Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 19 / 6

The forcing theorem has two central clauses. Definability: for each first-order formula $\phi$, the relation $p⊩\phi (\vec{\tau })$ on conditions and ground-model names is first-order definable in $M$. Truth lemma: for every $M$-generic $G$ and every tuple of names in $M$,Forcing is monotone under strengthening and agrees with the generic-extension semantics. Together with the generic model construction, $M[G]$ is a transitive model of ZFC containing $M$ with the same ordinals. The definability and truth clauses are the auxiliary parts used when proving individual extension axioms.

The canonical forcing name $\dot{a}=\check{a}$ is defined recursively byIf a weakest condition $1_P$ is provided, one may instead pair each $\check{b}$ only with $1_P$. The all-conditions version works for an arbitrary nonempty forcing without that extra convention. Induction on rank gives ${\check{a}}^G=a$ for every generic filter, since each element $b\in a$ is activated by some condition of the nonempty $G$. Thus all ground-model objects have canonical names.

A precise form of the generalized delta-system lemma is this. Let $\mu <\theta$ be infinite cardinals, with $\theta$ regular and uncountable, and suppose $|[\alpha {]}^{<\mu }|<\theta$ for every $\alpha <\theta$. Every family $\mathcal{A}$ of $\theta$ distinct sets, each of cardinality less than $\mu$, contains a subfamily $\mathcal{B}$ of size $\theta$ and a set $r$ such thatThe members form a delta-system with root $r$. A sufficient usual arithmetic hypothesis is ${\alpha }^{\mu }<\theta$ for all infinite cardinals $\alpha <\theta$. The finite-set case $\mu =\omega$ needs only regular uncountable $\theta$, since finite subsets of each $\alpha <\theta$ form a family of size less than $\theta$. This is the form used for the finite-support collapse below.

Choose a name $\dot{f}\in M$ for $f$ and a condition $p\in G$ forcing that it is a function $\alpha \to \beta$. The ordinals $\alpha ,\beta$ belong to $M$, by forcing preserves ordinals. For each $\zeta <\alpha$, choose in $M$ a maximal forcing antichain $A_{\zeta }$ in the cone above $p$, every member deciding $\dot{f}(\zeta )$ as an ordinal below $\beta$. Conditions deciding an ordinal-valued name are dense, by the forcing theorem.

Let $y(\zeta )$ be the set of values decided by members of $A_{\zeta }$. The chain condition gives $|y(\zeta ){|}^M<\kappa$, and Choice and Replacement in $M$ assemble all these sets into a function $y\in M$ with domain $\alpha$. Since $p\in G$, genericity ensures that $G$ meets each deciding forcing antichain above $p$; equivalently enlarge it to a global maximal forcing antichain by conditions incompatible with $p$. Its chosen value is the actual $f(\zeta )$. ThereforeThis is the possible-values lemma for chain-condition forcing. All size bounds in its construction are internal to the ground model.

Suppose a ground cardinal $\lambda \ge \kappa$ became smaller in the extension. There would be a surjection $f:\alpha \to \lambda$ for some ordinal $\alpha <\lambda$. A finite domain cannot map onto an infinite ordinal, so use part (a) to cover $f$ by a ground function $y$ of small value sets. Its range is contained in $U={\bigcup }_{\zeta <\alpha }y(\zeta )$.

If $\lambda >\kappa$, the ground cardinality of $U$ is at most $|\alpha {|}^M\cdot \kappa <\lambda$. If $\lambda =\kappa$, regularity bounds a union of fewer than $\kappa$ sets of size less than $\kappa$ by a cardinal less than $\kappa$. In either case $U$ is a proper ground subset of $\lambda$, and the same missing ordinal remains missing in the extension, contradicting surjectivity.

Hence all ground cardinalities at least $\kappa$ are preserved. Ground bijections also remain bijections, so old noncardinals cannot turn into cardinals. The assertion does not prevent collapse of smaller cardinals, and consequently does not by itself preserve the aleph index of $\kappa$.

Let $p\in G$ force that $\dot{D}$ is a diamond sequence. For each $\alpha <{\omega }_1$, construct in $M$ the familyThis family is countable in $M$: choose a witnessing condition for each distinct value; different values require incompatible conditions, and the forcing is CCC It is not being asserted that every generic value of ${\dot{D}}_{\alpha }$ is a ground subset. Only values forced equal to a ground subset enter $E_{\alpha }$.

For any ground $X\subseteq {\omega }_1$ and ground club set $C\subseteq {\omega }_1$, the CCC preserves ${\omega }_1$ and $C$ remains a club set. Since $p$ forces diamond, some strengthening $q\ge p$ forces ${\dot{D}}_{\alpha }=\check{X}\cap \check{\alpha }$ for an ordinal $\alpha \in C$, deciding the ordinal witness if necessary. Thus $X\cap \alpha \in E_{\alpha }$. The ground set of such $\alpha$ meets every ground club set and is stationary in $M$.

Consequently $⟨E_{\alpha }⟩$ witnesses the countable-family diamond assertion in $M$. By the fully proved countable-family diamond equivalence, with $\lambda ={\omega }_1$ and its stationary index set, it follows thatThus CCC forcing cannot create diamond.

A diamond sequence on a regular uncountable $\lambda$ implies $2^{{\aleph }_0}\le \lambda$. For every $x\subseteq \omega$, regard $x$ as a subset of $\lambda$. A correct guess above $\omega$ is exactly $x$, so the sequence's at most $\lambda$ guesses include every real.

Force with $\mathrm{Add}(\omega ,{\lambda }^{+})$ using finite conditions. The delta-system lemma and agreement on the finite root prove the countable chain condition. It preserves cardinals and the regularity of $\lambda$, while adding at least ${\lambda }^{+}$ distinct reals. The reals on distinct coordinates differ on a dense set of conditions, so their number is indeed at least this ground cardinal. In the extension,This gives the requested relative consistency over a model with the indicated regular uncountable cardinal, with the usual forcing-theoretic consistency interpretation.

Work internally in the ground model and write $P=\mathrm{Levy}({\aleph }_0,\lambda )$. Conditions are finite partial functions with the coordinate-wise value bounds in the PDF, ordered by inclusion, so compatible conditions agree on their common domain and their union is a common strengthening.

Suppose there were $\lambda$ many pairwise incompatible conditions. Each fixed finite domain carries fewer than $\lambda$ possible functions: it uses finitely many ordinals below $\lambda$, and each coordinate has fewer than $\lambda$ choices. Regularity therefore lets us select $\lambda$ distinct domains. The finite-set form of the generalized delta-system lemma gives a subfamily of size $\lambda$ with common intersection $r$. There are fewer than $\lambda$ possible value assignments to the finite root $r$. Regularity gives a further subfamily of size $\lambda$ agreeing on the root. Any two of its conditions have a union in $P$, contradicting incompatibility. Thus $P$ has the $\lambda$-chain condition. Strong inaccessibility is more than is needed for this finite-support argument; regular uncountability suffices.

For each $0<\alpha <\lambda$ and $n<\omega$, requiring $(n,\alpha )$ to be in the domain is dense. For each $\xi <\alpha$, requiring $\xi$ to occur as a value at some $(n,\alpha )$ is also dense: choose a fresh $n$ and extend the condition. Hence the generic union defines, for each such $\alpha$, a surjection $g_{\alpha }:\omega \to \alpha$. Every ground ordinal below $\lambda$ is therefore countable in $M[G]$.

By the chain condition and the cardinal-preservation argument, $\lambda$ remains an uncountable cardinal in the extension; the same small-value argument preserves its regularity as well. All ordinals are unchanged. Since every smaller ordinal is countable and $\lambda$ is not,This is the finite Lévy collapse to omega-one. The dense-set construction proves the collapse below $\lambda$, while the chain condition is what prevents collapsing $\lambda$ itself.