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For a forcing atom $p$ in a ground-model forcing order, $G_p$ is a ground-model generic filter. It contains $p$ and is upward closed. For $q,r\in G_p$, choose $q^{\prime }\le p,q$ and $r^{\prime }\le p,r$. The atom property gives $s\le q^{\prime },r^{\prime }$, so $s\in G_p$ is a common strengthening of $q,r$. Thus $G_p$ is a filter in an ordered set. Every dense subset of a forcing order has some $d\le p$, and $d\in G_p$. The defining compatibility predicate is bounded to the ground-model set of conditions, so axiom schema of separation forms $G_p$ inside the ground model. For a nonminimal atom, this filter can strictly contain the principal filter above $p$.

For forcing names $\sigma ,\tau$ and a generic filter $G$, ${\sigma }^G\in {\tau }^G$ exactly when some $p\in G$ syntactically forces $\sigma \in \tau$. Assuming the equality truth lemma, interpreted membership supplies an active pair $(\rho ,r)\in \tau$ and a condition in $G$ forcing $\sigma =\rho$; a common strengthening forces membership. Conversely the defining equality witnesses occur dense below a forcing condition forcing membership. The dense-below generic meeting lemma supplies one in $G$, and the equality truth lemma recovers interpreted membership.

The canonical forcing name for a ground-model set $x$ is $\check{x}=\{(\check{y},1):y\in x\}$. Its evaluation of a forcing name under every generic filter is $x$.

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.

If a forcing is $\kappa$-closed in the ground model, then it adds no functions from a ground ordinal $\alpha <\kappa$ into a ground set $B$. Recursively decide each value inside the ground model and take common stronger bounds at limit stages and after the final step. Conditions deciding a whole ground function are dense below any condition asserting this type of function. The dense-below generic meeting lemma ensures the generic filter meets that dense subset of a forcing order. One arbitrary bound need not be in the filter in an ordered set.

If a ground-model order is $\lambda$-closed, it adds no ordinal-valued functions with domain $\mu <\lambda$. Recursively decide each name value inside the ground model, using closure at intermediate limit stages and once more at the end. Below each starting condition, conditions deciding the entire function are dense. A generic filter meets them by the dense-below generic meeting lemma. The recursion must be internal to the ground model, since its closure property only covers its own sequences.

If $p\in G$ and a ground-model set $D$ is dense below a forcing condition $p$, then the generic filter $G$ meets $D$. Add all incompatible forcing conditions with $p$ to $D$. The enlarged set is globally dense: a condition compatible with $p$ first has a common strengthening, then a further strengthening in $D$. Genericity meets the enlargement, and directedness excludes its incompatible part.

A subset $D$ of a forcing order $P$ is dense if every $p\in P$ has a stronger condition $q\le p$ in $D$. This order-density condition is the one used by a generic filter; it is distinct from density in an arbitrarily specified topology.

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 $p$.

A filter is a nonempty upward-closed subset $G$ of a partial order such that every two members have a common lower bound in $G$. With standard notation for forcing, lower means stronger, so these are exactly the directed filters used to define a generic filter. The projection of a product-generic filter uses these properties before genericity of its factor filters is proved.

For an infinite ground-model cardinal number $\kappa$, use finite partial functions from $\omega$ to $\kappa$, ordered by reverse inclusion. The union of a generic filter is a surjection $\omega \to \kappa$. The order has size $\kappa$, hence the ${\kappa }^{+}$-chain condition, and preserves cardinals at least ${\kappa }^{+}$.

A forcing name is a recursively built set of pairs $(\sigma ,p)$, where $\sigma$ is a lower-rank forcing name and $p$ is a forcing condition. Ground-model names are interpreted using a generic filter to form the generic extension.

A formula holds in a generic extension exactly when some condition in the generic filter forces it for names of its parameters.

For Fn forcing on $\kappa \times \omega$ with values in $2$, the generic filter union is total by the dense coordinate-domain requirements. For each pair of distinct rows, the set of conditions giving them opposite bits at some column is dense: choose a column untouched in both finite rows. Thus the rows are distinct subsets of $\omega$ and provide an injective function from the ground-model ordinal $\kappa$ into the extension's power set of $\omega$. This construction alone does not prove preservation of its ground-model cardinal number status.

If $G$ is a generic filter over a countable transitive model $M$, the generic extension $M[G]$ consists of the interpretations by $G$ of all forcing names in $M$.

If a generic filter $G$ for an atomless forcing order belonged to its ground model, then the complement $\mathbb{P}\setminus G$ would be a dense ground-model set. A condition outside $G$ is already there; a condition in $G$ has two incompatible strengthenings, at least one outside its directed filter. Genericity would then require meeting the complement, a contradiction.

For the condition-first forcing name convention, pair every condition with each of two names $\tau ,{\tau }^{\prime }$. Under any nonempty generic filter, the evaluation of a forcing name is then $\{{\tau }_G,{\tau }_G^{\prime }\}$, the unordered pair of the two values. Equal values give a singleton. With the name-first convention, reverse the two components of each syntactic pair.

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.

A forcing name for $\mathbb{P}$ is a set of ordered pairs $(\sigma ,p)$ where $p\in P$ and $\sigma$ is itself a forcing name. This recursive definition is made well-founded by assigning the forcing name rank $\sup \{\mathrm{nrk}(\sigma )+1:(\sigma ,p)\in \tau \}$. Names over the ground model are those names belonging to $M$; conditions remain ground-model objects. A name describes which recursively interpreted elements are activated by the generic filter.