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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.

Work in ZFC. The move alphabet need not be countable; choice permits the selections of moves, strategies and witnesses used below. Assume $A\ne \varnothing$; with the usual convention that a player unable to move loses, the empty alphabet is a trivial terminal game. We prove Borel determinacy theorem by strengthening the assertion to the existence of game coverings that unravel payoffs. Simply asserting that determined payoffs are closed under countable unions would not be a proof.

It is useful to allow a tree of legal finite plays with terminal nodes explicitly labelled as losses for either player. For an infinite branch, I wins exactly when it belongs to the payoff $C$. Projection of a game tree is length-preserving and respects prefixes. At a projected original terminal node the outcome is inherited, but the new tree may also have artificial terminal losses.

A covering of an infinite game has such a projection $\pi :S\to T$ and maps $\Phi$ taking a strategy on $S$ to a strategy of the same player on $T$. Its crucial lifting condition is this: each maximal play $x$ on $T$ following $\Phi (\sigma )$ has a $\sigma$-compatible lift $y$ on $S$, either projecting exactly to $x$ with its inherited terminal outcome, or projecting to a prefix of $x$ and ending in a loss for the player using $\sigma$. Its strategy maps must be local in depth: choices up to a given depth use only the upstairs strategy up to that depth. A $k$-covering preserves the initial $k$ levels and choices. If ${\pi }^{-1}(C)$ is clopen, call it an unravelling of a game payoff. A winning upstairs strategy cannot have a compatible premature losing lift; hence any such covering transfers winning strategies downstairs, for the payoff pulled back by $\pi$.

First, clopen determinacy with terminal losses holds on any set-sized tree. At a node where all infinite extensions have one fixed winner, the other player can win only by forcing a terminal loss for that winner. Decide that reachability game by the winning-position attractor in an infinite game: begin with the desired losing terminals, add the attacking player's positions with a successor already in the attractor and the defending player's positions with every successor in it, and take unions at limit stages. Entry ranks give a strategy reaching the target; outside the attractor the defender avoids it. If there are no infinite extensions the same reachability analysis still applies. At every remaining position there are infinite extensions in both $C$ and its complement. This remaining tree is well-founded, since an infinite branch through it would contradict clopenness at that branch. Well-founded recursion now assigns a winner upwards from the already decided nodes, choosing a winning successor when the player to move has one. These choices give an actual winning strategy.

Next we construct an announcement-and-challenge covering of a closed payoff. Let $C\subseteq [T]$ be closed, and choose an even depth $l>k$. Play initially follows $T$ unchanged. At a nonterminal position $p$ of length $l$, replace I's next move $a$ by $(a,X)$, where $X$ is any subset of the following set $Z_{p,a}$ of exit positions: $r$ strictly extends $pa$, is nonterminal, and is a shortest such extension along its branch satisfying $[T_r]\cap C=\varnothing$. If $[T_{pa}]\cap C=\varnothing$ already, the first nonterminal extension beyond $pa$ counts as an exit. Distinct exits form a prefix antichain. An original terminal move always keeps its original outcome.

Player II now either accepts and makes an ordinary next move $b$, or challenges an $r\in X$ and makes the move $b$ prescribed by that $r$. On acceptance, continue the original play until the first exit $r$: it is an artificial win for I if $r\in X$, and an artificial loss for I otherwise. On a challenge, both players are forced to follow $r$ until it is reached, after which the original game resumes without the acceptance stopping rule. The projection forgets the announcements and challenge flags.

An infinite accepted play avoids all exits. Every sufficiently long prefix of its projection therefore has a continuation in $C$, so closedness implies that the projection is in $C$. Every infinite challenged play passes through an exit cylinder disjoint from $C$. Thus among infinite plays the pulled-back payoff is decided by II's acceptance or challenge: it is clopen. This remains true when a particular branch reaches an inherited terminal position before either case produces an infinite play.

Here is the strategy-transfer proof, which also fixes the artificial terminal labels. Given an I-strategy $\sigma$ upstairs, use its announced ordinary move $a$ downstairs and initially simulate an accepting response. If the projected play never exits, this is an exact lift. If its first exit $r$ is not in the announced $X$, the accepting lift ends in a loss for I, as allowed. If $r\in X$, switch to the simulation in which II challenged this particular $r$. The earlier ordinary prefix is unchanged: all moves through $r$ in the challenged simulation are forced and agree with the observed prefix. Beyond $r$ follow $\sigma$ in that simulation. This gives an exact compatible lift. Original terminal outcomes are inherited throughout.

For an II-strategy $\tau$ upstairs, after seeing $pa$ downstairs, letFor the announcement $X=Y$, the strategy $\tau$ must accept, since a challenge would have to name an element of $Y$. Use its accepting ordinary response downstairs and simulate that acceptance until an exit. If the exit $r$ belongs to $Y$, the accepting lift is an artificial loss for II, as required. If $r\notin Y$, choose an announcement $X_r$ for which $\tau$ challenges $r$, and switch to that simulation. Its first ordinary response and all the intervening forced moves agree with the actual prefix $r$; after $r$ use $\tau$ there. This gives an exact lift. Choices of $X_r$ use choice. No switch uses unseen future moves, and the definition of $Y$ uses only the strategy's responses at the fixed announcement level. Thus the maps are local in depth. The unchanged initial segment makes this a $k$-covering.

We now establish the stronger assertion: every Borel payoff on every such game tree admits a $k$-unravelling for every finite $k$. Closed payoffs were just handled. Complements need no new construction, since the complement of a clopen pulled-back payoff is clopen.

For countable unions, suppose $C={\bigcup }_{n<\omega }{C}_n$ and the stronger assertion is known for each constituent and for its continuous pullbacks. Starting with $T_0=T$, choose a covering $T_{n+1}\to T_n$ that unravels the current pullback of $C_n$ and preserves at least the first $k+n$ levels and choices. Previously unravelled payoffs remain clopen under further continuous projections. There is a depth-stabilizing inverse limit of game coverings: each finite level eventually ceases to change, and these stable levels form a tree $T_{\infty }$.

For completeness, the limit retains the covering property, not just the topology. Compose the projections and the depth-local strategy maps; their finite restrictions stabilize. To lift a play following the limit-projected strategy, lift it successively, coherently, from $T_n$ to $T_{n+1}$. If all lifts are exact, the stabilized finite prefixes give a compatible play on $T_{\infty }$. If a premature losing terminal lift occurs, every further lift is either exact or another losing terminal prefix, so its length cannot increase; lengths and then its finite position stabilize. This produces the permitted losing terminal lift in the limit. The choices needed to select successive lifts are available in ZFC. Projections, inherited terminal outcomes and depth-local strategy maps consequently make $T_{\infty }\to T$ a genuine $k$-covering.

On $T_{\infty }$, the pullbacks of all the $C_n$ are clopen, so their union is open. Its complement is closed. Apply the closed-payoff construction once more to unravel this complement; it unravels the union too. Composition gives the desired covering of $T$. This proves the induction step. Induction on a well-founded construction of a Borel set, whose operations are complementation and countable union starting from open sets, proves the assertion for every Borel payoff. Continuous preimages of a constituent have the same type of construction, so the induction hypothesis used at successive covers is justified. No countability of the move alphabet was used; all enlarged alphabets and trees remain sets.

Finally apply this result to $T=A^{<\omega }$. The unravelled clopen game has a winning strategy by the first argument. The covering lifting condition transfers it to the original game. ThereforeThis is a theorem in the usual background set theory with choice; it does not assert unrestricted determinacy for arbitrary, non-Borel payoffs.

Throughout the forcing arguments, a larger condition is stronger, as in the original paper. Ground-model cardinal calculations are internal to the indicated model. All three definition alternatives are supplied in each group that permits choosing two.

With this paper's order convention, larger conditions are stronger. A generic filter $G\subseteq P$ over $M$ is nonempty, closed toward weaker conditions, and directed toward stronger conditions: if $p,q\in G$, some $r\in G$ satisfies $r\ge p,q$. It also meets every $D\in M$ which $M$ regards as dense in $\mathbb{P}$.

For a countable transitive $M$, such a filter containing any prescribed condition can be built by enumerating its dense sets and successively choosing stronger conditions in them. The definition does not require $G\notin M$: for an atomic forcing a generic filter may already belong to $M$.

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.

Let $\kappa =({\omega }_2{)}^M$ be the fixed ground ordinal. We prove that $S$ remains stationary in this ordinal. The chain condition for forcing ensures that $\kappa$ remains regular: possible values of each coordinate of an ordinal-valued name form a ground set of size less than $\kappa$, by a maximal deciding forcing antichain. A hypothetical cofinal map with domain below $\kappa$ would have its range covered by fewer than $\kappa$ such small sets, hence bounded by regularity in $M$.

Let $\dot{C}$ name a club set in $\kappa$, and take $p\in G$ forcing this. For each $\alpha <\kappa$, choose in $M$ a maximal forcing antichain above $p$ deciding the least point of $\dot{C}$ strictly above $\alpha$. Its set of possible values has size less than $\kappa$, so choose a ground bound $h(\alpha )<\kappa$ larger than all of them. The ground setis a club set by the usual countable closure iteration and regularity. For every $\delta \in D$, $p$ forces $\dot{C}\cap \delta$ unbounded in $\delta$, and closedness forces $\delta \in \dot{C}$. Hence $p⊩\check{D}\subseteq \dot{C}$. In $M$, choose $\delta \in S\cap D$; that same ordinal belongs to $S\cap C$ in the extension. Stationarity at the fixed ground $\kappa$ is preserved.

There is an important qualification to the printed ${\omega }_2$ notation. If it is recomputed internally as $({\omega }_2{)}^{M[G]}$, the assertion is false without preservation of smaller cardinals. Finite partial maps from $\omega$ to $({\omega }_1{)}^M$ form a forcing of size $({\aleph }_1{)}^M$, hence have the $({\aleph }_2{)}^M$-chain condition, but collapse $({\omega }_1{)}^M$ to countable. Then $({\omega }_2{)}^M=({\omega }_1{)}^{M[G]}$, while $({\omega }_2{)}^{M[G]}$ is larger. The old $S$ is bounded in this new ${\omega }_2$, so cannot be stationary there. The proved statement uses the fixed ground ordinal, or alternatively requires the lower-cardinal preservation needed to retain its aleph index.

Use the equivalent tree formulation of the Suslin hypothesis: there is no normal, well-pruned Suslin tree of height ${\omega }_1$, with countable levels and no uncountable chains or antichains. This is equivalent to the linear-order formulation that every complete dense order without endpoints satisfying the countable chain condition for a linear order is separable.

If such a tree $T$ existed, use its nodes as forcing conditions, with a higher extension stronger. It is CCC because its antichains are countable. For every $\alpha <{\omega }_1$, the set $D_{\alpha }=\{t:\mathrm{ht}(t)\ge \alpha \}$ is dense because the tree is well-pruned. The assertion ${\mathrm{MA}}_{{\aleph }_1}$ is precisely that a CCC forcing and a family of at most ${\aleph }_1$ dense sets admit a filter meeting all of them.

Apply it to these dense sets. A directed filter in a tree is a chain: two compatible nodes are comparable, since both lie among the well-ordered predecessors of a common extension. Meeting every $D_{\alpha }$ makes this chain cofinal, contrary to the defining absence of uncountable chains in a Suslin tree. Consequently

Start with a ground model containing a normal splitting Suslin tree $T$, and force with $\mathrm{Add}(\omega ,{\omega }_2)$, the finite partial functions from ${\omega }_2\times \omega$ to $2$. Such a starting model is relatively consistent with ZFC: the constructible universe has $◊$, and $◊$ implies the existence of a Suslin tree.

This Cohen forcing is Knaster. In an uncountable family of finite conditions, the delta-system lemma yields an uncountable family whose domains have one common root; thinning to common values on that finite root makes its members pairwise compatible. If $P$ is Knaster and $T$ is CCC, the product $P\times T$ is CCC: thin any uncountable family to pairwise compatible $P$ coordinates, then use CCC to find two compatible $T$ coordinates. Thus $P$ forces that the old $T$ is still CCC A name for an uncountable antichain would otherwise produce an uncountable antichain in this product by deciding its nodes.

Countable levels and the tree's splitting property persist, and the splitting argument from the previous part rules out a new cofinal branch in a CCC tree. Therefore the old $T$ is still Suslin. Meanwhile the forcing adds at least ${\aleph }_2$ distinct reals, with all cardinals preserved, so $2^{{\aleph }_0}\ge {\aleph }_2$.

We obtain a model with a Suslin tree and $\neg \mathrm{CH}$. Hence existence of a Suslin tree does not imply the Continuum hypothesis. This uses Knaster forcing preserves Suslin trees, not a claim that arbitrary CCC forcing preserves them.

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.