $h\in {\omega }^{\omega }\cap M$. The setis dense: from $(s,f)$ replace $f$ by $n\mapsto \max \{f(n),h(n)+1\}$. Choose $(s,f)\in G\cap D_h$. Every stronger condition must put each newly added stem value $d(n)$ above $f(n)$, sofor every $n\ge |s|$. Thus $d$ is a dominating real over $M$.

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Conditions with the same finite stem in Hechler forcing are compatible: $(s,f)$ and $(s,g)$ have the common stronger condition $(s,\max \{f,g\})$. Since there are only countably many finite stems, Hechler forcing is sigma-centered and hence has the countable chain condition for forcing. It therefore preserves ${\aleph }_1^M$.

In $M$, the Continuum hypothesis givesEvery real in $M[G]$ has a nice name for a real, and the countable chain condition bounds the number of such names bywhere the last equality uses the ground-model continuum hypothesis. The extension still contains all ground-model reals, already ${\aleph }_1$ many, soThus forcing once with $\mathbb{D}$ preserves the continuum hypothesis.

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