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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 $p$.

A forcing collapses a ground-model cardinal $\kappa$ if it forces a bijection from a smaller ordinal onto the ground ordinal $\kappa$. The ordinal itself is preserved; its being an initial ordinal is lost.

For a regular uncountable cardinal number $\kappa$, a forcing order has the $\kappa$-chain condition when every antichain has cardinality below $\kappa$. It preserves cardinals and cofinalities at least $\kappa$.

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.

A forcing is cofinality-preserving over a transitive model $M$ if every generic extension has the same cofinality for every ground ordinal. The values are compared as ordinals, which forcing preserves.

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.

Forcing constructs a generic extension $M[G]$ of a model of a first-order theory of set theory from a partially ordered set $\mathbb{P}$ in $M$. A condition specifies partial information about a new object; in the standard order, $q\le p$ means that $q$ specifies stronger information. A generic filter $G$ meets every dense subset of a forcing order belonging to $M$. The elements of $M[G]$ are obtained by evaluating forcing names from $M$ using $G$.

For Cohen forcing, conditions are finite partial functions from $\omega$ to $\{0,1\}$. Their union is a new binary sequence. Dense requirements make the sequence total and ensure that it differs from each old binary sequence. When $M$ is a countable transitive model, one can enumerate its dense sets externally and successively choose stronger conditions meeting them. This construction is conditional on having such a model; consistency of ZFC alone does not assert the existence of a countable transitive model.

The forcing theorem makes this method useful for proofs: the forcing relation is definable in the ground model, and truth in the generic extension is witnessed by a condition in the generic filter.

A subset of a forcing order whose distinct members have no common stronger extension. A maximal forcing antichain has a compatible member for every condition. For forcing by nodes of a set-theoretic tree, ordered by extension, this coincides with a tree antichain.

The ordinal assigned recursively by $\mathrm{nrk}(\tau )=\sup \{\mathrm{nrk}(\sigma )+1:(\sigma ,p)\in \tau \}$. It measures dependency on subnames and justifies generic evaluation by recursion. The ordinary rank of a set also accounts for the coding of the ordered pairs and the forcing conditions; it need not equal this dependency rank of a set.

Every uncountable subset of a Knaster forcing has an uncountable pairwise compatible subset. The property implies the countable chain condition for forcing. Finite Cohen forcing has this property by the delta-system lemma and agreement on the finite root. A Knaster forcing times any CCC forcing is CCC.

If $T$ is a normal splitting Suslin tree and $P$ is Knaster, then $P\times T$ is CCC, so $P$ forces that the ground tree is still CCC. Countable levels and splitting persist. A new cofinal branch would give an uncountable antichain by splitting, so none is added. This permits adding many Cohen reals while retaining a Suslin tree, and distinguishes Knaster preservation from an unjustified preservation claim for every CCC forcing.

For a cardinal number $\kappa$, ${\mathrm{MA}}_{\kappa }$ says that for every forcing with the countable chain condition for forcing and every family of at most $\kappa$ dense subsets, some filter in an ordered set meets them all. Full Martin axiom asserts this for all $\kappa <2^{{\aleph }_0}$. The strict upper bound is part of the definition.

A filter is generic over its ground model exactly when it meets each ground-model maximal forcing antichain in one point. Extensions of a maximal antichain form a dense set, while any ground dense set contains an antichain maximal in the whole forcing. Directedness permits at most one antichain member in a filter.

Over a Generalized continuum hypothesis ground model, adding one Cohen real preserves every infinite power $2^{\lambda }={\lambda }^{+}$. For the countable forcing, nice subset forcing names for $\lambda$ number at most $(2^{{\aleph }_0}{)}^{\lambda }=2^{\lambda }$ in the ground model. Old subsets give the matching lower bound and the countable chain condition for forcing preserves cardinals. Starting from the constructible universe, the new real is nonconstructible while the constructible levels remain unchanged.

An inner model of ZF is a transitive class $M$ containing every ordinal, with inherited membership, such that each ZF axiom holds with all quantifiers restricted to $M$. Here transitivity means $x\in y\in M\Rightarrow x\in M$. For a proper class this is an axiom-by-axiom assertion, not an internal truth predicate for the whole class. An inner model can have fewer subsets of a given set than the surrounding universe; it need not have the same cardinals or the same continuum function.

Inner models prove relative consistency. If a theory $T$ proves that a definable class $M$ satisfies every axiom of $S$, an inconsistency proof from $S$ could be relativized to $M$ and become an inconsistency proof from $T$. Thus $\mathrm{Con}(T)\Rightarrow \mathrm{Con}(S)$. This reasoning concerns formal interpretations and needs no assumption that a given model of $T$ is externally well-founded.

The central example is the constructible universe. Definewhere $\mathrm{Def}(X)$ contains exactly the subsets definable over $(X,\in )$ using parameters from $X$. Satisfaction for a set structure is definable, so this recursion is available in ZF. Induction shows that the levels are transitive and increasing: if $a\in L_{\alpha }$, the subset $\{x\in L_{\alpha }:x\in a\}$ is $a$, so $a\in L_{\alpha +1}$. The ordinal part of $L_{\alpha }$ is $\alpha$, giving all ordinals in $L$.

For completeness, the main axiom mechanisms explain why this example works. Finite set operations on constructible parameters are definable at finitely many further stages. For a fixed finite family of formulas and parameters, close a sequence of levels under their least constructible witness stages and take its union; induction on formulas gives a reflecting level. Separation of a constructible set then occurs at the next definability stage. For replacement, ambient replacement bounds the construction stages of the unique constructible outputs; a reflecting level above that bound makes the range definable. For the internal power set of $a\in L$, use ambient separation to form $\mathcal{P}(a)\cap L$, bound the construction stages of its members, and at a sufficiently high level define the family of all subsets of $a$ in that level. Thus the internal power set is constructible even though some ambient subsets may not be. Foundation is inherited by transitivity, and infinity comes from $L_{\omega }$ and its successor stages.

There is a canonical well-order of $L$: order by first construction stage, then by the least definition code and finite parameter tuple, using the previously constructed well-orders for parameters. Each set in $L$ can be well-ordered by the restriction of this set-like class order; hence $L$ satisfies AC. Moreover the hierarchy computed inside $L$ is the same hierarchy, because definability over a fixed set structure and limit unions are absolute. Thus $L$ satisfies $V=L$. The further cardinal analysis of this hierarchy, encapsulated in the constructible universe theorem, gives GCH. ConsequentlyThis proves the relative consistency of choice, CH and GCH; in particular none can be refuted by ZF if ZF is consistent. The assertions here are internal to $L$: they do not assert CH in the ambient universe.

Inner models also isolate definable information. The relative constructible universe built with a set parameter retains that parameter and all ordinals while discarding unrelated sets. Such constructions help compare relative consistency strengths and identify which hypotheses survive passage to smaller universes; preservation of a large-cardinal property must be checked separately, not inferred merely from transitivity.

There is an important limitation, the constructible-universe obstruction to a uniform inner-model proof of failure of choice. Every inner model $M$ of ZF contains the ambient $L$. Indeed $L_{\alpha }^M=L_{\alpha }$ by induction: the same set structure has the same definable subsets at a successor stage, and both hierarchies take the same union at a limit. If the ambient universe satisfies $V=L$, this forces $M=V$. Thus inner models cannot uniformly produce a model of $\neg \mathrm{AC}$ or $V\ne L$ from every ZF universe. To prove the opposite independence directions one needs an outer-model or other interpretation technique, such as forcing or the non-well-founded symmetry construction in Question 2.

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.