Models of Set Theory in which Nonconstructible Reals First Appear at a Given Projective Level

Models of set theory are defined, in which nonconstructible reals first appear on a given level of the projective hierarchy. Our main results are as follows. Suppose that n ≥ 2 . Then: 1. If it holds in the constructible universe L that a ⊆ ω and a ∉ Σ n 1 ∪ Π n 1 , then there is a generic extension of L in which a ∈ Δ n + 1 1 but still a ∉ Σ n 1 ∪ Π n 1 , and moreover, any set x ⊆ ω , x ∈ Σ n 1 , is constructible and Σ n 1 in L . 2. There exists a generic extension L in which it is true that there is a nonconstructible Δ n + 1 1 set a ⊆ ω , but all Σ n 1 sets x ⊆ ω are constructible and even Σ n 1 in L , and in addition, V = L [ a ] in the extension. 3. There exists an generic extension of L in which there is a nonconstructible Σ n + 1 1 set a ⊆ ω , but all Δ n + 1 1 sets x ⊆ ω are constructible and Δ n + 1 1 in L . Thus, nonconstructible reals (here subsets of ω ) can first appear at a given lightface projective class strictly higher than Σ 2 1 , in an appropriate generic extension of L . The lower limit Σ 2 1 is motivated by the Shoenfield absoluteness theorem, which implies that all Σ 2 1 sets a ⊆ ω are constructible. Our methods are based on almost-disjoint forcing. We add a sufficient number of generic reals to L , which are very similar at a given projective level n but discernible at the next level n + 1 .


Introduction
Problems of definability and effective construction of mathematical objects have always been in the focus of attention during the development of mathematical foundations. In particular, Hadamard, Borel, Baire, and Lebesgue, participants of the discussion published in [1], in spite of significant differences in their positions regarding problems of mathematical foundations, emphasized that a pure existence proof and a direct definition (or an effective construction) of a mathematical object required are different mathematical results, and the second one of them does not follow from the first. Problems of definability and effectivity are considered in such contemporary monographs on foundations as [2][3][4][5]. Moschovakis, one of founders of modern set theory, pointed in [6] (p. xiv), that the central problem of descriptive set theory and definability theory in general [is] to find and study the characteristic properties of definable objects.
The general goal of the research line of this paper is to explore the existence of effectively definable structures in descriptive set theory on specific levels of the projective hierarchy. One of the directions here is the construction of set theoretic models, in which this or another problem is decided, at a predefined projective level n, differently than it is decided in L, Gödel's constructible universe, or, that is equivalent, by adding the axiom of constructibility, dubbed V = L.
Such set theoretic models are usually defined as generic extensions of L itself. Any such a generic extension leads to consistency and independence results in set theory, because if a sentence Φ holds in L or in a generic extension of L then Φ is consistent with the axioms of ZFC, the Zermelo-Fraenkel set theory (with the axiom of choice AC).
As a first, and perhaps most immediately interesting problem of this sort, in this paper, we consider the problem of the existence of effectively definable (that is, occurring in one of lightface classes Σ 1 n of the projective hierarchy) but nonconstructible reals. It follows from Shoenfield's absoluteness theorem [7] that every (lightface) Σ 1 2 set x ⊆ ω belongs to L. Generic models, in which there exist nonconstructible reals on effective levels of the projective hierarchy higher than Σ 1 2 , were defined in the early years of forcing; see a brief account in [8]. This culminated in two different generic extensions [9,10] containing a nonconstructible Π 1 2 singleton, hence, a ∆ 1 3 set a ⊆ ω . (We are concentrated on generic extensions of L in this paper, and therefore leave aside another research line, related to models with large cardinals, with many deep and fruitful results connected, in particular, with properties of Π 1 2 singletons, see e.g., [11][12][13]). Then it was established in [14] that for any n ≥ 2 there is a generic extension of L in which there exists a nonconstructible ∆ 1 n+1 real a ⊆ ω , but all Σ 1 n sets x ⊆ ω are constructible. Our motivation here is to further extend this research line. The next three theorems are the main results in this paper. Theorem 1. If n ≥ 2 and b ⊆ ω , b / ∈ Σ 1 n ∪ Π 1 n , then there is a generic extension of L in which b ∈ ∆ 1 n+1 but still b / ∈ Σ 1 n ∪ Π 1 n , and moreover, any set x ⊆ ω , x ∈ Σ 1 n , is constructible and Σ 1 n in L.
Theorem 1 shows that being at a certain lightface projective level is hardly an intrinsic property of a constructible real, unless it is already at that level in L. The theorem definitely fails for n = 1 since being ∆ 1 2 is an ablosute property of a real by the Shoenfield absoluteness theorem.
Theorem 2. If n ≥ 2, then there exists a generic extension of the universe L in which it is true that (i) there is a nonconstructible ∆ 1 n+1 set a ⊆ ω , but all Σ 1 n sets x ⊆ ω are constructible and Σ 1 n in L ; (ii) we can strengthen (i) by the requirement that V = L[a] in the extension. Theorem 3. If n ≥ 2 then there exists an extension of L in which there is a nonconstructible Σ 1 n+1 set a ⊆ ω but all ∆ 1 n+1 sets x ⊆ ω are constructible and ∆ 1 n+1 in L.
The common denominator of Theorems 2 and 3 is that nonconstructible reals can first appear at a given lightface projective class strictly higher than Σ 1 2 , in an appropriate generic extension of L. The lower limit Σ 1 2 is motivated by the Shoenfield absoluteness theorem. The generic models, which we define to prove the main theorems, make use of modifications of the almost-disjoint forcing by Jensen-Solovay [9].
Some other recent results can be mentioned here, which resemble Theorems 1-3 in that they give models in which a particular property of some kind holds at a certain pre-selected level of the projective hierarchy. Yet they are different in that they use modifications of Jensen's minimal Π 1 2 singleton forcing [10] and its finite-support products first considered by Enayat [15], as well as its collapse-style modification by Abraham [16], rather than the almost-disjoint forcing. • A model defined in [17], in which, for a given n ≥ 2, there is a (lightface) Π 1 n Vitali equivalence class in the real line R (that is, a set of the form x + Q in R ), containing no OD (ordinal definable) elements, and in the same time every countable Σ 1 n set consists of OD elements. • A model in [18], in which, for a given n ≥ 2, there is a Π 1 n singleton {a}, such that a codes a collapse of ω L 1 , and in the same time every Σ 1 n set a ⊆ ω is still constructible.
• A model defined in [19], in which, for a given n ≥ 2, there is a Π 1 n non-OD-uniformizable planar set with countable cross-sections, and at the same time, every Σ 1 n set with countable cross-sections is OD-uniformizable.

Organization of the Paper
Our plan of the proofs of the main results will be to construct, in L, a sequence of forcing notions P(ν), ν < ω 1 , satisfying the following three key conditions. 1. P(ν) are sufficiently homogeneous and independent of each other in the sense that, for any ν 0 , there are no P(ν 0 )-generic reals in a (∏ ν =ν 0 P(ν))-generic extensions of L. 2. The property of a real x being P(ν)-generic over L is Π 1 n as a binary relation, where n ≥ 2 is a number chosen in Theorems 1-3. 3. A condition which makes P(ν)-generic reals for different values ν < ω 1 undistinguishable from each other below the Π 1 n definability level (at which they are distinguishable by condition 2).
Each P(ν) will be a forcing notion of almost-disjoint type, determined by a set U(ν) ⊆ ω ω . To make the exposition self-contained, we review some basic details related to almost-disjoint forcing, finite-support products, and related generic extensions, taken mainly from [9], in Sections 2 and 3.
Having the construction of P(ν), ν < ω 1 , accomplished in Section 4, the proof of, e.g., Theorem 1 (Section 7.1) is performed as follows. Let b ∈ L, b ⊆ ω be chosen as in Theorem 1 for a given n ≥ 2. We consider a P-generic extension L[G] of L, where P = ∏ i<ω P(i). Let a i ⊆ ω be the P(i)-generic real generated by the i th projection G i of G ; these reals are nonconstructible and Then it is true in L[{a i } i∈z ] by condition 1, that b = {k < ω : there exist P(2k)-generic reals} = {k < ω : there are no P(2k + 1)-generic reals} , so using condition 2, we easily get b ∈ ∆ 1 n+1 in L[{a i } i∈z ]. A similar construction (but with b being generic over L) was carried out in the early years of forcing in [9] for n = 2, which is the least possible value. In the case n = 2, the fact, that all Σ 1 2 sets x ⊆ ω in the extension belong to L and are Σ 1 2 in L, is guaranteed by the Shoenfield absoluteness theorem.
If n ≥ 3, then the Shoenfield absoluteness argument does not work, of course. Still we can argue that any lightface Σ 1 n set x ⊆ ω in L[{a i } i∈z ] belongs to L by the general forcing theory, because the product forcing P z = ∏ i∈z P(i) ∈ L is homogeneous by condition 1. However this does not immediately imply the lightface definability of b in L, as P z is defined via z, hence via b. To solve this difficulty, we make use of condition 3 to prove another absoluteness property: Σ 1 n formulas turn out to be absolute between L[{a i } i∈z ] and the entire extension L[G] = L[{a i } i<ω ], which is an P-generic extension of L. Here P = ∏ i<ω P(i) is a parameter-free definable forcing in L, leading to the parameter-free definability of b in L. There are two issues here that need to be explained.
First, how to secure condition 3 in a sufficiently effective form. To explain the main technical device, we recall that by [9] the system of forcing notions P(ν) is the result of certain transfinite ω 1 -long construction of assembling it from countable fragments in L. The construction can be viewed as a maximal branch in a certain "mega-tree", say T , whose nodes are such countable fragments, and each of them is chosen to be the Gödel-least appropriate one over the previous one. The complexity of this construction is ∆ 1 2 in the codes, leading in [9] to the Π 1 2 definability of the property of being generic, as in condition 2, in case n = 2.
To adapt this construction for the case n ≥ 3, our method requires us to define a maximal branch in T that intersects all dense sets in T of class Σ 1 n−1 . Such a construction is carried out in Section 4. This genericity-like condition of meeting all dense Σ 1 n−1 sets, results in the Π 1 n definability of the property of being generic in condition 2, and also yields condition 3, since the abundance of order automorphisms of the "mega-tree" T (including those related to index permutations) allows to establish some homogeneity properties of a certain auxiliary forcing-style relation.
This auxiliary forcing-style relation, defined and studied in Sections 5 and 6. The auxiliary relation approximates the truth in P -generic extensions, as L[{a i } i∈z ] above, up to Σ 1 n formulas, but, unlike the ordinary P -forcing relation, is sufficiently homogeneous. In particular, it helps to obtain the mentioned absoluteness property. This will allow us to accomplish the proof of the main results, Theorem 1 together with part (i) of Theorem 2 in Section 7, part (ii) of Theorem 2 in Section 8, Theorem 3 in Section 9. The flowchart can be seen in Figure 1.
The flowchart can be seen in Figure 1. And we added the index and contents as Supplementary Materials for easy reading. ALMOST  CONCLUSION, SOME FURTHER RESULTS, Section 10
• card X is the cardinality of a set X, equal to the number of elements of X in case X is finite. • dom P = {x : ∃ y ( x, y ∈ P)} and ran P = {y : ∃ x ( x, y ∈ P)} -the domain and range of any set P that consists of pairs. • In particular if P = f is a function then dom f and ran f are the domain and the range of f . • Functions are identified with their graphs: if P = f is a function then X <ω is the set of all strings (finite sequences) of elements of a set X.

•
In particular ω <ω is the set of strings of natural numbers. • lh s < ω is the length of a string s. • s x is the string obtained by adjoining x as the rightmost term to a given string s. • s ⊂ t means that the string t is a proper extension of s. • ∅ = Λ is resp. the empty set and the empty string. • ω ω is the Baire space.

Almost Disjoint Forcing
In this section, we review basic definitions and results related to almost disjoint forcing, as well as some rarely used results related, for instance, to symmetries of almost disjoint forcing notions. Assumption 1. In this paper, we assume that L is the ground universe. Thus all forcing notions are defined in L while all generic extensions are those of L. (In fact many intermediate results remain true w. r. t. any ground universe.)

Almost Disjoint Forcing
We present this forcing in a form based on the fact that the set Fun of all functions f : ω → ω is almost disjoint in the sense that if f = g belong to Fun then the infinite sets { f m : m ∈ ω} and {g m : m ∈ ω} of finite strings have a finite intersection. Definition 1. Seq = ω <ω {Λ} = all finite non-empty strings of natural numbers. A recursive enumeration ω <ω = {s k : k ∈ ω} is fixed, such that s 0 = Λ, the empty string, and s k ⊆ s =⇒ k . Thus Seq = ω <ω {Λ} = {s k : k ≥ 1}. For any s = s k , we let num s = k ; in particular num Λ = 0. Fun = ω ω = all infinite sequences of natural numbers. A set X ⊆ Fun is dense iff for any s ∈ Seq there is f ∈ X such that s ⊂ f . Let S ⊆ Seq, f ∈ Fun. If the set S/ f = {n : f n ∈ S} is infinite then we say that S covers f , otherwise S does not cover f .
We underline that Λ, the empty string, does not belong to Seq. Given a set u ⊆ Fun in the ground universe, the general goal of almost disjoint forcing is to find a generic set S ⊆ Seq such that the equivalence holds for each f ∈ Fun in the ground universe. This goal will be achieved by a forcing P[u] introduced in Definition 4. In fact P[u] will be a part, determined by u, of a common reservoir P * .
Definition 2. P * is the set of all pairs p = S p ; F p of finite sets F p ⊆ Fun, S p ⊆ Seq. Elements of P * will sometimes be called (forcing) conditions. If p ∈ P * then put F ∨ p = { f n : f ∈ F p ∧ n ≥ 1}. The set F ∨ p is an infinite (or else F ∨ p = F p = ∅) tree in Seq, without terminal nodes.
Definition 3 (order). Let p, q ∈ P * . We define q ≤ p (q is stronger) iff S p ⊆ S q , F p ⊆ F q , and the difference S q S p does not intersect F ∨ p , that is, Thus any condition p ∈ P * is a pair that consists of a "finite" component S p and an "infinite" component F p . Either of the components is a finite set (possibly, empty), but S p consists of finite strings of integers while F p consists of infinite sequences of integers that will be called functions (from ω to ω ). Both components of a stronger condition q, naturally, increase, but strings t ∈ S q S p must satisfy t / ∈ F ∨ p -in other words, t is not a substring of any function (infinite sequence) f ∈ F p . If p ∈ P * then both ∅; F p and S p ; ∅ belong to P * and p ≤ S p ; ∅ , but p ≤ ∅; F p may fail. In fact p ≤ ∅; Lemma 1. Conditions p, q ∈ P * are compatible in P * iff 1) S q S p does not intersect F ∨ p , and 2) S p S q does not intersect F ∨ q . Therefore, any p, q ∈ P * are compatible in P * iff p ∧ q ≤ p and p ∧ q ≤ q.
Proof. The pair p ∧ q = S p ∪ S q ; F p ∪ F q is a condition in P * . Moreover if 1) and 2) hold then we have p ∧ q ≤ p and p ∧ q ≤ q, thus p, q are compatible. Now let us introduce a relativized version of P * . The parameter of relativization will be an arbitrary set u ⊆ Fun served as a reservoir of functions allowed to occur in sets F p .
. Thus in this case if conditions p, q are compatible in P * then they are compatible in P[u], too. Therefore, we will say that conditions p, q ∈ P * are compatible (or incompatible) without an indication which set P[u] containing both conditions is considered. Lemma 2. If u ⊆ Fun then P[u] is a ccc forcing.
Proof. If S p = S q then p and q are compatible by Lemma 1. However there are only countably many sets of the form S p .

Almost-Disjoint Generic Extensions
Fix, in L, a set u ⊆ Fun and consider a P[u]-generic extension L[G] of the ground (constructible by Assumption 1) set universe L, obtained by adjoining a P[u]-generic set G ⊆ P[u]. Put S G = p∈G S p ; thus S G ⊆ Seq. The next lemma reflects the idea of almost-disjoint forcing: elements of u are distinguished by the property of S G not covering f in the sense of Definition 1.

Lemma 3.
Suppose that u ⊆ Fun in the universe L, and G ⊆ P[u] is a set P[u]-generic over L. Then Define p so that F p = F q and S p = S q ∪ { f m}. Then p ∈ D f l and p q.) Let p ∈ D f l ∩ G . Then sup(S G / f ) > l . As l is arbitrary, S G / f is infinite.
(iii) Consider any p ∈ P [u]. Suppose that p ∈ G . Then obviously s p ⊆ S G . If there exists s ∈ (S G S p ) ∩ F ∨ p then by definition we have s ∈ S q for some q ∈ G . However, then p, q are incompatible by Lemma 1, a contradiction. Now suppose that p / ∈ G . Then there exists q ∈ G incompatible with p. By Lemma 1, there are two cases. First, there exists s ∈ (S q S p ) ∩ F ∨ p . Then s ∈ S G S p , so p is not compatible with S G . Second, there exists s ∈ (S p S q ) ∩ F ∨ q . Then any condition r ≤ q satisfies s / ∈ S r . Therefore s / ∈ S G , so S p ⊆ S G , and p is not compatible with S G . (

Lipschitz Transformations
Let Lip be the group of all ⊆-automorphisms of Seq; these transformations may be called Lipschitz by obvious association. Any λ ∈ Lip preserves the length lh of finite strings, that is, lh s = lh (λ · s) for all s ∈ Seq. Define the action of any transformation λ ∈ Lip on: Lemma 4 (routine). The action of any λ ∈ Lip is an order-preserving automorphism of P * . If u ⊆ Fun and p ∈ P[u] then λ · p ∈ P[λ · u].

Lemma 5.
Suppose that u, v ⊆ Fun are countable sets topologically dense in Fun, and p ∈ P[u], q ∈ P[v]. Then there is λ ∈ Lip and conditions p ∈ P[u], p ≤ p and q ∈ P[v], q ≤ q, such that λ · u = v, and λ · p = q -therefore conditions λ · p and q are compatible in P[v].
Proof. Put bas r = {s(0) : s ∈ S r } ∪ { f (0) : f ∈ F r } for any r ∈ P * ; bas r ⊆ ω is finite. Let M < ω satisfy bas p ∪ bas q ⊆ M . Because of density, for any i < M there exist f i ∈ u and g i ∈ u such that f i (0) = i and g i (0) = M + i . For any f = g ∈ Fun, let N( f , g) be the largest n with f n = g n. We will define enumerations u = { f k : k < ω} and u = {g k : k < ω}, without repetitions, which agree with the above definition for k < M and satisfy N( f k , f l ) = N(g k , g l ) for all k , l , and g k (0) = f k (0) for all k ≥ M . As soon as this is accomplished, define λ ∈ Lip as follows. Consider any s ∈ Seq of length m = lh s. As u is dense, s = f k m for some k . Put λ(s) = g k m. Clearly λ · u = u , and in particular λ · f k = g k for all k , and hence Now to define q put r = λ · p. Then r ∈ P[v], and bas r = β · bas p ⊆ ω M by ( * ), since bas p ⊆ M . Therefore, bas r ∩ bas q = ∅ because bas q ⊆ M as well. It follows that conditions r and q are compatible in P[v], and hence condition q = r ∧ q (that is, S q = S r ∪ S q and X q = X r ∪ X q ) belongs to P[v], and obviously q ≤ q. Pretty similarly, to define q, we put r = λ −1 · q ∈ P[u], thus bas r ⊆ ω M , bas r ∩ bas p = ∅, conditions r , p are compatible, condition p = p ∧ r (that is, S p = S p ∪ S r and X p = X p ∪ X r ) belongs to P[u], and p ≤ p. Note that q = λ · r and r = λ · p by construction. It follows that q = r ∧ q = λ · (p ∧ r) = λ · p , as required.
To define f k and g k by induction, suppose that k ≥ M , f 0 , . . . , f k−1 and g 0 , . . . , g k−1 are defined, and N( f i , f j ) = N(g i , g j ) holds in this domain. Consider any next function f ∈ u { f 0 , . . . , f k−1 }, and let it be f k . There are functions g ∈ Fun satisfying N( f j , f k ) = N(g j , g) for all j < k . This property of g is determined by a certain finite part g m. By the density the set v contains a function g of this type. Let g k be any of them. In the special case when N( f j , f k ) = 0 for all j < k (then k ≥ 2M ), we take any g k ∈ v satisfying N( f j , f k ) = 0 for all j < k and g k (0) = f k (0).

Substitution Transformations
The next lemma (Lemma 6) will help to prove that the forcing notions considered are sufficiently homogeneous. Assume that p, q ∈ P * satisfy the following requirement: We define a transformation H p q acting as follows. Let p ∈ P * , p ≤ p. Then by definition S p ⊆ S p , F p ⊆ F p , and S p ∩ F ∨ p = S p (by (2)). We put H p q (p ) = q := S q , F q , where F q = F p and S q = (S p S p ) ∪ S q . Thus the difference between S q and S p lies entirely within the set F ∨ p = F ∨ q , and in particular S q has S q there while S p has S p there.

Almost Disjoint Product Forcing
Here we review the structure and basic properties of product almost-disjoint forcing over L and corresponding generic extensions of L. In order to support various applications, we make use of ω 1many independent forcing notions.

Product Forcing, Systems, Restrictions
We begin with ω 1 -products of P * after which we consider more complicated forcing notions. Definition 5. Let I = ω 1 . This is the index set for the forcing products considered below. Let P * be the product of I copies of the set P * (Definition 2), with finite support. That is, P * consists of all functions p : |p| → P * such that the set |p| = dom p ⊆ I is finite.
Note that writing p c or U c, it is not assumed that c ⊆ |p|, resp., c ⊆ |U|.

Regular Forcing Notions
Unfortunately, product forcing notions of the form P[U] (U being a system in L) do not provide us with all the definability effects we need. We will make use of certain more complicated forcing notions K ⊆ P * in L. To explain the idea, let a system U ∈ L satyisfy |U| = ω . Let G ⊆ P[U] be generic over L. The sets S G (ν) = S G(ν) = p∈G S p (ν) ⊆ Seq then belong to L[G], and in fact The first idea is to make use of U c, but oops, clearly c / ∈ L, and consequently U c / ∈ L and P[U c] / ∈ L, so that many typical product forcing results do not apply in this case. The next definition attempts to view the problem from another angle.

Definition 8 (in L).
A set K ⊆ P * is called a regular subforcing if: (1) if conditions p, q ∈ K are compatible then p ∧ q ∈ K ; (2) if p, q ∈ K then p |q| ∈ K -but it is not assumed that p ∈ K necessarily implies p c ∈ K for an arbitrary c ⊆ |p|; (3) if p, q ∈ P * , q ≤ p, and |q| = |p| exactly, then p ∈ K implies q ∈ K ; (4) for any condition p ∈ P * , there exist: a condition p * ∈ P * and a set d ⊆ |p * | such that p * ≤ p, F p * (ν) = F p (ν) for all ν ∈ |p|, F p * (ν) = ∅ for all ν ∈ |p * | |p|, p * d ∈ K , and every condition q ∈ K , q ≤ p * d, satisfies |q| ∩ |p * | = d, and hence q is compatible with p * and with p.
In this case, if U is a system then define K[U] = K ∩ P[U]. In particular, if simply K = P * then P * [U] = P * ∩ P[U] = P[U]. Example 1 (trivial). If c ⊆ I in the ground universe L, then P * c is a regular forcing. To prove (4) of Definition 8 let p * = p and d = |p| ∩ c.
Example 2 (less trivial). Consider the set K of all conditions p ∈ P * such that |p| ⊆ ω and if ν ∈ |p|, ν ≥ 1, then s ν ∈ S p (0). We claim that K is a regular subforcing.
Theorem 4 (in L). The partially ordered set P * , and hence each P[U], and generally each regular subforcing of P[U] (for any system U ) satisfies CCC (countable antichain condition).
Proof. Suppose towards the contrary that A ⊆ P * is an uncountable antichain. We may assume that there is m ∈ ω such that |p| = m for all p ∈ A. Applying the ∆-lemma argument, we obtain an uncountable set A ⊆ A and a finite set w ⊆ I with card w < m strictly, such that |p| ∩ |q| = w for all p = q in A . Then A = { p w : p ∈ A } is still an uncountable antichain, with |p| = w for all p ∈ A , easily leading to a contradiction (see the proof of Lemma 2).
Lemma 7 (in L). If K ⊆ P * is a regular forcing and U is a system then K[U] = K ∩ P[U] is a regular subforcing of P[U].
To show how (4) of Definition 8 works, we prove Lemma 8 (in L). If U is a system and K ⊆ P[U] is a regular subforcing of P[U] then any set D ⊆ K pre-dense in K remains pre-dense in P[U].
Proof. Consider any p ∈ P[U]. Let p * ∈ P[U] and d ⊆ |p * | satisfy (4) of Definition 8. In particular, p * ≤ p and p * d ∈ K . By the pre-density, there is a condition q ∈ D compatible with p * d. Then by (1) of Definition 8 there is a condition r = q ∧ (p * d) ∈ K such that r ≤ q and r ≤ p * d. Then r is compatible with p by the choice of p * and d.

Outline of Product and Regular Extensions
We consider sets of the form P[U], U being a system in L, as well as regular subforcings K ⊆ P[U], as forcing notions over L. Accordingly, we will study P[U]-generic and K-generic extensions L[G] of the ground universe L. Define some elements of these extensions.
If U is a system in L, the ground universe, then any P[U]-generic set G ⊆ P[U] splits into the family of sets G(ν), ν ∈ I , and each G(ν) is P[U(ν)]-generic. Lemma 9. Let U be a system and K ⊆ P[U] be a regular subforcing in the ground universe L. Let G ⊆ P[U] be a set P[U]-generic over L. Then: Proof. (ii) This follows from Lemma 8.
Let us show that G ∩ K = (G c) ∩ K ; this proves G ∩ K ∈ L[G c]. Indeed if p ∈ G ∩ K then by definition |p| ⊆ c = |G ∩ K|, therefore p ∈ G c, as required.
(iv) This is clear since we have G ∩ K = G =ν ∩ K in the case considered. (i) First of all, G = ∏ ν G(ν) by the product-forcing theorem. Then, each G(ν) is recovered from the associated S G (ν) by means of a simple uniform formula, see the proof of Lemma 3(i).

Names for Sets in Product and Regular Extensions
For any set X we let N X be the set of all P * -names for subsets of X . Thus N X consists of all sets τ ⊆ P * × X . Let SN X (small names) consist of all at most countable names τ ∈ N X .
We define dom τ = { p : ∃ x ( p, x ∈ τ)}, |τ| = {|p| : p ∈ dom τ} for any name τ . Say that a name τ is below a given p ∈ P * if all p ∈ dom τ satisfy p ≤ p. For any set K ⊆ P * , we let N X (K) be the set of all names τ ∈ N X such that dom τ ⊆ K , and accordingly SN X (K) = N X (K) ∩ SN X (small names). In particular, we'll consider such sets of names as SN X (P[U]) and SN X (P[U] c). Names in N X (K) for different sets X will be called K-names. Accordingly, names in SN X (K) for different sets X will be called small K-names.

Example 3 (some names). Let
∈ P * be the empty condition, that is, | | = ∅. This is the weakest condition in any P[U]. If X is a set in the ground universe thenX = { , x : x ∈ X} is a K-name for any regular forcing K ⊆ P * , andX[G] = X for any set G containing .
We will typically use breve-names likeX for sets in the ground universe, and dot-names (like . x ) for sets in generic extensions.
Suppose that K ⊆ P * . Let G = { p, p : p ∈ K}. (In principle, G depends on K but this dependence will usually be suppressed.) Clearly G ∈ N K (K) (but G / ∈ SN K (K) unless K is countable), and in addition G[G] = G for any ∅ = G ⊆ K . Thus G is a name for the generic set G ⊆ K .
Similarly, G c = { p, p : p ∈ K c} (c ⊆ I ) is a name for G c (see Definition 9).

Names for Functions
For any sets X , Y let N X Y be the set of all P * -names for functions X → Y ; it consists of all τ ⊆ P * × (X × Y) such that the sets τ " x, y = { p : p, x, y ∈ τ} satisfy the following requirement: if y = y , p ∈ τ " x, y , p ∈ τ " x, y , then p, p are incompatible.
Let dom τ = x,y τ " x, y and |τ| = {|p| : p ∈ dom τ}. As above, SN X Y consists of all at most countable names τ ∈ N X Y . For any set K ⊆ P * , we let N X Y (K) be the set of all names τ ∈ N X Y such that dom τ ⊆ K , and accordingly SN X x, y is pre-dense in K for any x ∈ X . A name τ ∈ N X Y (K) is K-full below some p 0 ∈ K , iff all sets τ "x are pre-dense in K below p 0 , that is, any condition q ∈ K , q ≤ p 0 , is compatible with some r ∈ τ x (and this holds for all x ∈ X ).
Note that N X Y (K) ⊆ N X×Y (K), and accordingly SN X Y (K) ⊆ SN X×Y (K). Thus all names in N X Y (K) and in SN X Y (K) are still K-names in the sense above.

Corollary 1 (of Lemma 8, in L).
If U is a system, K ⊆ P[U] is a regular subforcing, X, Y any sets, and τ is a name in N X Y (K), then τ is K-full (resp., K-full below p ∈ K ) iff τ is P[U]-full (resp., P[U]-full below p ).
Suppose that τ ∈ N X Y . Call a set G ⊆ P * minimally τ-generic iff it is compatible in itself (if p, q ∈ G then there is r ∈ G with r ≤ p, r ≤ q), and intersects each set of the form τ "x , x ∈ X . In this case put Y occur (for various sets X, Y ), and a set G ⊆ P * is minimally τ-generic for any name τ in ϕ, then accordingly ϕ[G] is the result of substitution of τ[G] for each name τ in ϕ.
Claim 1 (obvious). Suppose that, in L, X, Y are any sets, p ∈ K ⊆ P * and τ ∈ N X Y (K) is K-full (resp., K-full below p ). Then, any set G ⊆ K , K-generic over L (resp., K-generic over L and containing p ), is minimally τ-generic.
Definition 11 (equivalent names). Names τ, µ ∈ SN ω ω (P * ) are called equivalent iff conditions q, r are incompatible whenever q ∈ τ " m, j and r ∈ µ " m, k for some m and j = k . (Recall that τ " m, k = { p : p, m, k ∈ τ}.) Similarly, names τ, µ are equivalent below some p ∈ P * iff the triple of conditions p, q, r is incompatible (that is, p ∧ q ∧ r is not ≤ than at least one of p, q, r ) whenever q ∈ τ " m, j and r ∈ µ " m, k for some m and j = k .
Claim 2 (obvious). Suppose that, in L, p ∈ K ⊆ P * , and names µ, τ ∈ SN ω ω (K) are equivalent (resp., equivalent below p ). Then, for any G ⊆ K both minimally µ-generic and minimally τ-generic (resp., and a countable antichain, and, for any p ∈ A, τ p ∈ SN ω ω (K) is a name K-full below p 0 . Then there is a K-full name τ ∈ SN ω ω (K), equivalent to τ p below p for any p ∈ A.
Proof. Let B be a maximal (countable) antichain in the set of all conditions q ∈ K incompatible with p 0 . Then A ∪ B is a countable maximal antichain in K . We let τ consist of: 1) all triples r ∧ q, k, m , such that q ∈ A and r, k, m ∈ τ q , and 2) all triples q, k, 0 , such that q ∈ B and m ∈ ω .

Names and Sets in Generic Extensions
For any forcing P, let || − − P denote the P-forcing relation over L as the ground model.

Theorem 5.
Suppose that U is a system and K ⊆ P[U] a regular subforcing in L. Let G ⊆ K be a set Kgeneric over L. Then: (i) if p ∈ K and ϕ is a closed formula with K-names as parameters, then (ii) It follows from general forcing theory that there is a K-full name σ ∈ N X Y (K), not necessarily countable, such that f = σ [G]. Then all sets Q A is countable and pre-dense in K below p. Apply Lemma 10 to get a name τ as required.  It follows by the definition of K that the set |G | = p∈G |p| satisfies |G | ⊆ ω , contains 0, and if ν ≥ 1 then ν ∈ |G | iff s ν ∈ S G (0). Therefore, by Lemma 9 and Theorem 5, the sets G(0) and S G (0) belong to

Transformations Related to Product Forcing
There are three important families of transformations of the whole system of objects related to product forcing. Two of them are considered in this Subsection.

Lemma 12.
If α ∈ Lip I then p −→ π · p is an order preserving bijection of P * onto P * , and if U is a system then we have p ∈ P Corollary 2 (of Lemma 5). Suppose that U, V are countable systems, |U| = |V|, and p ∈ P[U], q ∈ P[V]. Then there is a transformation α ∈ Lip I such that conditions α · p and q are compatible in P[V].
Suppose that U , p, q ∈ P[U], H p q are as in Theorem 6. Extend the action of H p q onto names and formulas. Recall that a name τ ∈ N X Y is below p iff p ≤ p holds for any triple p , n, k ∈ τ .
• If X , Y are any sets and τ ∈ N X Y is a name below p then put If ϕ is a formula with names below p as parameters then H p q (ϕ) denotes the result of substitution of H p q (τ) for any name τ in ϕ.
Forcing notions of the form P[U] are quite homogeneous by Theorem 6. The next result is a usual product forcing application of such a homogeneity.
We can w. l. o. g. assume that p, q satisfy (3) above (otherwise extend p, q appropriately). Define P, Q, H p q as in Definition 12 and Theorem 6.
Let G ⊆ Q be a generic set containing p. Assuming w. l. o. g. that G ⊆ P, the set H = { H p q (p ) : p ∈ G} ⊆ Q will be generic as well by Theorem 6, and q ∈ H . Therefore ϕ [G] is true in since G e, H e can be obtained from each other via maps coded in L. This is a contradiction.
(iii) This is a particular case.

Corollary 3.
Under the assumptions of Theorem 7, suppose that X , Y are arbitrary sets in L, p ∈ Q, Proof. We can assume that |p| ⊆ d by Theorem 7(iii), thus p = p d ∈ K . It follows from Theorems 5(ii) and 7(i) that there exist: a (countable) antichain A ⊆ K maximal below p, and, for any , such that every q ∈ A forces τ = τ q , as in the proof of Theorem 5(iv).

Basic Forcing Notion and Basic Generic Extension
The proofs of Theorems 1-3, that follow in Sections 7-9, will have something in common. Namely the generic extensions we employ to get the results required will be parts of a basic extension, introduced and studied in this section. To define the extension, we'll define (in L as the ground universe) an increasing sequence { M ξ , U ξ } ξ<ω 1 of pairs of certain type-a Jensen-Solovay sequence, since this construction goes back to [9]-and make use of a forcing notion of the form P[U], where U = ξ<ω 1 U ξ . It turns out that if such a sequence is n-complete, in sense that it meets all sets of n-complexity within the whole tree of possible constructions, then the truth of analytic formulas up to level n in corresponding generic extensions has a remarkable connection with the forcing approximations studied in Section 5. This will allow us to convert the homogeneity of the construction of Jensen-Solovay sequences into a uniformity of the corresponding generic extensions, expressed by Theorem 13. Recall that V = L assumed in the ground universe by Assumption 1.

Jensen-Solovay Sequences
If U V are systems then by definition P[U] ⊆ P[V] holds. However this is not necessarily a suitably good notion. For instance a dense set X ⊆ P[U] may not be pre-dense in P[V], thus if G ⊆ P[V] is a generic set then the "projection" G ∩ P[U] is not necessarily P[U]-generic. Yet there is a special type of extension of systems, introduced by Jensen and Solovay [9], which preserves the density. This method is based on the requirement that the functions in Fun that occur in V but not in U must be generic over a certain model that contains U .
Recall that ZFC − is ZFC minus the Power Set axiom, see Section 5.1 below. Let ZFC − 1 be ZFC − plus the axioms V = L and "every set is at most countable".
Let sJS, small pairs, consist of all M, U ∈ JS such that M |= ZFC − 1 and M (then U as well) is countable. Define the extension relations: It would be a vital simplification to get rid of M as an explicit element of the construction, e.g., by setting U * U iff U U and there is a CTM M containing U and such that U M U . Proof. Prove that the set F = ν∈|U| (U (ν) U(ν)) is multiply Cohen generic over M . Consider a simple case when f ∈ U (ν) U(ν) and g ∈ U (µ) U (µ), where ν, µ ∈ |U|, and prove that f , g is Cohen generic over M . (The general case does not differ much.) By definition, f is Cohen generic over M and g is Cohen generic over M . Therefore, g is Cohen generic over It remains to apply the product forcing theorem.
Proof. The same idea as in the proof of Lemma 13.

Stability of Dense Sets
Assume that M, U ∈ sJS and D is a pre-dense subset of P[U] (say, a maximal antichain). If U is another system satisfying U U , then it may well happen that D is not maximal in P[U ]. The role of the multiple genericity requirement (a) in Definition 13, first discovered in [9], is to somehow seal the property of pre-density of sets already in M for any further extensions. This is the content of the following key theorem. The product forcing arguments allow us to extend the stability result to pre-dense sets not necessarily in M , as in items (ii), (iii) of the following theorem.
Theorem 8. Assume that, in L, M, U ∈ sJS, U is a disjoint system, and U M U . If D is a pre-dense subset of P[U] (resp., pre-dense below some p ∈ P[U] ) then D remains pre-dense in P[U ] (resp., pre-dense in P[U ] below p ) in each of the following three cases: Proof. We consider only the case of sets D pre-dense in P[U] itself; the case of pre-density below some p ∈ P[U] is treated similarly.
(i) Suppose, towards the contrary, that a condition p ∈ P[U ] is incompatible with each q ∈ D . As D ⊆ P[U], we can w. l. o. g. assume that |p| ⊆ |U|.
Our plan is to define a condition p ∈ P[U], also incompatible with each q ∈ D , contrary to the pre-density. To maintain such a construction, consider the finite string f = f 1 , . . . , f m of all elements f ∈ Fun occurring in ν∈|p| F p (ν) but not in U . It follows from U M U that f is Cohen-generic over M . Further analysis shows that p being incompatible with D is implied by the fact that f meets a certain M -countable family of Cohen-dense sets. Therefore, we can simulate this in M , getting a string g ∈ M which meets the same Cohen-dense sets, and hence yields a condition p ∈ P[U], also incompatible with each q ∈ D .
This argument was first carried out in [9] in full generality, where we address the reader. However, to present the key idea in sufficient detail in a somewhat simplified subcase, we assume that (1) To make the argument even more transparent, we suppose that (2) X = { f , g}, where f = g and the pair f , g is Cohen generic over M . (The general case follows the same idea and can be found in [9]; we leave it to the reader.) Thus The plan is to replace the functions f , g by some functions f , g ∈ U(ν) so that the incompatibility of p with conditions in D will be preserved.
It holds by the choice of p and Lemma 1 that and D 1 depends on f , g via F p (ν). (See Section 3.1 on notation.) The equality Note that each A ∈ A is a finite subset of Seq, so we can reenumerate A = { A k : k < ω} in M and rewrite ( †) as follows: As the pair f , g is Cohen-generic, there is a number m 0 such that ( ‡) is forced over M by σ 0 , τ 0 , where σ 0 = f m 0 and τ 0 = g m 0 . In other words, A k ∩ S( f , g ) = ∅ holds for all k whenever f , g is Cohen-generic over M and σ 0 ⊂ f , τ 0 ⊂ g . It follows that for any k and strings σ, τ ∈ Seq extending resp. σ 0 , τ 0 there are strings σ , τ ∈ Seq extending resp. σ, τ , at least one of which extends one of w ∈ A k . This allows us to define, in M , a pair of f , g ∈ Fun such that σ 0 ⊂ f , τ 0 ⊂ g , and for any k at least one of f , g extends one of w ∈ A k . In other words, we have which contradicts the pre-density of D .
(ii) The above proof works with M[G] instead of M since the set X as in the proof is multiple Cohen generic over M[G] by the product forcing theorem.
is a Cohen generic extension of M . Following the above, let ν ∈ |U|, ν = ν 0 . By the definition of the set F = F p (ν) U(ν) is multiply Cohen generic not only over M but also over M [H]. This allows to carry out the same argument as above.

Digression: Definability in HC
The next subsection will contain a transfinite construction of a key forcing notion in L relativized to HC. Recall that HC is the collection of all hereditarily countable sets. In particular, HC = L ω 1 in L. In matters of related definability classes, we refer to e.g., Part B, Chapter 5, Section 4 in [20], or Chapter 13 in [21], on the Lévy hierarchy of ∈-formulas and definability classes Σ X n , Π X n , ∆ X n for any set X , and especially on Σ HC n , Π HC n , ∆ HC n for X = HC in Sections 8 and 9 in [22], or elsewhere. In particular, Σ HC n = all sets X ⊆ HC, definable in HC by a parameter-free Σ n formula. Σ HC n = all sets X ⊆ HC definable in HC by a Σ n formula with sets in HC as parameters.
Something like Σ HC n (x), x ∈ HC, means that only x is admitted as a parameter, while Σ HC n (M), where M ⊆ HC is a transitive model, means that all x ∈ M are admitted as parameters. Collections like Π HC n , Π HC n (x), Π HC n (M) are defined similarly, and ∆ HC n = Σ HC n ∩ Π HC n , etc.. The boldface classes are defined as follows: Σ HC n = Σ HC n (HC), Π HC n = Π HC n (HC), ∆ HC n = ∆ HC n (HC).

Remark 2.
It is known that the classes Σ HC n , Π HC n , ∆ HC n are equal to resp. Σ 1 n+1 , Π 1 n+1 , ∆ 1 n+1 for sets of reals, and the same for parameters and boldface classes. This well-known result was explicitly mentioned in [23] (Lemma on p. 281), a detailed proof see Lemma 25.25 in [21], or Theorem 9.1 in [22].

Remark 3.
Recall that < L is the Gödel wellordering of L, the constructible universe.
It is known that the restriction < L HC is a ∆ HC 1 relation, and if n ≥ 1, p ∈ ω ω is any parameter, and R(x, y, z, . . . ) is a finitary ∆ HC n (p) relation on HC then the relations ∃ x < L y R(x, y, z, . . . ) and ∀ x < L y R(x, y, z, . . . ) (with arguments y, z, . . . ) are ∆ HC n (p) as well.

Complete Sequences and the Basic Notion of Forcing
Say that a pair M, U ∈ sJS solves a set D ⊆ sJS iff either M, U ∈ D , or there is no pair M , U ∈ D extending M, U . Let D solv be the set of all pairs M, U ∈ sJS which solve D . Let us prove the existence of complete sequences.

Proof.
Define pairs M ξ , U ξ , ξ < ω 1 , by induction. Let U 0 be the null system with |U 0 | = ∅, and M 0 be the least CTM of ZFC − 1 . If λ < ω 1 is limit then put U λ = ξ<λ U ξ and let M λ be the least CTM of To check the definability property use the fact mentioned by Remark 3 in Section 4.3.
Thus P is the product of sets P(ν) = P[U(ν)], ν ∈ I , with finite support. We proceed with a couple of simple lemmas. (ii) if ν ∈ I then U(ν) is uncountable and topologically dense in ω ω , and if ν = µ belong to I then U(ν) ∩ U(µ) is empty; (iii) any set D ∈ M ξ , D ⊆ P ξ , pre-dense in P ξ (resp., pre-dense in P ξ below some p ∈ P ξ ), is pre-dense in P (resp., pre-dense in P below p ); (iv) any name τ ∈ M ξ ∩ SN ω ω (P ξ ), P ξ -full (resp., P ξ -full below some p ∈ P ξ ), is P-full (resp., P-full below p ); (v) if G ⊆ P is a set P-generic over the ground universe L then the set G ξ = G ∩ P ξ is P ξ -generic over M ξ .

Proof.
To prove (i) use Lemma 15. Both claims of (ii) hold by Definition 13. To prove (iii) and (iv) use Corollary 5. Finally, (v) follows from (iii). Now let us address definability issues.
The sets P and SN ω ω (P) (P-names for functions in Fun) are ∆ HC n−1 . The set of all P-full names in SN ω ω (P) is ∆ HC n−1 .
Proof. The sequence { M ξ , U ξ } ξ<ω 1 is ∆ HC n−1 by definition, hence the relation f ∈ U(ν) is Σ HC n−1 . On the other hand, if f ∈ Fun belongs to some M ξ then f ∈ U(ν) obviously implies f ∈ U ξ (ν), leading to a Π HC n−1 definition of the relation f ∈ U(ν). To prove the last claim, note that by Corollary 5 if a name τ ∈ SN ω ω (P ξ ) ∩ M ξ is P ξ -full then it remains P-full.

Basic Generic Extension and Regular Subextensions
Recall that an integer n ≥ 2 and sets U ξ , M ξ , U, P ξ , P are fixed in L by Definition 16. These sets are fixed for the remainder.
Suppose that, in L, K ⊆ P is a regular subforcing. If G ⊆ P is a set P-generic over L then G ∩ K is K-generic over L by Lemma 9(vi), and hence L[G ∩ K] is a K-generic extension of L. The following formulas Γ i (i ∈ I ) will give us a useful coding tool in extensions of this form: This is based on the next two results. Recall that |G ∩ K| = p∈G∩K |p|. Proof. The set W = { ν, f : ν ∈ I ∧ f ∈ U(ν)} is ∆ HC n−1 in L, by Lemma 16, and hence so is Theorem 10. Suppose that, in L, K ⊆ P is a regular subforcing. Let G ⊆ P be P-generic over L. Then: , and there is no sets S ⊆ Seq in L[G ∩ K] satisfying Γ ν (S).
Proof. (i) This is a corollary of Lemma 9(vi).
(ii) Suppose towards the contrary that some S ∈ L[G ∩ K] satisfies Γ ν (S). Note that S ∈ L[G =ν ] by Lemma 9(iv). Now we can forget about the given set K . It follows from Theorem 5(iii) (with K = P =ν ), that there is a name τ ∈ SN Seq (P =ν ) such that S = τ[G =ν ]. There is an ordinal ξ < ω 1 satisfying τ ∈ M ξ and τ ∈ SN Seq (P ξ =ν ). Then S = τ[G ξ =ν ], where G ξ = G ∩ P ξ is P ξ -generic over M ξ by Corollary 6, and hence S belongs to M ξ [G ξ =ν ]. Note that U(ν) is uncountable by Corollary 6(ii), and hence F = U(ν) U ξ (ν) is uncountable. Let f ∈ F . Then f is Cohen generic over the model M ξ by Corollary 6. On the other hand G ξ =ν is , it follows that there is a string s ∈ Seq, s ⊂ f , such that S contains no strings extending s. Take any µ ∈ I , j = ν. By Corollary 6(ii), there exists a function g ∈ U(µ) U ξ (µ), g / ∈ U(ν), satisfying s ⊂ g. Then, max (S/g) = ω by Γ ν (S). However, this is absurd by the choice of s.

Corollary 7.
Suppose that, in L, K ⊆ P is a regular forcing. Let G ⊆ P be a set P-generic over L. Then Proof. Claim (i) follows from the theorem, because by the regularity we have G ∩ K ∈ L[G =ν ] for all ν / ∈ |G ∩ K|. Claim (ii) immediately follows from Lemma 17. To prove (iii) note that, by (i) and (ii), it holds in L[G ∩ K] that the set |G ∩ K| is defined by a Σ HC n formula ∃ S Γ ν (S) in HC.

Forcing Approximations
Here we define and study here an important forcing-like relation forc. It will give us control over various phenomena of analytic definability in the generic extensions considered.
We continue to assume V = L in the ground universe by Assumption 1.

Models and Absolute Sets
To consider transitive models of weaker theories, we let ZFC − be ZFC minus the Power Set axiom, with the schema of Collection instead of replacement, and AC in the form of well-orderability of every set. See [24] on ZFC − in detail.
Let ZFC − 1 be ZFC − plus the axioms V = L and "every set is at most countable". Let W ⊆ HC. By definition, a set X ⊆ HC is ∆ HC 1 (W) iff there exist a Σ 1 formula σ(x) and a Π 1 formula π(x), with sets in W as parameters, such that in particular, we have σ HC (x) ⇐⇒ π HC (x) for all x . However, generally speaking, this does not imply that X ∩ M ∈ ∆ M 1 (W), where M ∈ HC is a countable transitive model (CTM). The goal of the next two definitions is to distinguish and formalize this kind of absoluteness.

Definition 17.
If for a given ∆ HC 1 (W) set X , there exists such a pair of formulas, containing only parameters in W and satisfying (4) and ∀ x ∈ M (σ M (x) ⇐⇒ π M (x)) for all countable transitive models M |= ZFC − containing all parameters that occur in σ and/or in π , then we say that X is absolute ∆ HC 1 (W). In this case, if M is as indicated then the set X ∩ M is ∆ M 1 (W) via the same pair of formulas. In particular, any ∆ HC 0 (W) set is absolute ∆ HC 1 (W) by obvious reasons.

Formulas
Here we introduce a language that will help us study analytic definability in P[U]-generic extensions, for different systems U , and their submodels.

Definition 19.
Let L be the 2nd order Peano language, with variables of type 1 over ω ω . If K ⊆ P * then an L(K) formula is any formula of L, with some free variables of types 0, 1 replaced by resp. numbers in ω and names in SN ω ω (K), and some type 1 quantifiers are allowed to have bounding indices B (i.e., ∃ B , ∀ B ) such that B ⊆ I is finite or countable. Note that |ϕ| ⊆ ||ϕ|| ⊆ I provided ϕ is an L(P * ) formula. If a set G ⊆ P * is minimally ϕ-generic (i.e., minimally τ-generic w. r. t. every name τ ∈ NAM ϕ, in the sense of Section 3.5), then let the valuation ϕ[G] be the result of substitution of τ[G] for any name τ ∈ NAM ϕ, and changing each quantifier ∃ B x , ∀ B x to ∃ (∀ ) x ∈ ω ω ∩ L[G B] respectively, while index-free type 1 quantifiers are relativized to ω ω ; ϕ[G] is a formula of L with real parameters, and with some quantifiers of type 1 explicitly relativized to certain submodels of L[G].
An arithmetic formula in L(K) is a formula with no quantifiers of type 1 (names in SN ω ω (K) as in Definition 19 are allowed). If n < ω then let a LΣ 1 n (K), resp., LΠ 1 n (K) formula be a formula of the form

Forcing Approximation
The next definition invents a convenient forcing-type relation forc for pairs M, U in sJS and formulas ϕ in L(K[U]), associated with the truth in K[U]-generic extensions of L, where K ⊆ P * is a regular forcing. Recall that K[U] = K ∩ P[U] whenever K ⊆ P * is a regular forcing and U is a system.
and similarly defined Forc K w (Π 1 n ), where it is assumed that w ∈ ω ω and K ⊆ P * is a regular forcing and an absolute ∆ HC 1 (w) set.
Theorem 11 (in L). Let w ∈ ω ω and K ⊆ P * be a regular forcing and an absolute ∆ HC 1 (w) set. Then:

Advanced Properties of Forcing Approximations
The following lemma works whenever the domain K ⊆ P * (a regular forcing) of conditions p Step LΠ 1 n → LΣ 1 n+1 . Let ψ(x) be a LΠ , M) formula. Suppose towards the contrary that p K forc M U ϕ holds, but p K forc M U c ϕ fails, so that there exist a pair M , V ∈ sJS and a condition q ∈ K[V], such that M, U c M , V , q ≤ p, and q K forc M V ϕ ¬ . Then q K forc M V c ϕ ¬ by the inductive hypothesis. Note that |q| ⊆ c by the choice of K , but not necessarily |V| ⊆ c. Proof. It suffices to consider the case of Π 1 1 formulas; the induction steps LΠ 1 k → LΣ 1 k+1 and LΣ 1 k → LΠ 1 k are rather easy. Suppose that ϕ is LΠ 1 1 and p K forc M U ϕ(τ 1 , . . . , τ m ). Suppose that G ⊆ K[U] is a set K[U]-generic over M , and p ∈ G . We claim that τ [G] = µ [G] for all ; this obviously implies the result required. Suppose that this is not the case. Then, by definition, there exist numbers m and j = k and conditions q ∈ G ∩ (τ " m, j ) and r ∈ G ∩ (µ " m, k ). Then p, q, r must be compatible (as elements of the same generic set), which is a contradiction. : q ≤ p}, a set A ∈ M , A ⊆ P is a maximal antichain in P, and q K forc M U ϕ for all q ∈ A. Then p K forc M U ϕ.
Proof. If ϕ is a LΠ 1 1 formula then the result follows from (F2) of Definition 20 and known properties of the ordinary forcing over M . Now let ϕ be Π 1 k , k ≥ 2. Suppose towards the contrary that p K forc M U ϕ fails. Then there exist: a pair M , U ∈ sJS extending M, U , and a condition r ∈ K[U ], r ≤ p, such that r K forc M U ϕ ¬ . Note that A remains a maximal antichain in the set Q = {q ∈ P[U] : q ≤ p} (bigger than P above), by Lemma 8. Therefore, A is still a maximal antichain in Q = {q ∈ P[U ] : q ≤ p}, by Theorem 8(i), hence a maximal antichain in P = {q ∈ K[U ] : q ≤ p}. It follows that r is compatible in K[U ] with at least one condition q ∈ A. However, r K forc M U ϕ ¬ while q K forc M U ϕ, easily leading to a contradiction with Lemma 19.

Transformations and Invariance
Here we show that, under certain assumptions, the transformations of the first two groups defined in Section 3.7 preserve forcing approximations forc. This is not an absolutely elementary thing: there is no way to reasonably apply transformations to transitive models M involved in the definition of forc . What we can do is to require that the transformations involved belong to the models involved. This leads to certain complications of different sort.
Family 1: permutations. First of all we have to extend the definition of the action of π in Section 3.7 to include formulas. Suppose that c, c ⊆ I . Define the action of any π ∈ BIJ c c onto formulas ϕ of L(P * ) such that ||ϕ|| ⊆ c : -to get π ϕ substitute π · τ for any τ ∈ NAM ϕ and π · B for any B ∈ IND ϕ.
Proof. Under the assumptions of the lemma, in particular, the requirement of c, c , π being absolute ∆ HC 1 (M), π acts as an isomorphism on all relevant domains and preserves all relevant relations between the objects involved. Thus M, π ·U , π · K , π · p, π ϕ still satisfy Definition 20(F1), and ||πϕ|| ⊆ c , |π ·U| ⊆ c , π · K ⊆ P * c . (For instance, to show that π ·U still belongs to M , note that the set |U| ⊆ c belongs to M , thus π |U| ∈ M , too, since π is an absolute ∆ HC 1 (M) function.) This allows to prove the lemma by induction on the complexity of ϕ.

Family 2: Lipschitz transformations.
We extend the action of α ∈ Lip I to formulas of L(P * ): to get π ϕ substitute π · τ for any τ ∈ NAM ϕ but do not change the quantifier indices B.
Note that the action of any α ∈ Lip I ∩ M on systems, conditions, names, and formulas, as defined there, is absolute ∆ HC 1 (M). This allows to prove the next invariance lemma similarly to Lemma 22, which we leave for the reader.

Elementary Equivalence Theorem
This section presents further properties of P-generic extensions of L and their subextensions, including Theorem 13 and its corollaties on the elementary equivalence of different subextensions.

Assumption 2.
We continue to assume V = L in the ground universe. Below in this section, a number n ≥ 2 is fixed, and pairs M ξ , U ξ , the system U = ξ<ω 1 U ξ , the forcing notions P ξ = P[U ξ ] and P = P[U] = ξ<ω 1 P ξ are as in Definition 16 for this n.

Further Properties of Forcing Approximations
Coming back to the complete sequence of pairs M ξ , U ξ introduced by Definition 16, we consider the auxiliary forcing relation forc with respect to those pairs. We begin with the following definition.
Definition 21 (in L). Let K ⊆ P * be a regular forcing. Recall that for any ξ < ω 1 . Let p K forc ξ ϕ mean p K forc M ξ U ξ ϕ-then by definition K has to be an absolute ∆ HC 1 (M ξ ) set, by the way. We let p K forc ∞ ϕ mean: ∃ ξ < ω 1 (p K forc ξ ϕ).
The following is an easy consequence of Lemma 18.
Lemma 24 (in L). Let K ⊆ P * be a regular forcing. Assume that ϕ is a closed formula in LΠ 1 . Then: (i) if p K forc ξ ϕ and ξ ζ < ω 1 , q ∈ K[U ζ ], q p, then q K forc ζ ϕ ; (ii) p K forc ∞ ϕ and p K forc ∞ ϕ ¬ contradict to each other; : q ≤ p} is a maximal antichain in Q, and q K forc ∞ ϕ for all q ∈ A, then p K forc ∞ ϕ .

Proof. (iii) As
A is a countable set, there exists an ordinal ξ < ω 1 such that q K forc ξ ϕ for all q ∈ A. Apply Lemma 21.
Lemma 25 (in L). Assume that K ⊆ P * is a regular forcing, ϕ is a closed formula in LΠ 1 , all names in ϕ are K[U]-full below p, and finally w ∈ ω ω and K is absolute ∆ HC 1 (w). Then: Proof. (i) As any name is a countable object, there is an ordinal η < ω 1 such that p ∈ K[U η ], w ∈ M η , and all names in ϕ belong to M η ∩ SN ω ω (K[U η ]); then all names in ϕ are K[U η ]-full below p, of course. As k < n, the set D of all pairs M, U ∈ sJS that extend M η , U η and there is a condition q ∈ K[U], q p, satisfying q K forc M U ϕ ¬ , belongs to Σ HC n−2 by Theorem 11. Therefore, by the n-completeness of the sequence { M ξ , U ξ } ξ<ω 1 , there is an ordinal ζ , η ζ < ω 1 , such that M ζ , U ζ ∈ D solv . (By the way, this is the only use of the n-completeness!) We have two cases.
Case 2: there is no pair M, U ∈ D extending M ζ , U ζ . Prove p K forc ζ ϕ. Suppose otherwise. Then by the choice of η and (F4) of Definition 20 there exist a pair M, U ∈ sJS extending M ζ , U ζ , and a condition q ∈ K[U], q ≤ p, such that q K forc M U ϕ ¬ . Then M, U ∈ D , a contradiction. (ii) Suppose that there is no condition q ∈ K[U], q ≤ p, with q K forc ∞ ϕ ¬ . Then by (i) the set Q = {q ∈ K[U] : q ≤ p ∧ q K forc ∞ ϕ} is dense in K[U] below p. Let A ⊆ Q be a maximal antichain. It remains to apply Lemma 24(iii).

Relations to the Truth in Generic Extensions
According to the next theorem, the truth in the generic extensions considered is connected in the usual way with the relation forc ∞ up to the n-th level of analytic hierarchy. Recall that V = L is assumed in the ground universe.
Theorem 12. Assume that, in L, K ⊆ P * is a regular forcing, ϕ is a closed formula in LΠ 1 -full, w ∈ ω ω , and K is an absolute ∆ HC 1 (w) set. Let G ⊆ P be a P-generic set over L. Then: ] and strictly k < n holds, then ∃ p ∈ G ∩ K (p K forc ∞ ϕ).

The formulas ϕ[G], ϕ[G ∩ K] coincide under the assumptions of the theorem.
Proof. (ii) We argue by induction on the complexity of ϕ.
The case of LΠ 1 1 formulas. Let ϕ be a closed formula in LΠ 1 1 (K[U]). As names in the formulas considered are countable objects, there is an ordinal ξ < ω 1 such that w ∈ M ξ and ϕ is a LΠ 1 ] as well by the Shoenfield absoluteness.
Step LΣ 1 k → LΠ 1 k , 2 ≤ k < n. Prove the theorem for a LΠ 1 k (K[U]) formula ϕ, assuming that the result holds for ϕ ¬ . If ϕ [G] is false in L[G] then ϕ ¬ [G] is true. Thus by the inductive hypothesis, there is a condition p ∈ G such that p K forc ∞ ϕ ¬ . Then q K forc ∞ ϕ for any q ∈ G is impossible by Lemma 24(ii). Conversely, suppose that p K forc ∞ ϕ fails for all p ∈ G ∩ K . Then by Lemma 25 Finally prove (i) for a formula ϕ := ∃ x ψ(x), ψ being LΠ 1 n (K[U]). Suppose that p ∈ G ∩ K[U] and p K forc ∞ ϕ. Then there is a name τ ∈ SN ω ω (K[U]), K[U]-full below p and such that p K forc ∞ ψ(τ). We can w. l. o. g. assume that τ is totally K[U]-full, by Lemmas 10 and 20. We have to prove that the formula ψ(τ) [G], that is, However, ψ(τ) ¬ is a Σ 1 n formula. Therefore, by the first claim of the lemma, there is a condition q ∈ G ∩ K such that q K forc ∞ ψ(τ) ¬ . However, p K forc ∞ ψ(τ) and p, q are compatible (as they belong to the same generic set). This contradicts Lemma 24(ii).

Consequences for the Ordinary Forcing Relation
For any forcing P ∈ L, we let || − − P be the ordinary P-forcing relation over L as the ground universe. In particular || − − P is the P-forcing relation over L. L). Under the assumptions of Theorem 12, let p ∈ K[U]. Then:
(iii) Let G ⊆ P be P-generic over L, and p ∈ G .
, and hence there is r ∈ G ∩ K such that r K forc ∞ ϕ, by Theorem 12. However, then p, r are compatible (as members of G ), hence q = p ∧ r still is a condition, and q ∈ K[U].
(iv) If p K forc ∞ ϕ fails, then by Lemma 25(ii) there is a condition q ∈ K[U], q ≤ p, such that However, then, by (iii), there is a condition q ∈ K[U], q ≤ r , q K forc ∞ ϕ ¬ , as required.
The next corollary evaluates the complexity of the ordinary forcing relations || − − K [U] . The result is related to formulas in classes LΠ 1 n and higher.

Elementary Equivalence Theorem
According to Theorem 10, sets S satisfying Γ i (S) are different for different indices i ∈ I , and the difference can be determined, in the extensions of the form L[G z], at the level Π HC n−1 by Corollary 7, that is, Π 1 n (see Remark 2 in Section 4.3). On the other hand, the extensions considered remain rather amorphous w. r. t. lower levels of definability, as witnessed by the following key theorem.
Theorem 13. Suppose that, in L : d ⊆ I , w ∈ ω ω , sets b, c ⊆ d = I d have equal cardinality, d is uncountable, K ⊆ P * d is a regular forcing, Ψ(y) is a Π 1 n−1 formula with parameters in ω ω ∩ L[G ∩ K], and K , b, c, d are absolute ∆ HC 1 (w) sets. Let G ⊆ P be P-generic over L. Then, if there is a real y ∈ ω ω ∩ L[G ∩ K, G b] such that Ψ(y) holds in L[G ∩ K, G d ], then there exists y ∈ L[G ∩ K, G c] such that Ψ(y ) holds in L[G ∩ K, G d ] .
Recall that ∆ HC 1 (w) means that w is admitted as the only parameter. The assumption that d is uncountable, can be avoided at the cost of extra complications, but the case of d countable is not considered below. The proof makes use of the transformations introduced in Section 3.7.
Proof. As all cardinals are preserved in L[G], we w. l. o. g. assume that b, c are countably infinite (or finite of equal cardinality) in L. Suppose towards the contrary that . Replace each parameter z in Ψ(x) by such a name τ z in L, and let ψ(x) be the LΠ 1 n−1 (K[U]) formula obtained. Then |ψ| ⊆ d. Further, the set is a regular forcing, and L[G ∩ K, G b] = L[G ∩ K ]. Choose y by (A). Once again, Theorem 5(ii), yields a K [U]-full name τ y ∈ SN ω ω (K [U]) such that y = τ y [G]. The name τ y is small, hence the set |τ y | ⊆ d ∪ b is countable (in L). We let d 0 = |τ y | ∩ d; the set B = d 0 ∪ b is still countable and |τ y | ⊆ B. Thus the formula ∃ B y ψ(y) [G] is true in L[G ∩ K, G d ].
Now let Q = { p ∈ P * : p d ∈ K} = K × (P * d ). Thus Q is a regular forcing, and L[ ]. Therefore ∃ B y ψ(y) [G] is true in L[G ∩ Q[U]] by the above. It follows by Theorem 12(ii) that there is a condition p ∈ G ∩ Q such that p Q forc ∞ ∃ B y ψ(y), and, by (B), we can also assume that p Q[U]-forces ¬ ∃ C y ψ(y) over L where C = d 0 ∪ c. Further, in L, there exists an ordinal ξ < ω 1 such that where M = M ξ and U = U ξ , and in addition the countable sets Now we can assume that both sets |U| (d ∪ b) and |U| (d ∪ c) are infinite. (Otherwise take a suitably bigger ξ .) Then there is a bijection f ∈ M , f : |U| onto −→ |U|, such that f d is the identity and f [b] = c. Define a bijection π ∈ BIJ I I such that π |U| coincides with f and π (I |U|) is the identity. Let q = π · p and V = π ·U . Acting by π on (7), we obtain, by Lemma 22, Comments: 1) π · Q = Q since π d is the identity by construction and K ⊆ P * d; 2) π · B = π[B] = f [B] = C by construction; 3) π · ψ(x) is ψ(x) because |ψ| ⊆ d and π d is the identity. Note that V ∈ M is a system with |V| = π · |U| = |U|, and p ∈ U , q ∈ V , U d = V d and q d = p d by the choice of π and f . In addition, U , V are countable systems in M |= ZFC − 1 . Corollary 2 yields a transformation α ∈ Lip I in M such that |α| = |U| = |V|, α · V = U , conditions q = α · q ∈ Q[U] and p are compatible, and α d is the identity (as U d = V d and p d = q d). However, then α · Q = Q, and α(∃ C x ψ(x)) coincides with ∃ C x ψ(x) since |ψ| ⊆ d. Therefore q Q forc M U ∃ C y ψ(y) by (8) and Lemma 23. This implies q Q forc ∞ ∃ C y ψ(y). We conclude that q Q[U]-forces ∃ C y ψ(y) over L, by Corollary 8(i). However, q is compatible with p and p forces the negation of this sentence. The contradiction completes the proof. If k = 2 then the result holds by the Shoenfield absoluteness theorem. It remains to carry out the step k → k + 1 (k < n). Let ϕ(x) be a Π 1 k formula with parameters in L[G ∩ K, G c]; we have to prove the result for the Σ 1 k formula ∃ x ϕ(x), assuming k < n. First of all, as the cardinals are preserved, there is a set δ ∈ L, δ ⊆ d , countable in L and such that all parameters of ϕ belong to L[G ∩ K, G δ]. Let d = d ∪ δ and K = { p ∈ P * d : p d ∈ K}; we can identify K with K × (P * δ), of course. Then, in L, K is a regular forcing, K ⊆ P * d , and all parameters of ϕ belong to L[G ∩ K ]. Now suppose that ∃ x ϕ(x) holds in L[G ∩ K, G d ], the bigger of the two models of the lemma. Let this be witnessed by a real . As the cardinals are preserved, there is a set b ∈ L, b ⊆ (d ) , countably infinite in L and such that x 0 belongs to L[G ∩ K , G b ]. Since c is uncountable, there exists a set c ∈ L, c ⊆ (d ) ∩ c, countably infinite in L. By the choice of δ, there is a real w ∈ ω ω ∩ L such that the sets K , d , c , b are absolute ∆ HC 1 (w ) in L. By Theorem 13, there is a real y 0 ∈ L[G ∩ K , G c ] such that ϕ(y 0 ) holds in L[G ∩ K , G (d ) ] = L[G ∩ K, G d ], and then in L[G ∩ K, G c] by the inductive assumption.
Note that if say c is uncountable but b countable, and d is countable, then Theorem 13 fails by means of the formula "there is a real x such that all reals belong to L[x, G ∩ K]", and G ∩ K is equiconstructible with a real in this case. Question 1. It would be very interesting to figure out whether Theorem 13 and Corollary 10 hold also for sets b, c not necessarily constructible.
The following corollary presents a partial positive result. A set z ⊆ I = ω L 1 is bounded iff there is α < ω L 1 such that z ⊆ α. Corollary 11. Suppose that G ⊆ P is P-generic over L, and z ⊆ I is a set unbounded in I , locally constructible in the sense that z ∩ α ∈ L for all α ∈ I , and all L-cardinals are preserved in L

Application 1: Nonconstructible ∆ 1 n Reals
In this section, we proveTheorems 1 and 2(i). Theorem 1 provides change of definability of reals situated in the ground set universe L, in generic extensions of L. Thus, any real a / ∈ Σ 1 n ∪ Π 1 n in L can be placed exactly at ∆ 1 n+1 in an appropriate (almost disjoint) extension of L. Theorem 2 contains several results for nonconstructible reals. The proofs of these results will make use of various results in Sections 5 and 6, in particular, a result (Theorem 11) related to definability of relevant forcing relations. Assumption 3. We continue to assume V = L in the ground universe. We fix an integer n ≥ 2, for which Theorems 1 and 2 will be proved, and make use of a system U and the forcing notion P = P[U] as in Definition 16; both U and P belong to L.
Thus c ⊆ ω ⊆ I = ω 1 , c ∈ L, K ⊆ P * is a regular forcing. Let G ⊆ P be a P-generic set over L. Then the set s = s k such that 1) 2k ∈ |p| or 2k − 1 ∈ |p|, and 2) s / ∈ F ∨ p (0) (to make sure that p * ≤ p). Now we let d contain 0, all numbers 2k ∈ |p * | such that s k ∈ S p * (0), and all numbers 2k − 1 ∈ |p * | such that s k ∈ F ∨ p (0) S p * (0). (Compare to Example 2 in Section 3.2!) The rest of the lemma follows from Lemma 9.
Thus extensions of the form L[G z G ] considered here are exactly K[U]-generic extensions of L. To check that those extensions satisfy Theorem 2(i), we prove the following Claims 5 and 6. The first of them is entirely similar to Claim 3, so the proof is omitted (left to the reader).
The proof of this claim involves the following lemma.
This proves Claim 6, of course, by Theorem 11. Now let us check (11) itself; this will be similar to the proof of (9) in Section 7.1.
However, this implies the right-hand side of (9). Now assume that M, U ∈ sJS, p ∈ K[U], and p K forc M U ϕ(m). Suppose towards the contrary that ϕ(m) is false in L[G ∩ K], so that there is a condition q ∈ G ∩ K such that q || − − K[U] ¬ ϕ(m). However, this contradicts Lemma 27.
(Claim 6 and Theorem 2(i) modulo Lemma 27) 7.3. Nonconstructible ∆ 1 n+1 Real, Part 2 We continue the proof of Theorem 2(i). The proof of Lemma 27 that follows below makes use of transformations in BIJ ω ω (bijections of ω ) and those in the set Lip ω = {α ∈ Lip I : |α| ⊆ ω}, essentially the ω-product of Lip. Yet this will be somewhat more complicated than the proof of Theorem 1 above, because in this case K is not preserved under the action of arbitrary transformations in BIJ ω ω and Lip ω . This is why we have to consider certain combinations of those transformations.
Not all π ∈ BIJ ω ω are K-preserving, and neither is any α ∈ Lip ω with α 0 = the identity. Yet there are plenty of K-preserving transformations in Σ.
Proof (Lemma 27). Suppose towards the contrary that both q || − − K[U] ¬ Φ and p K forc M U Φ. By the way we can w. l. o. g. assume that |U| ⊆ ω , by Lemma 19, and moreover, that |U| = ω exactly. (Otherwise extend U by U(ν) = Q for all ν ∈ ω |U|, where Q = all eventually-0 functions f ∈ Fun.) There is an ordinal . Apply Lemma 28 in M ξ . It gives a K-preserving transformation σ = α • π ∈ M ξ such that σ ·U = V and the conditions r = σ · p and q (both in K[V]) are compatible. On the other hand, we have r K forc M ξ V Φ by Lemmas 22 and 23, and hence r K forc Corollary 8(i). However, r is compatible with q, and q forces the opposite, a contradiction.

Application 2: Nonconstructible Self-Definable ∆ 1 n Reals
Note that the set a as in Theorem 2(i) is definable in the generic extension of L, considered in Section 7.2, by means of other reals in that extension, including those which do not necessarily belong to L[a]. Claim (ii) of Theorem 2 achieves the same effect with the advantage that a is definable inside L[a].
The key idea (originally from [9] Section 4) can be explained as follows. Recall that a set of the form a 0 = a G (0) was made definable in a generic extension of the form L[G z G ] by means of the presence/absense of other sets of the form S G (ν), ν < ω , in L[G z], see Sections 7.2 and 7.3. Our plan will now be to make each of the according sets a G (ν) ∈ L[G z] (note that a G (ν) ⊆ ω {0}, see Definition 9), as well as the whole sequence of them, ∆ 1 n+1 -definable in L[G z]. In order to do this, we need to develop a suitable coding construction.

Assumption 4.
We continue to assume V = L in the ground universe. We fix an integer n ≥ 2, for which Theorem 1(ii) will be proved, and make use of a system U and the forcing notion P = P[U] as in Definition 16; both U and P belong to L.
8.1. Nonconstructible Self-Definable ∆ 1 n+1 Reals: The Model Here we begin the proof of Theorem 2(ii). Recall that ω ω = {s k : k < ω} is a fixed recursive enumeration of strings of natural numbers, such that s 0 = Λ, the empty string, and s k ⊆ s k =⇒ k ≤ k . Let k i = num (s k i), thus s k i = s k i . Then we have: • Each set L(k) = { k i : i < ω} ⊆ ω is countably infinite, k < min i k i , k = k =⇒ L(k) ∩ L(k ) = ∅ and i = j =⇒ k i = k j , and finally each m ≥ 1 is equal to k i for exactly one pair of indices of i, k < ω .
Define a partial order on ω so that i k iff s i ⊂ s k . Obviously k k i for all i , k ∈ ω , and 0 is the -least element.
For any sequence a = {a k } k<ω of sets a k ⊆ ω , we define a set ζ a ⊆ ω so that: 2) if k ∈ ζ a then, for every i : if i ∈ a k then k 2i ∈ ζ a and k 2i+1 / ∈ ζ a , but if i / ∈ a k then k 2i / ∈ ζ a and k 2i+1 ∈ ζ a ; 3) if k / ∈ ζ a then k i / ∈ ζ a for all i .
The next theorem obviously implies Theorem 2(ii).
n , then x ∈ L and x is Σ 1 n in L.
Proof (will continue towards the end of Section 7). Our arguments will be a more elaborate version of arguments in Sections 7.2, 7.3. We'll make use of the set K of all conditions p ∈ P * ω such that for all i and k : (compare to (A), (B) in Section 7.2), and the related set K[U] = K ∩ P.

Lemma 29.
K is a regular forcing in L. If G ⊆ P is a set P-generic over L then G] , and accordingly L[ Proof. As above, the nontrivial item of the regularity property is (4) of Definition 8. Suppose that p ∈ P * . Then |p| ⊆ ω is finite. Let δ be the least -initial segment of ω satisfying |p| ⊆ δ; δ is finite as well. Define p * ∈ P * so that |p * | = δ and F p * (k) = F p (k) for all k , but the sets S p * (k) may be strictly bigger than the corresponding sets S p (k). The definition of S p * (k) goes on by -inverse induction on k ∈ δ. If k ∈ δ is -maximal in δ then obviously k ∈ |p|, and we put S p * (k) = S p (k). Assume that k ∈ δ is not -maximal in δ, and the value of p * ( k m ) = S p * ( k m ) ; F p ( k m ) is defined for all m such that k m ∈ δ. Put S p * (k) = S p (k) ∪ S, where S consists of all strings s = s i such that (a) k 2i+1 ∈ |p * | = γ or k 2i ∈ |p * |, and (b) s / ∈ F ∨ p (k) (to make sure that p * ≤ p). By definition, |p * | = δ, and if i, k ∈ ω and at least one of the numbers k 2i+1 , k 2i belongs to δ, then the string s i belongs to F ∨ p * (k) ∪ S p * (k). Now we define a set d ⊆ δ so that the decision whether a number k ∈ δ belongs to d is made by direct -induction. We put 0 ∈ d. Suppose that some k ∈ δ already belongs to d. We define: . A simple verification that p * and d satisfy Definition 8(4) is left to the reader. Further, the set Prove, e.g., the first equivalence. Suppose that k 2i+1 ∈ |G ∩ K|. Then k 2i+1 ∈ |p| for some p ∈ K in G , and we have s i ∈ F ∨ p (k) S p (k) by (B), so that s i / ∈ S G (k) and accordingly i / ∈ a G (k), thus by definition k 2i+1 ∈ ζ a [G] . Suppose conversely that k 2i+1 ∈ ζ a [G] . Then by definition i / ∈ a G (k), hence s i / ∈ G G (k). This must be forced by some p ∈ K ∩ G , and, as k ∈ |G ∩ K|, we can assume that k ∈ |p|. However, in this case forcing s i / ∈ G G (k) means by necessity that just s i ∈ F ∨ p (k) S p (k), so there exists a stronger condition p ∈ K ∩ G with k 2i+1 ∈ |p |. We conclude that k 2i+1 ∈ |G ∩ K|. The proof of claim (ii) follows in the next two subsections.

Remark 5.
A slightly more elaborate argument, like in the end of Section 4 in [9], shows that even more G] ] satisfyings the following requirements: (a) 0 ∈ ζ , and if k / ∈ ζ then k 2i / ∈ ζ and k 2i / ∈ ζ for all i ; The conjunction of them amounts to a Π 1 n definition of {ζ } in L[ζ].

Key Lemma
As in Section 7.2, Claim (ii) of Theorem 14 is a consequence of the following lemma (the key lemma from the title), the proof of which will end the proof of theorems 14 and 2(ii).

Lemma 30 (in L).
Suppose that M, U ∈ sJS, p ∈ K[U], q ∈ K[U]. Let Φ be any closed parameter-free Σ 1 n formula. Then it is impossible that simultaneously q || − − K[U] ¬ Φ and p K forc M U Φ.
Following Definition 22, a transformation σ ∈ Σ (see Remark 4 in Section 7.3 on Σ) is called Kpreserving, if p ∈ K ⇐⇒ σ · p ∈ K for all p ∈ P * ω . Clearly the regular forcing K here is different (and way more complex in some aspects) than K in Section 7.3. The following lemma is analogous to Lemma 28.
Lemma 31 (in L). Suppose that U , V are countable systems with |U| = |V| = ω , and p ∈ K[U], q ∈ K[V]. Then there is a K-preserving transformation σ ∈ Σ such that σ · U = V , and the conditions σ · p and q are compatible.
Proof. The proof resembles the proof of Lemma 28, but is somewhat more complicated. Essentially, we'll have a ramified ω-long iteration in which the construction employed in Lemma 28 will be just one step. We define -cones C k = {i ∈ ω : k i} and C k = C k ∪ {k} for any k ∈ ω .
Proof. Note that α k 0 is a ⊆-preserving bijection of the set Seq of all finite non-empty strings of integers. Let f = f α k 0 : ω onto −→ ω be the associated permutation of integers, so that f (i) = j iff α 0 (s i ) = s j . Let the transformation π = π[α k 0 ] be the identity outside of the strict -cone C k 0 ; in particular, π(k 0 ) = k 0 . Beyond this, put π( k 0 2i ) = k 0 2 f (i) and π( k 0 2i+1 ) = k 0 2 f (i)+1 for all i . Now, if k ∈ C k 0 and π(k) = k is defined then put π( k 2m ) = k 2m for all m. (Claim) k µ ] ∈ BIJ ω ω as in assumption (II) of Section 8.3 -thus π µ is the identity outside of C k µ ; -a K-preserving superposition ρ µ = π µ • α µ , equal to the identity outside of the extended cone C k µ , as in assumption (III) of Section 8.3.
Define Π m and T m by (12) above. Put U m = T m · U and p m = T m · p. By Lemma 5, there is a transformation α k m+1 ∈ Lip such that α k m+1 · U m (k m+1 ) = V(k m+1 ) and the conditions α k m+1 · p m (k m+1 ) and q(k m+1 ) are compatible.
After we have defined α k m ∈ Lip by induction on m, let's take the transformation α = {α k } k<ω ∈ Lip ω as the input of the construction in Section 8.3. The latter gives us a permutation Π ∈ BIJ ω ω of Claim 8, such that the superposition σ = α • Π ∈ Σ is K-preserving. It remains to check that 1) σ ·U = V and that 2) σ · p and q are compatible conditions.

Assumption 5.
We continue to assume V = L in the ground universe. We fix an integer n ≥ 2, for which Theorem 3 will be proved, and make use of a system U and the forcing notion P = P[U] as in Definition 16; both U and P belong to L.
9.1. Nonconstructible Σ 1 n+1 Reals: The Model The most obvious idea as of getting an extension required is to slightly modify the proof of Theorem 2(ii) in the following direction. Suppose that G ⊆ P be P-generic over L, and let S G (ν) and a ν = a G (ν) = {k ≥ 1 : s k ∈ S(i)} be defined as in Definition 9. We proved (see the proof of Theorem 2(i) above) that if by (10) of Section 7.2 then the set a 0 is ∆ 1 n+1 in L[G z], and the part {2k + 2 : k ∈ a 0 } of z is responsible for a 0 being Σ 1 n+1 in L[G z] (by means of the equality a 0 = {k : ∃ S Γ 2k+2 (S)}) while the part {2k + 1 : k / ∈ a 0 } is responsible for a 0 being Π 1 n+1 in L[G z] (by means of the equality a 0 = {k : ¬ ∃ S Γ 2k+1 (S)}). As now the second part is not needed, one might hope that if y is defined by then L[G y] will be a model for Theorem 3. At least a 0 will be Σ 1 n+1 in L[G y] by exactly the same reasons. However we have not been able to prove the second part of the theorem, i.e., that all reals ∆ 1 n+1 in L[G y] belong to L. The point of difficulty is the following hypothesis: Conjecture 1. Under the assumptions above, if m / ∈ y = y G then any parameter-free Σ 1 n+1 formula true in L[G y] is true in L[G y, a m ] as well.
We definitely cannot expect the conjecture to be true for formulas with parameters in L[G y] (the smaller model) since if p ∈ L[G y], p ⊆ ω codes the sequence {a i } i∈y then Fun ⊆ L[p] is true in L[G y] but false in L[G y, a m ].
We have a near-counterexample to Conjecture 1: the formula ∃ x (Γ 0 (x) ∧ Fun ⊆ L[x]) of class Σ 1 n+1 (assuming n ≥ 3) holds in L[a 0 ] and fails in L[a 0 , a 1 ]. The set y = {a 0 } is definitely not of the form (13), so this is not literally a counterexample, yet it casts doubts on the approach based on (13). Now we describe the extension involved in the proof of Theorem 3. The model we define will be a submodel of the whole extension L[G], where G is P-generic over L, and a set y of (13) is involved in the definition. We let where a 0 = a G (0) (then Y ∈ L[a 0 ] L) and y G is defined by (13). The goal is to prove that L[G Y] witnesses Theorem 3 with a = a 0 . The task splits in two claims: Claim 10 is established just as similar claims above, so we leave it for the reader. Let us concentrate on Claim 11. We make use of the set K 0 of all conditions p ∈ P * ω such that if k ≥ 1 and k ∈ |p|, then s k ∈ S p (0) (= Example 2 in Section 3.2); (15) as well as the related sets: K = K 0 × (P * (I ω)) = { p ∈ P * : p ω ∈ K 0 }, K 0 [U] = K 0 ∩ P, and accordingly K[U] = K ∩ P.

Lemma 32.
It is true in L that: K 0 and K are regular forcings and absolute ∆ HC 1 sets, and if z ⊆ I contains 0 then the restrictions K z, K 0 z are regular forcings, too.
If G ⊆ P is a set P-generic over L then G ∩ K = G ∩ K[U] is a set K[U]-generic over L, Proof. To check (4) of Definition 8 for K 0 see Example 2 in Section 3.2. To prove, that the set K 0 z = { p ∈ K 0 : |p| ⊆ z} (z ∈ L, z ⊆ ω ) is regular, argue as in Example 2 in Section 3.2. The rest of the lemma is easy: apply Lemma 9.

Key Lemma
Here we establish the following key lemma. Recall that sets y G , Y G are defined by (13) and (14).

Lemma 33.
Suppose that G ⊆ P is P-generic over L, and y G = {0} ∪ a G (0), y ⊆ ω , the symmetric difference δ = y ∆ y G is finite, and 0 / Let u = y y G and v = y G y; thus δ = u ∪ v. Then v ⊆ a G (0) but u ∩ a G (0) = ∅ by the definition of y G . In other words, the finite disjoint sets S u = {s k : k ∈ u} and S v = {s k : k ∈ v} satisfy S v ⊆ S G (0) but S u ∩ S G (0) = ∅. It follows that there is a condition p ∈ G ∩ K[U] such that |p| = {0}, S v ⊆ S p (0), and S u ⊆ F ∨ p (0) S p (0). We can increase F p (0) if necessary for S p (0) ⊆ F ∨ p (0) (a technical requirement) to hold. Now let q be a condition obtained by the following modification of p: still |q| = {0} and F q (0) = F p (0) (therefore, q belongs to K[U] together with p), and S q (0) = (S p (0) (3) in Section 3.7. Therefore the map (Definition 12) is an order isomorphism of P onto Q by Theorem 6, acting so that: We conclude that H p q also is an order isomorphism of P ∩ P onto Q ∩ P by ( * ), and hence the set As for the elementary equivalence claim, note first of all that the common part L[G (y G ∩ y), G (I ω)] of the two models also is a K[U]-generic extension of L by the above. (Take y G ∩ y as a new y.) Thus in fact it suffices to prove that under the assumptions of the theorem if j ∈ ω y G then L[G y G , a G (j), G (I ω)] is an elementary extension of L[G y G , G (I ω)] w. r. t. all Σ 1 n formulas. Let Φ be a closed Σ 1 n formula with parameters in L[G y G , G (I ω)]. It can be deduced, using either Theorem 5(ii) or directly the CCC property of P (Theorem 4) that there is an ordinal γ, It follows from Lemma 32 that K = K d is a regular forcing, and in fact G γ ⊆ K since j / ∈ y G . Moreover, by definition all of K 0 , K , K , d, b, c are absolute ∆ HC 1 (w) sets in L for some w ∈ ω ω . Therefore by Corollary 10 Φ is simultaneously true in L[G ∩ K , G b] and in L[G ∩ K , G c]. However, and similarly L[G ∩ K , G c] = L[y G , a G (j), G (I ω)], as required.

Second Key Lemma
In continuation of the proof of Claim 11, we establish another key lemma (Lemma 35). Suppose that Here I ω ⊆ d c is constructible while y G ∩ u ⊆ d c is finite and hence constructible as well. We conclude by Theorem 7(i) that p 0 c P-forces "L[G Y G ] |= ∀ m (ϕ(m) ⇐⇒ ¬ ψ(m))". However, c ∩ |p 0 | = {0}, so we are done.
Following the lemma, fix a condition p 0 ∈ G satisfying |p 0 | = {0} and (II). Proof. It will be technically easier to establish the result in the following form equivalent to the original form by Theorem 5(i): Assume that this fails; then there exist two conditions p, q ∈ P stronger than p 0 and satisfying: We can assume that |p| = |q| = {0}; otherwise apply Lemma 34 to formulas ϕ and ¬ ϕ. Strengthening p, q, if necessary, we can w. l. o. g. assume that (a) F q (0) = F p (0) and S p (0) ∪ S q (0) ⊆ F ∨ p (0) = F ∨ q (0). (= (3) in Section 3.8.) Working towards a contradiction, we w. l. o. g. assume that, in addition to (a), the following holds: (b) the symmetric difference S p (0) ∆ S q (0) contains a single element s ∈ Seq.

Final Argument
Here we finish the proof of both Claim 11 in Section 9.1 and Theorem 3. Suppose that G ⊆ P is a set P-generic over L, Y = Y G , and a set x ⊆ ω in L[G Y], formulas ϕ, ψ, and a condition p 0 satisfy assumptions (I), (II) above. Then, by Lemma 35, where, in L, A ⊆ K[U] × ω is a Σ HC n set such that A(p, m) ⇐⇒ p || − − K[U] ϕ(m) for all p ∈ K[U] and m (Corollary 9). It follows that, in L, x is Σ HC n (p 0 ), hence Σ 1 n+1 (w) (see Remark 2 in Section 4.3), where w ∈ L ∩ ω ω is a suitable code of p 0 .
To eliminate p 0 , consider the set Q of all conditions p ∈ K[U] such that |p| = |p 0 | and S p (ν) = S p 0 (ν) for all ν ∈ |p| = |p 0 |. Note that K[U] = K ∩ P is a set of the same complexity as P, that is, ∆ HC n−1 , and hence so is Q because |p 0 | and all S p 0 (ν), ν ∈ |p 0 | are finite sets. It follows that Q is ∆ HC n−1 . We now claim that, in L, x = {m ∈ ω : ∃ p ∈ Q A(p, m)} ; this obviously yields x being lightface Σ 1 n+1 in L. Indeed ⊆ follows by taking p = p 0 ∈ Q and applying (16). Now suppose that p ∈ Q and A(p, m), that is, p || − − K[U] ϕ(m). Recall that p 0 decides ϕ(m) by Lemma 35. However, p 0 || − − K[U] ¬ ϕ(m) is impossible since any condition in Q is compatible with p 0 . Therefore p 0 || − − K[U] ϕ(m) as required. Thus x ∈ Σ 1 n+1 in L is established. That the complementary set ω x is Σ 1 n+1 as well is verified the same way, using the formula ψ instead of ϕ. (Theorem 3)

Conclusions and Some Further Results
With proofs of the main theorems accomplished, in this final section some further results are briefly discussed, which we plan to achieve and publish elsewhere.

Separation
This is another application of submodels of the same basic model. Recall that given a class K of pointsets, the separation principle K-Sep claims that any two disjoint K -sets in the same space can be separated by a set in K ∩ K , where K consists of all complements of K -sets. The separation principle was introduced by N. Luzin. Luzin proved (see [25]) that Σ 1 1 -Sep holds, and then P. Novikov [26,27] demonstrated that Π 1 1 -Sep fails, while at the second projective level, the other way around, Π 1 2 -Sep holds but Σ 1 2 -Sep fails. As for higher projective levels, the separation problem belongs to a considerable list of problems related to the projective hieharchy in Luzin's book [25], Chapter V. Further development of set theory showed that Luzin's problems are very hard to solve. Some of them are now known to be independent of the Zermelo-Fraenkel set theory ZFC, while some others are still open in different aspects, but it is known that adding Gödel's axiom of constructibility V = L solves most of them. In particular, V = L implies [28,29] that Π 1 n -Sep holds but Σ 1 n -Sep fails for all n ≥ 3-similarly to the classical case n = 2. It follows that the statement ∀ n ≥ 3 ( Π 1 n -Sep ∧ ¬ Σ 1 n -Sep) is consistent with ZFC, and the problem is then to find a model in which we have Σ 1 n -Sep and/or ¬ Π 1 n -Sep (opposite to the state of affairs in L) for one or several or all indices n ≥ 3. This was the content of problems P 3029 and 3030 in the survey [8] of early years of forcing.
This turns out a very difficult question, and still open in its general forms, especially w. r. t. Σ 1 n -Sep. (Compare to Problem 9 in [30], Section 9.) As for the ¬ Π 1 n -Sep side, there are indications in the set-theoretic literature, that generic extensions, where both Σ 1 n -Sep and Π 1 n -Sep fail, are constructed by L. Harrington for n = 3 (see 5B.3 in [6]) and for arbitrary n ≥ 3 (see [8] and [31], p. 230). These results were indeed announced in Harrington's handwritten notes (Addendum A in [32]), with brief outline of some key arguments related mainly to case n = 3 and based on almost-disjoint forcing. There are no such results in Harrington's published works, assumed methods in their principal part (arbitrary n) are not used even for any other results, and separability theorems in this context are not considered. An article by Harrington, entitled "Consistency and independence results in descriptive set theory", which apparently might have contained these results, was announced in the References list in [31], to appear in Ann. of Math., 1978, but in fact it has never been published.
The following conjecture concludes Addendum A of Harrington's note [32]: In fact (we believe) there is a model of ZFC in which Separation fails for all of the following at once: Σ 1 n , Π 1 n , 3 ≤ n < ω , Σ m n , Π m n , 1 ≤ n < ω , 2 ≤ m < ω . ( Σ m n , Π m n are classes arising in the type-theoretic hierarchy).
The hypothesis is partially confirmed by the following our theorem (to appear elsewhere).
Friedman concludes [38] with a modified version of the above problem, given as Problem 87 : find a model of ZFC + "for any reals x, y, we have: if x ∈ L[y] then x is ∆ 1 3 in y". (18) This was solved in the positive by David [40], yet so far it is unknown whether this result generalizes to higher classes ∆ 1 n , n ≥ 4, or ∆ 1 ∞ . We also note that problems (17) and (18) were known in the early years of forcing, see, e.g., problems P 3110, 3111, 3112 in [8].

Axiom Schemata in 2nd Order Arithmetic
Different axiomatic systems in second-order arithmetic Z 2 is widely represented in modern research, in particular, in the context of reverse mathematics and other sections of proof theory. See e.g., Simpson [5] (Part B), and numerous articles, and from older sources-for example, Kreisel [41], where the choice of subsystems is called the central problem. These systems are obtained by joining a particular combination of comprehension schema CA, countable choice AC, dependent choice DC, transfinite induction TI and recursion TR , etc., to the basic theory, say ACA 0 . The schemata can be specifieded by the complexity of the core formula in the Kleene hierarchy, as well as by allowing or prohibiting parameters. (For the importance of parameters, see [41], section III.) The relationships between the subsystems have been actively studied. In particular, it is known that Σ 1 n+1 -CA is strictly stronger, than Σ 1 n -CA, and the same for AC and DC. Proofs of these results in e.g., [5,Chapter VII] use the fact that the schema at a higher quantifier level allows to get strictly more countable ordinals, than the schema at a lower level, but in essence, it is utilized that the (n + 1)th level schema proves the consistency of the nth level schema.
A few more complex results are known, where the compared systems are equiconsistent, despite the increase in quantifier complexity in the schemata, so the consistency argument doesn't work. It such a case one has to resort to set theoretic methods. This is the old result of A. Levy [42] that Σ 1 3 -AC does not follow from CA, as well as a recent theorem in [43] saying that Σ 1 3 -DC does not follow from AC; both are obtained using complex generic models of ZF without the full axiom of choice. The task of our further research in this direction will be to prove consistency theorems that demonstrate the importance of both the quantifier complexity and the presence of parameters in the Z 2 schemata.
Here CA* is the parameter-free part of the comprehension schema CA. Thus, both the quantifier complexity and the presence of parameters are essential for the deductive power of the comprehension schema in second-order arithmetic.

Remark 6.
We are grateful to one of the reviewers for pointing out possible connections of our research with some questions of fuzzy set theory [44,45], yet this issue cannot be considered for a short time.