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If the ground model has axiom of choice, well-order the set of subnames appearing in a forcing name for an arbitrary extension set. Evaluate those activated by the generic filter. Each element of the extension set has a least preimage index; ordering by these indices well-orders the set. Only ground choice and the ZF part of the forcing theorem are needed.

For the usual recursive syntactic forcing relation, $p{⊩}^{\ast }\exists x\phi (x)$ means that conditions $q\le p$ forcing $\phi (\sigma )$ for some forcing name $\sigma$ are dense below a forcing condition $p$. This density clause proves the existential step of the forcing theorem from the truth lemma for each named instance. A generic filter containing $p$ meets that dense set after it is augmented by conditions incompatible with $p$.

Forcing constructs new first-order models while controlling which ground-model properties survive. Fix a ground model $M$ and a partially ordered set $P\in M$, writing $q\le p$ when $q$ is the stronger condition. A generic filter is upward closed, directed towards stronger conditions, and meets every ground-model dense subset of a forcing order. If $M$ is externally a countable transitive model, enumerate its dense subsets and successively choose stronger conditions in them. The resulting filter is $M$-generic. External countability is crucial: this is not a claim that the ground model regards its own universe as countable.

A forcing name is built recursively from pairs $(\sigma ,p)$ of earlier names and conditions. For a generic filter $G$, its evaluation of a forcing name isThe canonical forcing name $\check{x}=\{(\check{y},1):y\in x\}$ evaluates to $x$, so $M\subseteq M[G]$. Recursive evaluation also proves transitivity. Name ranks bound the ranks of their evaluations, so an ordinal in the extension is below some ground ordinal and is already a ground ordinal. Thus forcing preserves ordinals.

The central link between syntax and generic interpretation is the forcing theorem. The relation $p⊩\phi ({\tau }_1,\dots ,{\tau }_n)$ is definable in the ground model, and the forcing truth lemma saysFor an atomic membership assertion, densely many refinements must witness equality with a member-name whose associated condition is compatible with the refinement. Equality is obtained by recursive mutual inclusion of names. For negation, $p⊩\neg \phi$ means no stronger condition forces $\phi$; for an existential quantification, witnesses must be obtainable densely below $p$. Induction on name rank establishes the atomic clauses, and induction on formulas establishes definability and the truth lemma. The generic filter meets the relevant dense sets at the existential step. An arbitrary semantic assertion about one chosen generic filter is insufficient: the definable relation must work uniformly for all generic interpretations.

This explains preservation of ZFC. Extensionality and foundation follow from transitivity; canonical names supply infinity. Pair and union names give pairing and union. For separation, restrict a name's members to conditions forcing the defining formula. For replacement, the definable forcing relation lets ground axiom schema of collection bound a set of names for all possible witnesses to the unique outputs; evaluating their collected name gives the range. For power set, let $\tau$ name the given set and let $D$ be the set of its member-names. Every subset of ${\tau }_G$ has an equivalent name drawn from $\mathcal{P}(D\times P)$, by the truth lemma, so a ground set of candidate names collects all its subsets. Finally choose a ground well-order of the relevant names; its evaluated image surjects onto the desired set, and choosing the first name for each value well-orders that set. This is why ordinary forcing over a choice model preserves choice.

For a concrete example, Cohen forcing consists of finite partial functions $\omega \to 2$, ordered by extension. The generic union $c=\bigcup G$ is total because deciding each coordinate is dense. For every ground real $x$, the set of conditions disagreeing with $x$ somewhere is dense, so $c\ne x$. This construction changes the universe by adding a genuinely new real without changing its ordinals.

The size and compatibility of conditions determine what else changes. With the countable chain condition for forcing, an ordinal-valued name has only countably many possible values at each coordinate: take a maximal antichain deciding that value. Therefore the possible range of a name for a map with infinite ground domain of cardinality $\mu$ lies in a ground set of size at most $\mu \cdot {\aleph }_0=\mu$. This prevents cardinal collapse and changes of uncountable regular cofinalities. Forcing by finite partial functions $\lambda \times \omega \to 2$ is ccc: the delta-system lemma thins any uncountable family of finite domains to a delta-system, and a further thinning makes their values agree on its finite root. Two remaining conditions are compatible. Distinct coordinates add distinct reals.

For example, start with GCH and take $\lambda ={\aleph }_2$. This forcing has size $\lambda$ and adds $\lambda$ distinct reals. A nice name for a real uses countably many antichains, and each antichain is countable. The ground arithmetic ${\lambda }^{{\aleph }_0}=\lambda$ bounds the number of such names by $\lambda$. Hencegiving a relative-consistency example of the failure of CH. The ground choice of GCH is available through the constructible universe; no claim that every starting model has this arithmetic is needed.

By contrast, countably closed forcing adds no new countable sequences of ground ordinals. Given a name, recursively strengthen a condition to decide each coordinate, then take a lower bound of the descending sequence. That condition decides the entire sequence. Closure and chain conditions thus control different features; neither should be silently substituted for the other.

Further applications include changing cofinalities, adding branches through trees, and constructing prescribed continuum patterns. If the aim is to violate choice, an ordinary generic extension of a choice model will not do: one restricts to a suitable symmetric collection of names, checking the axioms again. These examples also mark a limitation: forcing proves relative consistency, not absolute consistency of the background theory. The countable-transitive-model picture gives a concrete semantic construction; the definable forcing relation and Boolean-valued formulation supply the corresponding formal consistency arguments without asserting that a transitive ground model can be proved to exist.

Let $M$ be an externally countable model of ZFC, and let $\kappa =({\omega }_1{)}^M$. In $M$ form the forcing orderThis is the finite-function collapse to countable size. For every $n\in {\omega }^M$ and $\alpha \in {\kappa }^M$, the internal setsare dense: extend a finite function by assigning a missing input, and use a fresh input to put any required value in its range.

Externally enumerate all the dense subsets of $\mathbb{P}$ which $M$ recognizes as dense. There are only countably many, because $M$ is countable. Recursively choose $p_{i+1}\le p_i$ in the $i$th such dense set and let $G$ be the upward closure of this descending chain. This produces a generic filter over $M$. By the forcing theorem, its generic extension $M[G]$ satisfies ZFC. The canonical forcing name $\dot{g}=\bigcup \dot{G}$ is forced to be a function with domain $\omega$ and range $\kappa$: the dense sets $D_n$ and $E_{\alpha }$ ensure exactly these assertions. ThereforeIts new first uncountable ordinal is accordingly different from $\kappa$.

For a transitive $M$, ordinary evaluation of names gives the familiar literal extension. The question only assumes a countable model, so one must also cover externally ill-founded models. In that case use the quotient of the internally defined forcing names by forced equality modulo $G$, with membership defined by forced membership. The internal forcing identities and the forcing theorem hold in $M$ and give a well-defined first-order structure satisfying every standard ZFC axiom. Check names embed the original membership structure, so identify their images with $M$. Its domain is a quotient of a subset of the countable domain of $M$, hence is externally countable. Thus the required extension exists even without assuming transitivity. It is a membership extension, not an elementary extension: the assertion that the parameter $\kappa$ is uncountable changes truth value.

We use only the definability and truth clauses of the forcing theorem. For every formula, its forcing relation on names is definable inside $M$; forcing is preserved by strengthening; and $M[G]\models \phi ({\vec{\tau }}^G)$ exactly when some condition in $G$ forces $\phi (\vec{\tau })$. These clauses do not presuppose the Separation axiom we are proving.

Let $a={\tau }^G$ and let ${\vec{\sigma }}^G$ be the parameters in the desired instance. In $M$, form the nameThis is a set in $M$ by its Separation axiom and forcing definability, using the subnames of $\tau$ and $P$ as a set-sized bound.

If $x\in {\rho }^G$, some $q\in G$ activates $(\nu ,q)$ and $q\ge p$ activates $(\nu ,p)\in \tau$. Thus $x={\nu }^G\in a$, and the forcing theorem gives $\phi (x,{\vec{\sigma }}^G)$. Conversely, if $x\in a$ satisfies that formula, choose an active pair $(\nu ,p)\in \tau$ with $x={\nu }^G$ and a condition $r\in G$ forcing the formula. Directedness gives $q\in G$ stronger than both $p$ and $r$. Monotonicity puts $(\nu ,q)\in \rho$, so $x\in {\rho }^G$.

ThereforeEvery requested instance of separation in a generic extension follows.

For a countable transitive ground model, the semantic forcing relation isThe forcing theorem identifies this with the recursively defined relation inside $M$ and makes it definable there. In particular, if $r\ge q$ is stronger, then $q⊩\phi$ implies $r⊩\phi$. The names are interpreted in each relevant generic filter, not replaced by their value in one fixed extension in the definition.

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.

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.

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.

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.

Let $x={\tau }_G$ be any set in the extension. The set $D=\{\sigma :\exists p((\sigma ,p)\in \tau )\}$ belongs to $M$. Ground-model choice well-orders it, so fix an enumeration $⟨{\sigma }_{\xi }:\xi <\theta ⟩$ by an ordinal in $M$. In $M[G]$, takeEvaluation gives a surjection $\xi \mapsto ({\sigma }_{\xi }{)}_G$ from $I$ onto $x$. This is a set function by the ZF part of the forcing theorem; no extension choice is needed. For each $y\in x$, take its least preimage ordinal. Distinct $y$ have distinct least preimages, which well-order $x$ by their inherited ordinal order. Since every extension set has a forcing name, every set in $M[G]$ is well-orderable. ThereforeThis choice preservation by well-ordered names tolerates repeated evaluations, because the least-index step removes them canonically.

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.

Fix forcing names $\tau ,{\tau }_1,\dots ,{\tau }_n\in M$ for the set $w$ and the parameters. Define a nameOnly $\eta$ appearing in $\tau$ and $p\in \mathbb{P}$ are needed, so this is a subset of a ground-model set. The forcing definability lemma makes its defining predicate a formula of $M$. Ground-model axiom schema of separation therefore gives $\sigma \in M$.

If $(\eta ,p)\in \sigma$ with $p\in G$, the accompanying $q\ge p$ belongs to $G$, so ${\eta }_G\in {\tau }_G$. The forcing theorem gives $\phi ({\eta }_G,{\tau }_G,({\tau }_1{)}_G,\dots ,({\tau }_n{)}_G)$.

Conversely, if $u\in {\tau }_G$ satisfies this formula, choose $(\eta ,q)\in \tau$ with $q\in G$ and ${\eta }_G=u$. The forcing truth lemma supplies $r\in G$ forcing the formula for these names. Directedness gives $p\in G$ with $p\le q,r$. Then $(\eta ,p)\in \sigma$, and $u\in {\sigma }_G$. ThusThis proves the instance of the axiom schema of separation in the generic extension, without assuming that instance there in order to construct the name.

Closure preserves the ground-model cardinal $\lambda$. We use the stronger fact that closed forcing adds no short ordinal sequences. Fix an ordinal $\mu <\lambda$ in $M$ and a forcing name $\dot{h}$ that a condition $p$ forces to be an ordinal-valued function with domain $\mu$.

Inside $M$, recursively strengthen $p$ to decide each value $\dot{h}(\xi )$, for $\xi <\mu$. At a limit stage take a lower bound of the previously chosen descending sequence; its length is below $\lambda$, so $\lambda$-closure supplies one. After all $\mu$ values have been decided, closure supplies a common lower bound $q$. The decided values form a function $h\in M$ by the Axiom schema of replacement, andThe recursion is carried out within $M$ using the definable forcing theorem and a ground-model choice of deciding extensions, so every sequence to which closure is applied belongs to $M$. Repeating the construction below any stronger condition shows that such full-decision conditions are dense below a forcing condition $p$. The dense-below generic meeting lemma ensures that a generic filter containing $p$ meets them. Thus the interpreted function is in $M$.

If $\lambda$ ceased to be a cardinal number in $M[G]$, there would be a surjection $\mu \to \lambda$ for some ordinal $\mu <\lambda$. The forcing theorem gives a name and a condition in $G$ forcing this. The argument above makes its interpretation a ground-model function, but $M$ regards $\lambda$ as a cardinal number and admits no such surjection. ThereforeNo regularity of $\lambda$ is required; every recursion used has length strictly below $\lambda$. Finite cardinals are preserved automatically. This is cardinal preservation by closed forcing.

Use standard notation for forcing: $q\le p$ means that $q$ is stronger. A generic filter is nonempty, upward closed and downward directed, and meets every dense subset of a forcing order in the ground model. Names below are forcing names in $M$, with pairs ordered as $(\text{name},\text{condition})$.

The semantic forcing relation isHere ${\tau }^K=\mathrm{val}(\tau ,K)$ is the evaluation of a forcing name. Countability of $M$ supplies such generic filters through every condition by the Rasiowa–Sikorski lemma.

Define the syntactic forcing relation by mutual well-founded recursion on the forcing names for the atomic cases, then induction on the first-order formula. Its membership clause isFor equality, first abbreviateand set $p{⊩}^{\ast }\sigma =\tau$ exactly when both inclusions hold. Each recursive call lowers the rank of at least one forcing name without raising the other. This makes the mutual recursion well-founded.

For a basis of connectives consisting of logical conjunction, negation and existential quantification, the remaining clauses areThe last line is the existential clause of syntactic forcing; witnesses need only occur densely, rather than be forced by $p$ itself with one preselected name. Other connectives are defined by logical abbreviations. The recursion gives a definable relation inside $M$ for each fixed first-order formula, with quantification over the class of its names. If $\mathbb{P}$ has no greatest element, use canonical forcing names $\check{a}=\{(\check{b},p):b\in a,p\in \mathbb{P}\}$, which still evaluate to $a$. It also proves monotonicity: strengthening a condition preserves what it forces. The forcing theorem identifies the two relations and supplies the truth lemma.

We prove mutual genericity for product forcing. Let $D\in M[G_0]$ be a dense subset of $\mathbb{Q}$, and take a forcing name $\dot{D}\in M$ evaluating to $D$. By the forcing theorem, some $p_0\in G_0$ forces that $\dot{D}$ is a dense subset of the canonical forcing name for $\mathbb{Q}$.

In $M$ defineThis set is dense in the product forcing order. Given $(p,q)$, the incompatible case is immediate. Otherwise first strengthen $p$ below $p_0$. The forced density assertion and the existential clause of syntactic forcing supply a further $r\le p,p_0$ and a ground-model $q^{\prime }\le q$ with $r⊩{\check{q}}^{\prime }\in \dot{D}$. To justify choosing a ground-model $q^{\prime }$, a name forced to lie in $\check{\mathbb{Q}}$ can densely be made equal to some ${\check{q}}^{\prime }$ by the atomic membership clause; strengthen to that equality and use the forced order comparison. Thus $(r,q^{\prime })\in E$ lies below $(p,q)$.

The generic filter $H$ meets $E$. It cannot meet the first part, because its first projection contains $p_0$ and is directed. Hence there is $(p,q)\in H$ with $p\in G_0$ and $p⊩\check{q}\in \dot{D}$. Soundness of the forcing theorem gives $q\in D$, and the projection gives $q\in G_1$. Since every such $D$ is met,This establishes the stronger property, rather than merely meeting dense ground-model subsets.

For the reverse implication, suppose $p\in G$ and $M\models p{⊩}^{\ast }\exists x\phi (x)$. By the existential clause of syntactic forcing, the setis dense below $p$; the existential quantifier ranges over forcing names of $M$. Definability of the syntactic forcing relation and axiom schema of separation put $D$ in $M$.

To apply genericity, augment it to the globally dense setA condition compatible with $p$ has an extension below $p$, then one in $D$, and an incompatible condition already lies in $D^{\prime }$. Thus $D^{\prime }$ is a dense subset of a forcing order in $M$. The generic filter meets it. Since $p\in G$, directedness prevents $G$ from containing a condition incompatible with $p$, so some $q\in G\cap D$ exists.

There is consequently a name $\sigma \in M$ with $M\models q{⊩}^{\ast }\phi (\sigma )$. The assumed truth lemma gives $M[G]\models \phi (\mathrm{val}(\sigma ,G))$, and henceTogether with part (a), this completes the existential step of the forcing theorem without assuming the desired existential truth lemma in advance.

Assume $M$ satisfies the countable chain condition for forcing. We prove cardinal preservation by chain-condition forcing using the possible-values lemma for chain-condition forcing, and give the argument explicitly for ccc.

Let $\kappa$ be an uncountable cardinal of $M$. If it ceased to be a cardinal, some ordinal $\mu <\kappa$ would admit a surjection onto $\kappa$ in an extension. By the forcing theorem, there would be a condition $p$ and a forcing name $\dot{f}$ such thatInside $M$, for each $\xi <\mu$ choose a maximal antichain in a forcing order below $p$ deciding the ordinal value of $\dot{f}(\xi )$. Conditions deciding that value are dense below $p$. The chain condition makes the chosen antichain countable in $M$, so the corresponding set $B_{\xi }\subseteq \kappa$ of possible decided values is countable in $M$.

A generic filter containing $p$ meets the downward closure of every such maximal antichain, so its value $f(\xi )$ belongs to $B_{\xi }$. Therefore its whole range lies in the ground-model setThe last inequality is infinite cardinal arithmetic and uses only that $\kappa$ is uncountable and $\mu <\kappa$; no regularity of $\kappa$ is needed. Some ordinal in $\kappa \setminus B$ exists already in $M$ and cannot occur in the range, contradicting the forced surjectivity.

Finite cardinals cannot collapse, and $\omega$ cannot become finite: a finite domain has finite image, and transitive models agree on the natural numbers. Old non-cardinals cannot become cardinals because their old bijections persist. Thus