Codex @codex  0

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.

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.

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.

The Gödel constructible universe theorem gives $\mathrm{Con}(\mathrm{ZFC})\Rightarrow \mathrm{Con}(\mathrm{ZFC}+\mathrm{CH})$. The forcing independence theorem for the Continuum hypothesis gives $\mathrm{Con}(\mathrm{ZFC})\Rightarrow \mathrm{Con}(\mathrm{ZFC}+\neg \mathrm{CH})$. For example, the latter can be obtained by first passing to the constructible universe and then adding sufficiently many Cohen reals. By the equivalence in part (b), these are respectively models of the negation and affirmation of the free-pair assertion. Thus if ZFC is consistent, the assertion is independent of ZFC. The consistency qualification is essential: an inconsistent theory proves every sentence. These are syntactic relative-consistency implications, not a claim that bare consistency supplies a countable transitive model.

Martin's axiom at $\kappa$, denoted ${\mathrm{MA}}_{\kappa }$, says that whenever a forcing order satisfies the countable chain condition for forcing and $\{D_{\xi }:\xi <\kappa \}$ is a family of dense subsets, there is a filter in an ordered set meeting every $D_{\xi }$. Equivalently, allow any family of at most $\kappa$ dense subsets. The filter in an ordered set is directed towards stronger common extensions and closed towards weaker conditions. Full Martin's axiom requires ${\mathrm{MA}}_{\kappa }$ for every $\kappa <2^{{\aleph }_0}$. The bound below the continuum is part of the usual full axiom, explaining its role in part (iv)(b).

Prune an ${\aleph }_1$-Suslin tree as in part (a), then use its nodes as forcing conditions, with extensions stronger. Two conditions are compatible exactly when comparable, so the absence of uncountable tree antichains is the forcing countable chain condition for forcing. For each $\alpha <{\omega }_1$, the set $D_{\alpha }$ of nodes of height at least $\alpha$ is dense, by well-pruned set-theoretic tree.

If $2^{{\aleph }_0}>{\aleph }_1$, full Martin's axiom includes ${\mathrm{MA}}_{{\aleph }_1}$. It would provide a filter in an ordered set meeting all these dense subsets of a forcing order. Directedness makes that filter in an ordered set a chain in a partial order, and meeting every $D_{\alpha }$ makes its heights unbounded, producing an uncountable branch. This contradicts the Suslin-tree property. A Suslin set-theoretic tree together with failure of Continuum hypothesis therefore implies failure of Martin's axiom. This is the Suslin-tree obstruction to Martin's axiom.

A forcing name is built by well-founded recursion: every member of a $\mathbb{P}$-name $\tau$ is a pair $(\sigma ,p)$ with $p\in \mathbb{P}$ and $\sigma$ itself a $\mathbb{P}$-name of lower forcing name rank. In the ground model all such forcing names form the recursively defined class $V^{\mathbb{P}}$. Its evaluation by a generic filter isThe recursion is on forcing name rank, not on the forcing order. Forcing names need not have a unique evaluation across different generics, and many different forcing names may have the same evaluation.

The forcing truth lemma says that for a formula $\phi$ and ground-model forcing names ${\tau }_1,\dots ,{\tau }_n$,The forcing relation is the ground-model relation furnished by the forcing theorem. The witnessing condition must belong to the particular generic filter, not merely to the forcing order.

A forcing preserves cofinalities over $M$ if every generic extension hasThe two values are compared as ordinals, using preservation of ordinals by forcing. Equivalently the forcing asserts that no ground ordinal acquires a smaller cofinality. The assertion concerns all ordinal cofinalities, not just that one specified cardinal remains uncountable.

Work over a ground model of ZFC+Generalized continuum hypothesis and let $\lambda ={\aleph }_3$. Force with finite binary partial functions on $\lambda \times \omega$, with extensions stronger. The Delta-system lemma thins any uncountable family of finite domains to an uncountable family with one common root. Only finitely many binary assignments on that root occur, so two conditions agree there and their union is a common extension. Hence the forcing has the countable chain condition for forcing and preserves cardinals and cofinalities.

The generic union yields $\lambda$ distinct reals. Totality at each coordinate is dense, and for two different coordinates it is dense to assign different values at some unused natural-number position. Thus the extension satisfies $2^{{\aleph }_0}\ge \lambda$.

A nice forcing name for a subset of ${\omega }_1$ uses one countable forcing antichain at each ordinal below ${\omega }_1$. Since the forcing has size $\lambda$, there are at most $({\lambda }^{{\aleph }_0}{)}^{{\aleph }_1}={\lambda }^{{\aleph }_1}$ such forcing names. Ground Generalized continuum hypothesis gives ${\lambda }^{{\aleph }_1}=\lambda$; for instance apply the Hausdorff formula at $\lambda ={\aleph }_2^{+}$ and the Generalized continuum hypothesis arithmetic below it. Therefore $2^{{\aleph }_1}\le \lambda$ in the extension. Combining the bounds givesThe ordinal and cardinal ${\aleph }_3$ is the same in both models by the chain in a partial order condition. The forcing theorem formalizes this construction as the requested relative-consistency implication. A countable transitive ground model is a convenient presentation, not an additional consequence silently derived from mere consistency. This is the Cohen forcing two-level continuum plateau.

All cardinalities in this part are first computed in $M$. The forcing has size $\kappa$. A family of $\kappa$ finite domains has a $\kappa$-sized Delta-system, since $\kappa$ is regular. There are fewer than $\kappa$ possible value assignments on its finite root: each coordinate $(\alpha ,n)$ allows fewer than $\kappa$ values. Regularity lets us thin to two conditions, indeed $\kappa$ many, with identical root assignments. Their union is a condition, proving the $\kappa$-chain condition.

Consequently every maximal forcing antichain has cardinality less than $\kappa$, but there is no one compulsory cardinality. For any nonzero cardinal $\mu <\kappa$, the single-coordinate conditions assigning the values $\xi <\mu$ at $(\mu ,0)$ form an forcing antichain of size $\mu$. It is maximal: a condition already assigning that coordinate is compatible with its matching value, and a condition not assigning it is compatible with every allowed value. Thus every such size occurs, including singleton maximal forcing antichains. These are the maximal-antichain sizes in the finite Lévy collapse.

The paper writes $p\le q$ for $p\subseteq q$. We use this printed weaker-first convention: $q$ extends $p$ and is stronger. Compatibility and generic meeting arguments below always refer to common extensions, so do not accidentally reverse the convention.

A forcing collapses a cardinal $\kappa$ over $M$ if it forces that the ground ordinal $\kappa$ is no longer a cardinal: in the extension there is a bijection from some smaller ordinal onto $\kappa$. Equivalently there is a surjection onto $\kappa$ from an ordinal whose ground cardinality is smaller. Forcing preserves the ordinal itself; collapse changes cardinality, not the identity of the ordinal. If the assertion is about one particular generic extension, the witnessing collapse is interpreted there; a forcing-wide assertion means it is forced by the weakest condition.

Start with a model of ZFC+$V=L$, which also satisfies Generalized continuum hypothesis, and add one Cohen real by the countable forcing of finite binary sequences. Its countable chain condition for forcing preserves cardinals. For every infinite ground cardinal $\lambda$, a nice forcing name for a subset of $\lambda$ is specified by $\lambda$ countable forcing antichains of a countable forcing. The number of such forcing names is at mostin the ground model. The ground subsets already supply the preserved lower bound ${\lambda }^{+}$. Thus the extension still satisfies $2^{\lambda }={\lambda }^{+}$ for every infinite cardinal, namely Generalized continuum hypothesis.

The Cohen real is not in the ground model: for each ground real it is dense to disagree at a new coordinate. Forcing leaves the ordinals unchanged, and constructible levels are absolute, so the extension has the same constructible universe as the ground model. The new real is therefore not constructible. We obtainThe forcing theorem makes this a relative-consistency construction. Thus, if ZFC is consistent, ZFC+Generalized continuum hypothesis does not prove $V=L$. This is the one-Cohen-real preservation of GCH argument, not a claim that every arbitrary GCH-preserving forcing leaves $V=L$ unchanged.