A Model in Which Well-Orderings of the Reals First Appear at a Given Projective Level, Part II

We consider the problem of the existence of well-orderings of the reals, definable at a certain level of the projective hierarchy. This research is motivated by the modern development of descriptive set theory. Given n ≥ 3 , a finite support product of forcing notions similar to Jensen’s minimal-∆3 -real forcing is applied to define a model of set theory in which there exists a good ∆ 1 n well-ordering of the reals, but there are no ∆n−1 well-orderings of the reals (not necessarily good). We conclude that the existence of a good well-ordering of the reals at a certain level n ≥ 3 of the projective hierarchy is strictly weaker than the existence of a such well-ordering at the previous level n− 1. This is our first main result. We also demonstrate that this independence theorem can be obtained on the basis of the consistency of ZFC− (that is, a version of ZFC without the Power Set axiom) plus ‘there exists the power set of ω ’, which is a much weaker assumption than the consistency of ZFC usually assumed in such independence results obtained by the forcing method. This is our second main result. Further reduction to the consistency of second-order Peano arithmetic PA2 is discussed. These are new results in such a generality (with n ≥ 3 arbitrary), and valuable improvements upon earlier results. We expect that these results will lead to further advances in descriptive set theory of projective classes.


Introduction
This paper is written as a continuation of our earlier paper [1] under the same title, which thereby has to be viewed as Part I of this paper.
Problems related to the well-orderability of the real line R emerged in the early years of set theory. The axiom of choice AC implies that every set can be well-ordered, yet AC does not yield a concrete construction of any particular well-ordering of R. The famous discussion between Baire, Borel, Hadamard, and Lebesgue in [2] presents related issues widely discussed by mathematicians early in the 20th century.
Then, studies in descriptive set theory demonstrate that no well-ordering of R belongs (as a set of pairs) to the first-level projective classes Σ 1 1 , Π 1 1 , see e.g., Sierpinski [3]. This was a consequence of Luzin's theorem [4] saying that sets in Σ 1 1 ∪ Π 1 1 are Lebesgue measurable. (We refer to Moschovakis' monograph [5] in matters of both modern and early notation systems and early history of descriptive set theory. Yet, we may recall that Σ 1 1 consists of all continuous images of Borel sets in Polish spaces, Π 1 n consists of all complements of Σ 1 n sets, Σ 1 n+1 consists of continuous images of Π 1 n sets, and ∆ 1 n = Σ 1 n ∩ Π 1 n , for all n ≥ 1.) For the sake of brevity, we let WO(Γ) be the hypothesis saying: " There is a well-ordering of the real numbers which belongs to Γ as a set of pairs." Here, Γ is a given class of subsets of Polish spaces. Typical examples include projective classes Σ 1 n , Π 1 n , ∆ 1 n defined as above, and their effective subclasses resp. Σ 1 n , Π 1 n , ∆ 1 n , defined the same way but beginning with effective Borel sets, i.e., those that admit a Borel construction from an effectively (that is, computably) defined sequence of rational cubes.
Here, we can limit ourselves to classes ∆ 1 n and ∆ 1 n . Indeed, because if a well-ordering of the reals is say Σ 1 n then it is Π 1 n as well since x y is equivalent to x = y ∨ y x. Therefore, the result above can be summarized as ¬ WO(∆ 1 1 ). At the next projective level, Gödel [6] proved that WO(∆ 1 2 ) is consistent with the axioms of the Zermelo-Fraenkel set theory ZFC. This was established by a concrete definition of a ∆ 1 2 well-ordering L of the reals in the constructible universe L. Then, Addison [7] distinguished a crucial property of L now known as goodness. Namely, a ∆ 1 n -good wellordering is defined to be any ∆ 1 n well-ordering such that the class ∆ 1 n is closed under -bounded quantification. In other words, it is required that if P(y, x) is a binary ∆ 1 n relation on the reals, then the following relations Q(z, x) := ∃ y x P(z, y) and R(z, x) := ∀ y x P(z, y) belong to ∆ 1 n as well. The result by Gödel-Addison then claims that, in L, L is a ∆ 1 2good well-ordering of the reals. It follows that the existence of such a well-ordering is a consequence of the axiom of constructibility V = L, and hence it is consistent with ZFC. The ∆ 1 2 -goodness of L is behind many key results on sets of the second projective level, see for instance ([5], Section 5A).
As for the opposite direction, studies in the early years of modern set theory (see, e.g., Levy [8] and Solovay [9]) demonstrated that the non-existence statement, saying that there is no well-ordering of R definable by any set-theoretic formula with ordinal and real parameters (this includes Σ 1 ∞ = n Σ 1 n as a small part), is consistent as well. Modern research in connection with projective well-orderings touches on such issues as connections with forcing axioms [10,11], connections with large cardinals [12,13], connections with cardinal characteristics of the continuum [14,15], connections with the structure and properties of projective sets [16][17][18][19], and others. The following theorem contributes to this research field. The theorem is the first principal result of this paper. Theorem 1. Let n ≥ 3. Then, there exists a generic extension of L, in which: (i) WO(∆ 1 n ) is true, and moreover, there is a ω 1 -long ∆ 1 n -good well-ordering of the reals; (ii) WO(∆ 1 n−1 ) is false, that is, there are no ∆ 1 n−1 well-orderings of the reals, of any kind, i.e., not necessarily good.
Therefore, it is consistent that "WO(∆ 1 n ) holds, even by means of a ∆ 1 n -good well-ordering, and in the same time the stronger statement WO(∆ 1 n−1 ) fails".
As an immediate corollary of Theorem 1, we conclude that, for any n ≥ 3, the hypothesis WO(∆ 1 n−1 ) is strictly stronger than WO(∆ 1 n ) because there exists a model in which the latter holds whereas the former fails. Thus, the strict ascending condition ∆ 1 n−1 ∆ 1 n of the classes ∆ 1 n is adequately reflected in the property of the existence of a well-ordering of the reals in a given class.
Theorem 1 significantly strengthens a theorem in our previous paper [1], where we defined a generic extension of L in which there is a ∆ 1 n -good well-ordering but there do not exist ∆ 1 n−1 -good well-orderings of the reals. Thus, Theorem 1 improves the result in [1] by eliminating the goodness property in part (ii). This improvement required some crucial modifications in the proof of the theorem in part (ii) in this paper. Indeed in [1] we were able to use some well-known consequences of the ∆ 1 n−1 -goodness, in particular, the basis theorem saying that all non-empty Σ 1 n−1 sets of reals contain ∆ 1 n−1 elements.
This consequence is not available in the context of claim (ii) of the theorem since the ∆ 1 n−1 -goodness is not assumed. To circumvent this difficulty, this paper introduces an entirely new technique of working with the auxiliary forcing relation forc, developed in Sections 23-30 of this paper.
The other direction of the paper belongs to the context of the second-order Peano arithmetic PA 2 and related set and class theories. Theory PA 2 governs the interrelations between the natural numbers and sets of natural numbers, and is widely assumed to lay down working foundations for essential parts of modern mathematics including whatever is (or can be) developed by means of the theory of projective sets, see e.g., Simpson [20].
In particular, claims (i) and (ii) of Theorem 1 can be adequately presented by certain formulas of the language of PA 2 based on suitable universal formulas for classes Σ 1 n and Σ 1 n−1 . Therefore, for any given n ≥ 3, the statement (i) + (ii) of Theorem 1 is essentially a formula, say Φ n , of the language of PA 2 , whose consistency is established by the theorem. Thus, it becomes a natural problem to prove the consistency of Φ n as in Theorem 1 on the base of tools close to PA 2 , rather than (much stronger) ZFC tools. The next theorem, our second main result, solves this problem on the basis of ZFC − (minus stands for the absence of the Power Set axiom), which is a theory equiconsistent with PA 2 and thereby a substantial approximation towards PA 2 .
Further reduction to pure PA 2 will be the topic of our subsequent planned paper.

Outline of the Proof
Given n ≥ 3 as in Theorem 1, a generic extension of L, the constructible universe, was defined in [1], in which there exist ∆ 1 n -good well-orderings of the reals, but no ∆ 1 n−1good well-orderings. Here, to prove our main results, Theorems 1 and 2, we make use of a modified model. This model involves a product forcing construction in L, earlier applied in [18,21] for models with various effects related to the property of separation in the projective hierarchy, and also in [17] for a model in which the full basis theorem holds in the effective projective hierarchy (all non-empty Σ 1 ∞ sets of reals contain Σ 1 ∞ elements), in the absence of a Σ 1 ∞ well-ordering of the reals, for generic models with counterexamples to the countable axiom of choice AC ω and dependent choices DC in [22], to name a few examples.
Following the earlier papers [1,17,18], we make use of a sequence of forcings P(ξ), ξ < ω 1 , defined in L such that the product forcing P P = ∏ ξ P(ξ) adds a sequence of generic reals to L, uniformly Π 1 n−1 -definable in two arguments. Each forcing notion P(ξ) in this construction is a set of perfect trees T ⊆ 2 <ω , similarly to the Jensen minimal forcing defined in [23]. See more in ( [24], 28A) on Jensen's forcing. Infinite finite-support products of Jensen's forcing were first considered by Enayat [25], as demonstrated in [1], following this modification of Jensen-Enayat construction results in the existence of ∆ 1 ngood well-orderings in P P-generic extensions, thus witnessing (i) of Theorem 1.
Yet, a substantial modification of the Jensen-Enayat forcing construction is maintained in this paper, in order to get rid of using countable models of ZFC − (i.e., ZFC without the Power Set axiom). Different tools based on such models were used in earlier papers, e.g., in [1,18], in particular, for evaluating the complexity of various sets, leke e.g., the forcing notion itself. However, as one of our goals is to reduce the whole complexity of the construction of the models required, we have to remove models of ZFC − from our instrumentarium. Getting rid of models of ZFC − is thereby a principal technical achievement of this paper.
We begin in Sections 3-7 with a rather routine material related to arboreal forcings (those with perfect trees in 2 <ω as forcing conditions) and their countable finitesupport products called multiforcings, as well as finite tuples of trees called multitrees. The principal refinement relation π < < ϙ between multiforcings π, ϙ is introduced in Sec-tion 7. Roughly speaking, its meaning consists in the requirement that every multitree in MT(ϙ) (all multitrees related to ϙ) has to be meager in every multitree in MT(π). The second part of the paper (Sections 8-15) develops the background for the abovementioned technical achievement. It is based on the notion of sealing refinement π < < D ϙ ( D being a dense subset of MT(π)), which means that, besides π < < ϙ, every multitree p ∈ MT(ϙ) is covered by a finite collection of D-extendable multitrees in MT(π) (Definition 11). The following transitivity property takes place: if multiforcings π, ϙ, δ satisfy π < < D ϙ < < δ then (π ∪ cw ϙ) < < D δ, where ∪ cw is the component-wise union of multiforcings. We consider different types of dense sets to be sealed, including those which govern a kind of Cantor-Bendixson derivative procedure in Sections 12 and 13.
Corollary 12 summarizes the transitivity property as above for different versions of < < D . Theorem 4 proves the existence of sealing refinements. Theorem 5 provides consequences for generic extensions.
The next part of the paper (Sections [16][17][18][19][20][21][22] presents the key constructions involved in the proof of Theorem 1. We fix a natural number n ≥ 3 as in Theorem 1, and consider the constructible universe L as the ground model. Theorem 6 in Section 19 introduces a ω 1long < <-increasing sequence

Part I: Basic Constructions
Here, we present a rather routine material on arboreal forcing notions, i.e., those with perfect trees in 2 <ω in the role of forcing conditions. Then, in Section 6, we consider countable finite-support products of arboreal forcing notions, called multiforcings, as well as finite tuples of trees called multitrees. We introduce and study a principal refinement relation between arboreal forcing notions in Section 4 and between multiforcings in Section 7.

Trees and Arboreal Forcing Notions
Recall that 2 <ω is the set of all tuples (i.e., finite sequences) of 0, 1. If t ∈ 2 <ω and i = 0, 1, then t i is the extension of t by i taking the rightmost position. If s, t ∈ 2 <ω , then • s ⊆ t if and only if t extends s; • s ⊂ t if and only if s ⊆ t but s = t.
Generally, ⊂ denotes a strict inclusion (the equality "=" not allowed) in all cases in this paper, i.e., the same as . The non-strict inclusion is ⊆. The length of t is denoted by lh(t), and we put 2 n = {t ∈ 2 <ω : lh(t) = n}, the set all tuples of length n.
Trees in 2 <ω are considered. Thus, T ⊆ 2 <ω is a tree if t ∈ T =⇒ s ∈ T holds for all tuples s ⊂ t in 2 <ω . Then, the body if it is pruned and has no isolated branches; − We let PT contain all perfect trees ∅ = T ⊆ 2 <ω ; − If s ∈ T ∈ PT then we put T s = {t ∈ T : s ⊆ t ∨ t ⊆ s}; clearly T s ∈ PT as well.
If T ∈ PT then [T] is a perfect set in 2 ω .
this is equivalent to S ∩ T being finite. Then, S ⊥ T means the negation of S ⊥ T .
A set A ⊆ PT is an antichain if S ⊥ T holds for all S = T in A.
Definition 2 (arboreal forcing notions). A set P ⊆ PT is an arboreal forcing if u ∈ T ∈ P implies T u ∈ P. We define AF to be the set of all arboreal forcings P. Any P ∈ AF is: − Regular, if, for all trees S, T ∈ P, the intersection for some finite or countable antichain A ⊆ P-note that in this case the antichain A is unique and the forcing P has to be countable.
Note that every special arboreal forcing is regular. Definition 3 (perfect kernels). The perfect kernel of a tree T ⊆ ω <ω is the set ker(T) = {s ∈ T : there exists a perfect tree S with s ∈ S ⊆ T s }.
This is the largest perfect tree K ⊆ T .
Definition 4 (meet of perfect trees). If S, T ∈ PT then let S ∧ T = ker(S ∩ T).
The intersection S ∩ T may not even be pruned, but S ∧ T is a perfect (or empty) tree, Lemma 1. Let P be a regular arboreal forcing. Then, (i) If T 1 , . . . , T n ∈ P and X = [T 1 ] ∩ · · · ∩ [T n ] = ∅ then X is a finite union of sets of the form [S], S ∈ P, and then T = T 1 ∧ . . . ∧ T n = ker(T 1 ∩ · · · ∩ T n ) is a perfect tree equal to a finite union of trees in P, and [T 1 ] ∩ · · · ∩ [T n ] = [T].
(ii) is an easy corollary of (i).

Lemma 2.
If T ∈ P ∈ AF and S ∈ PT, T ⊆ S, then there exists a tree T ∈ P satisfying T ⊆ T and [T ] ∩ [S] = ∅.
Proof. Let T = T s , where s ∈ T S.

Refinements of Arboreal Forcings
In this section, we introduce the key notion of refinement of arboreal forcings. We remind that if P = P ; ≤ is any poset then a set D ⊆ P is: An arboreal forcing Q is a refinement of an arboreal forcing P, in symbol P < Q, if: (1) Q is dense in P ∪ Q, so that for any T ∈ P there is Q ∈ Q with Q ⊆ T ; (2) For any T ∈ Q we have T ⊆ fin P, meaning that there exists a finite D ⊆ P satisfying T ⊆ D, or equivalently [T] ⊆ S∈D [S]; (3) If T ∈ Q and S ∈ P then the intersection Thus, trees in the refinement Q define closed sets that are essentially smaller in the sense of category than the trees of the original arboreal forcing P do. Lemma 3. Assume that P < Q are arbitrary regular arboreal forcings. Then: (i) The union P ∪ Q is regular, too, and Q is open dense in P ∪ Q; (ii) If S ∈ P, T ∈ Q, and S ⊥ T , then S ∧ T is a finite union of trees T s ∈ Q, s ∈ T ; (iii) If S, S ∈ P, T ∈ Q, and T ⊆ S 1 ∩ S 2 , then there are trees R ∈ P and T ∈ Q satisfying T ⊆ T and T ⊆ R ⊆ S 1 ∩ S 2 .
Proof. To prove the regularity of P ∪ Q in (i), make use of (3). To prove (ii) apply (3) once again. Finally prove (iii). By Lemma 1(i), there are trees R 1 , . . . , R n ∈ P such that It follows by (ii) that there is a tuple s ∈ T such that T = T s ⊆ R i . We observe that T ∈ Q as Q is an arboreal forcing. Put R = R i .
Proof. If S ∈ P α and T ∈ P β , α < β, then P α < P β ; hence, [S] ∩ [T] is clopen in [T] by (3) above. This implies the regularity of P. The additional claims are elementary as well.

Sealing Refinements: Arboreal Forcings
Assume that P < Q. Then, a dense set D ⊆ P is not dense in P ∪ Q any more. Generally speaking, it may not even be pre-dense in P ∪ Q. Yet, it happens that there is a special type of dense sets called sealed dense that preserves pre-density under refinements, and a special type of refinements that turns dense sets into sealed dense sets.
The case of arboreal forcings considered here is a simplified introduction into the more important case of multiforcings in the next section.
, and 2) if S ∈ P then S ⊆ fin D.

Lemma 6.
Assume that D is a sealed dense set in P ∈ AF. Then, D is open dense.
Proof. To prove the density, assume that S ∈ P. Then, Then, there is a tuple t ∈ S such that T = S t ⊆ T i . Then, T ∈ P; hence, T ∈ D by the openness. Lemma 7. Assume that P < Q are arboreal forcings, and D is a sealed dense set in P. Then, Proof. Let U ∈ Q. By (2), in Section 4, the tree U is covered by a finite set of trees in P, hence, by a finite set T 1 , . . . , T n of trees T i ∈ D because D is sealed dense in P. Then, any It remains to note that each such V belongs to D ⇑Q.
Thus, the sealed denseness is preserved by the refinement operation. The next lemma shows that dense sets give rise to a sealed dense set by a certain kind of refinement. Definition 6. If P < Q are arboreal forcings and D ⊆ P, then P < D Q means that every tree T ∈ Q is covered by a finite union of trees in D.
Lemma 8 (see Lemma 5.4 in [18]). Assume that P, Q, R are arboreal forcings, D ⊆ P, and P < D Q < R. Then, P < D (Q ∪ R) and P ∪ Q < D R.
Given a multiforcing π , multitrees p, q are π-compatible, if there exists a multitree r ∈ MT(π) such that r p and r q, and otherwise are π-incompatible, in symbol p ⊥ π q. Sets A ⊆ MT of pairwise π-incompatible multitrees are π-antichains.
If multitrees are incompatible, then they are π-incompatible for any π . The next corollary shows that the inverse is true for regular multiforcings.
Corollary 2 (of Lemma 1(ii)). Let π be a regular multiforcing and p, q ∈ MT(π). Then, p, q are π-compatible if p, q are compatible as in Definition 9.
It follows that being an antichain is equivalent to being a π-antichain.
Definition 10 (component-wise meet of multitrees). Let π be a regular multiforcing. Say that a finite set of multitrees p 1 , . . . , p n ∈ MT(π) is compatible as a whole if for any index ξ ∈ i |p i |, we have i [p i (ξ)] = ∅. (Here, and below, it is understood that p i (ξ) = 2 <ω whenever ξ / ∈ |p i |.) In such a case, let us define a multitree p = cw i p i = p 1 ∧ cw . . . ∧ cw p n so that |p| = i |p i | and p(ξ) = i p i (ξ) = ker( i p i (ξ)) for all ξ ∈ |p|.
Corollary 4 (of Lemma 1(i)). Suppose that π is a regular multiforcing, and a finite set of multitrees p 1 , . . . , p n ∈ MT(π) is compatible as a whole. Then, p = cw i p i is a multitree, and [p] is a finite union of sets of the form [q], where q ∈ MT(π), |q| = |p|. Remark 1 (forcing). Let P ∈ AF be an arboreal forcing. We may treat P as a forcing notion, so that if T ⊆ T then T is a stronger condition. Clearly, P adjoins a real in 2 ω .
If π = P ξ ξ∈|π| ∈ MF is a multiforcing then the set MT(π), ordered as above, is accordingly viewed as a forcing notion which adjoins a generic sequence x ξ ξ∈|π| , where every x ξ = x ξ [G] ∈ 2 ω is a P ξ -generic real. Reals of the form x ξ [G] will be called principal generic reals in the extension by a MT(π)-generic set G.

Part II: Sealing Refinements
The first goal of this Part is to introduce a notion of sealing refinements for multiforcings, similar to the sealing refinements for arboreal forcings as in Section 5. This is a considerably more difficult case because obtaining adequate, working definitions both of the sealed density and the sealing refinements are somewhat less obvious. In particular, the notion of sealing refinement π < < D ϙ ( D being a dense subset of MT(π)), stipulates, that, besides π < < ϙ, every multitree p ∈ MT(ϙ) is covered by a finite collection of D-extendable multitrees in MT(π) (Definition 11). We consider different types of dense sets to be sealed, including those that govern a kind of Cantor-Bendixon derivative procedure in Sections 12 and 13.
Corollary 12 summarizes the transitivity property as above for different versions of < < D . Theorem 4 proves the existence of sealing refinements. Theorem 5 provides consequences for generic extensions. These are main results of Part II.

Sealing Refinements
Suppose that u is a multitree and D a set of multitrees. Define u ⊆ fin D, if there exists a finite subset D ⊆ D such that 1) |v| = |u| for all v ∈ D , and 2) [u] ⊆ v∈D [v]. Definition 11. Let π be a multiforcing and D ⊆ MT.
• A multitree p is D-extendable if there exists a multitree q ∈ D satisfying p = q |p|.
(1) D is sealed dense in MT(π) if D is open in MT(π) and u ⊆ fin D |u| ext (π) holds for every u ∈ MT(π).
We will use the notation D ⇑ϙ as in Definition 8 in the following lemmas.
Proof. (i) As the openness of D is given, prove the 'moreover' claim. Let u ∈ MT(π), X = |u|. Then, [u] ⊆ [p 1 ] ∪ . . . ∪ [p n ], where the multitrees p i ∈ MT(π) satisfy |p i | = X and are D-extendable. Then, u is compatible with at least one p i , and hence π-compatible by Corollary 2, so that there is a multitree v ∈ MT(π) with v p i , v u, and still |v| = X . It remains to be shown that v is D-extendable.
By the choice of p i , there exists a multitree q ∈ D with X ⊆ Y = |q| and q X = p i . Define a multitree w ∈ MT(π) so that |w| = Y , w X = v, and w (Y X) = q (Y X). Then, clearly, w q, and hence w ∈ D by the openness of D.
(iii) The openness of D ⇑ϙ in MT(ϙ) is obvious. To prove the sealed density, let u ∈ MT(ϙ). Since MT(π ∪ cw ϙ) is a product, we can assume that X = |u| ⊆ |π|. As where the multitrees p i ∈ MT(π) satisfy |p i | = X and are D-extendable; hence, there are p i ∈ D such that X ⊆ |p i | and p i = p i X .
For each p i , it follows by Corollary 7 that there exists a finite set V(i) of multitrees v ∈ MT(ϙ) satisfying still |v| = X and [u] Let v ∈ V(i) and q = p i , so that q ∈ D and p i = q X . As u p, there is a multitree w ∈ MT(ϙ) with w q and w X = v. Then, w ∈ D ⇑ϙ; therefore, w witnesses that v is (D ⇑ϙ)-extendable. This completes the proof that D ⇑ϙ is sealed dense in MT(ϙ).
To prove the pre-density of D in MT(π), let p ∈ MT(π). As π < < ϙ, there is u ∈ MT(ϙ), u p. There exists v ∈ D ⇑ϙ, v u, by the above. Then, v some q ∈ D. Thus, v witnesses that p, q are compatible; therefore, p, q are π-compatible by Corollary 2.
Thus, in the case of multiforcings, the sealed density is preserved by the refinement operation, and just a dense set D converts to a sealed dense set by the refinement < < D .

Two Examples
Here, we consider two important types of dense sets that can be made sealed dense.
If π is a multiforcing and p, q are π-incompatible multitrees in MT (not necessarily in MT(π)), then it is well possible that p, q become ϙ-compatible for another multiforcingϙ, even with π < < ϙ. To inhibit such a case, the following condition is introduced.
The following corollary reinterprets some key results above in terms of < < pq .
Proof. The proof is similar to Corollary 8. We make use of Lemma 10 and Lemma 12(i),(iv), in view of the fact that (N pq (π)⇑ϙ) ⊆ N pq (ϙ). As for the extra item (iii), we obviously have N pq (π) ⊆ N p q (π) provided p p and q q.

Real Names and Direct Forcing
In this section, a notational system for names of reals in 2 ω is introduced. It is appropriate for dealing with forcing notions MT(π). Definition 12. We let a real name be any c ⊆ MT × (ω × 2) such that the sets K c ni = { p ∈ MT : p, n, i ∈ c} satisfy the following condition: given n < ω , any p ∈ K c n0 , q ∈ K c n1 are incompatible, i.e., p ⊥ q (Definition 9). Let K c A real name c is small if every set K c n is finite or countable -then both the set |c| = n p∈K c n |p|, and c itself, are countable as well. Given a multiforcing π , a real name c is: is pre-dense (and then clearly open dense) in MT(π). − sealed π-complete, whenever each set K c n ⇑ π is sealed dense in MT(π). It is not assumed here that c ⊆ MT(π) × (ω × 2), or equivalently, K c n ⊆ MT(π), ∀ n.
Suppose that c is a real name. Say that a multitree p: Lemma 13. Let π be a multiforcing, p ∈ MT(π), n < ω , c a π-complete real name, T ∈ PT.
There exists i = 0, 1 and a multitree q ∈ MT(π), q p, which directly forces c(n) = i. There exists s ∈ T and a multitree q ∈ MT(π), q p, which directly forces c / ∈ [T s ].
The definition of direct forcing is associated with the following notion of genericity.
(2) If p, q ∈ G then there is r ∈ G with r p, r q. Say that G is π-generic over a given π-complete real name c, if in addition (3) G intersects every set of the form K c n ⇑ π , n < ω . In this case, we define a real c[G] ∈ 2 ω as follows: Lemma 14 (obvious). Suppose that π is a multiforcing and c is a π-complete real name. Let G ⊆ MT(π) be π -generic over c. If some p ∈ G directly forces c(n Example 4. If ξ < ω 1 , then let .
x ξ be a real name such that each set K .
x ξ ni consists of a single multitree P ξ ni , satisfying |P ξ ni | = {ξ } (a singleton), and P ξ x ξ is a small real name, π-complete for any multiforcing π . If a set G ⊆ MT(π) is π-generic over .
x ξ is a canonical name for x ξ [G].

Sealing Real Names and Avoiding Refinements
Here, we develop the idea of Definition 11 in the context of dense sets generated by real names.

Definition 14.
Let π < < ϙ be multiforcings and c be a real name. We define that ϙ seals c over π , in symbol π < < c ϙ, in case ϙ seals each set over π , i.e., π < < K c n ⇑ π ϙ, in the sense of Definition 11.
If π is a multiforcing then the forcing notion MT(π) adjoins a family of principal generic reals x ξ = x ξ [G] ∈ 2 ω , ξ ∈ |π|, where every x ξ is π(ξ)-generic over the ground set universe. Obviously many more reals are added. The next definition provides a sufficient condition for a π-complete real name c to generate not a real of the form x ξ .

Definition 15.
Suppose that π is a multiforcing and ξ ∈ |π|. A real name c is called nonprincipal at ξ over π , if the next set D c ξ (π) is open dense in MT(π): It will be demonstrated by Theorem 5(i) below that the non-principality at ξ implies that c is not a name of the real x ξ [G]. Moreover, the avoidance condition in the following definition will be demonstrated to imply that c is a name of a non-generic real.

Definition 16.
Let π be a multiforcing and Y ⊆ PT be a set of trees (e.g., Y = π(ξ) for some ξ ∈ |π|). A real name c is said to avoid Y over π , if for each tree Q ∈ Y , the set in the sense of Definition 11. Let π, ϙ be multiforcings, π < < ϙ, Y ⊆ PT be a set of trees. We write π < < c Y ϙ, if for each tree Q ∈ Y , ϙ seals the set D c Q (π) over π -that is formally π < < D c Q (π) ϙ. The relation π < < c Y ϙ will be applied mainly in case Y = ϙ(ξ) for some ξ ∈ |π|.
Theorem 11.1 in [18] demonstrates that if π is a small regular multiforcing, ξ ∈ |π|, and a real name c is non-principal at ξ over π (in the sense of Definition 15) then there is a special multiforcing ϙ with π < < c ϙ(ξ) ϙ (as in Definition 16). This fact will be used in the proof of Theorem 4 below. Lemma 15. Let π < < ϙ be regular multiforcings, Y ⊆ PT be a set of trees, c be a real name.
(i) If π < < c Y ϙ then c avoids Y over ϙ ; (ii) If c avoids Y over π then c avoids Y over ϙ as well.
by the avoidance assumption. Thus, D ⇑ϙ is a sealed dense set in MT(ϙ) by Lemma 10(iii). However, clearly,

Inductive Analysis of Well-Foundedness
Here, we accomplish some work related to the combinatorial description of forcing of well-founded trees. This will be applied in Part IV as a tool to define an auxiliary forcing relation for formulas in Σ 1 1 and Π 1 1 via the well-foundedness of certain trees. A set τ ⊆ MT × ω <ω is called a tree-name, if whenever s ⊂ t belong to ω <ω and p ∈ MT then p, t ∈ τ =⇒ p, s ∈ τ . Following Section 10, say that a multitree q ∈ MT directly forces s ∈ τ if q 0 , s ∈ τ for some q 0 ∈ MT(π) such that q 0 q.
Proof. It suffices to prove that just τ σ = τ ϙ ; all further inductive steps are similar. Recall that MT(ϙ) is open dense in MT(σ) by Corollary 6. It follows that one and the same set Thus, τ[G] is a tree is ω <ω because τ is a tree-name. For any tree T ⊆ ω <ω , let T be the pruned derivative, that consists of all s ∈ T that are not terminal nodes in T , and let T ∞ be the pruned kernel, the largest subtree S ⊆ T with no terminal nodes, that consists of all s ∈ T that belong to infinite branches B ⊆ T .

Lemma 17.
Assume that π is a multiforcing, τ ⊆ MT × ω <ω is a tree-name, and a set G ⊆ MT(π) is π -generic over τ . Then: Proof. (i) The contrary assumption results in the two following cases.
In particular, we have multitrees p ∈ G and p 0 such that p 0 , s ∈ τ π and p p 0 . By definition, the set W s p 0 (τ) has to be dense in MT(π) p 0 . Therefore, as G is generic over τ , and p ∈ G, some q ∈ W s p 0 (τ) belongs to G as well. By definition, q directly forces s j ∈ τ for some j < ω . Then, there is a multitree q 0 satisfying q q 0 and q 0 , s j ∈ τ . However, s j ∈ τ[G], contrary to the choice of s.
Case 2 : a tuple s j belongs to τ[G] but s does not belong to τ π [G]. Then, we have p 0 , s j ∈ τ , p ∈ G, p p 0 . It follows that p 0 , s ∈ τ π , and hence s ∈ τ π [G], contrary to the choice of s.
Corollary 11. Under the assumptions of the lemma, let p ∈ G.
(i) If p directly forces s ∈ τ ∞ π then τ[G] has an infinite chain containing s.
. This easily implies both items.

Absoluteness of the Derivative
The key result of this section will be to show that, under certain restrictions, the pruned derivative operation introduced in Section 12 is absolute with respect to refinements of the multiforcings involved. We need, however, to introduce another property of the form of sealing of dense sets, as in Definition 11. Definition 19. Let π be a multiforcing, and τ ⊆ MT × ω <ω is a tree-name, as in Section 12. Say that τ is sealed in π , if the following conditions hold: π , and D = + W s p (τ ν π , π) (see Section 12) is a set dense (then open dense as well) in MT(π), then D is sealed dense in MT(π).
Let ϙ be another multiforcing with π < < ϙ, so that ϙ is a refinement of π . Say that ϙ seals τ over π , symbolically π < < τ ϙ, if the next two conditions hold: (c) Just as (a) above; (d) If ν ≤ ν π (τ) and p, s ∈ τ ν π , and the set The following claims show the effect of < < τ in terms of Lemma 16.
Step 2: τ ϙ ⊆ τ π . Assume that p, s ∈ τ ϙ , so that p, s ∈ τ and the set W s p (τ, ϙ) is open dense in MT(ϙ) p . Then, the set W s p (τ, π)⇑ϙ is dense in MT(ϙ) p as well by (ii). We claim that the set W s p (τ, π) is dense in MT(π) p . Indeed, let p 1 ∈ MT(π) p . Then, there is q 1 ∈ MT(ϙ), q 1 p 1 . By (ii), there exists a pair q 2 p 2 of multitrees p 2 ∈ W s p (τ, π) and q 2 ∈ MT(ϙ) such that q 2 q 1 . Therefore, q 2 witnesses that the multitrees p 1 , p 2 in MT(π) are compatible, and hence, compatible right in MT(π) by Corollary 2. Thus, we have established that the set W s p (τ, π) is at least pre-dense, and then obviously dense in MT(π) p , as required.
(v) Prove (b) of Definition 19 for ϙ. In view of (iv) and Lemma 18, it suffices to only consider the case ν = 0, i.e., given p, s ∈ τ , and assuming that the set D = + W s p (τ, ϙ) is dense in MT(ϙ), we have to prove that D is sealed dense in MT(ϙ).
By definition, the set W s p (τ, ϙ) is dense (then open dense) in MT(ϙ) p . It follows by (ii) that the set W s p (τ, π)⇑ϙ is also dense in MT(ϙ) p . We conclude (see Step 2 above) that the set W s p (τ, π) itself is dense in MT(π) p . Then, the set E = + W s p (τ, π) itself is dense in MT(π)p. Therefore, we have π < < E ϙ because π < < τ ϙ is assumed. It follows by Lemma 10(iii) that E ↑ϙ is sealed dense in MT(ϙ). However, easily E↑ϙ ⊆ D by (ii). This ends the proof that D is sealed dense in MT(ϙ).
Proof. Our basic reference is Lemma 12(i)(iv), which has to be applied for those sets D involved in the definition of π < < < < M ϙ above (Definition 20).
This follows a refinement existence result.
Proof (sketch). The proof is based on some rather difficult results in [18] which we make use of here without proofs.
First of all, we can assume that M ∈ HC = all hereditarily countable sets, since all elements in M HC are irrelevant. Let M ⊆ HC be the (countable) set containing π, M, every ξ ∈ |π|, and every element of M. Let M + contain all sets X ⊆ HC, ∈-definable over HC, with sets in M allowed as parameters.
Definition 7.1 in [18] introduces the notion of M-generic refinements. Lemma 7.2 and Theorem 7.3 in [18] prove the existence of a special multiforcing ϙ, which satisfies |ϙ| = |π| and is an M-generic refinement of a given small regular multiforcing π . If ϙ is such, then Theorem 8.1 in [18] proves the relation π < < old D ϙ, and hence, π < < D ϙ, for all open dense sets D ∈ M + , D ⊆ MT(π). This implies (1)-(5) and (7) of Definition 20 because all dense sets involved there belong to M + by construction.
Finally, (6) of Definition 20 is separately established by Theorem 11.1 in [18]. We may note that (6) of Definition 20 differs from other items of this definition in that the list of the dense sets involved depends on the new multitree ϙ (the one claimed to exist). Therefore, it needs a special theorem in [18], namely Theorem 11.1.

Consequences for Generic Extensions
Lemma 19 shows that real names provide a suitable representation of reals in MT(π)generic extensions. Then, corollaries for non-principal names will be the subject of Theorem 5.

Lemma 19.
Assume that π is a regular multiforcing in the ground set universe V, and G ⊆ MT(π) is a MT(π)-generic set over V.
Proof. Claim (i) is a partial case of a general forcing theorem. To prove claim (ii), consider open dense set K c n ⇑ π = { p ∈ MT(π) : ∃ q ∈ K c n (p q)}, choose maximal antichains A n ⊆ K c n ⇑ π in those sets, note that each A n is countable by CCC, and finally, define Theorem 5. Suppose that π is a regular multiforcing, and ξ ∈ |π|. Then, the following holds.
(i) If MT(π) is a CCC forcing, G ⊆ MT(π) is a set generic over the ground set universe V, x ∈ 2 ω is a real in V[G], and x = x ξ [G], then there exists a small π-complete real name c ⊆ MT(π) × (ω × 2), non-principal at ξ over π , satisfying x = c[G] .
Proof. (i) By a general forcing theorem, there exists a π-complete real name c such that x = c[G] and MT(π) forces that c = x ξ [G]. We can assume that c is small, by Lemma 19 (as MT(π) is CCC). Let us prove that c is non-principal at ξ over π , meaning that the set is open dense in MT(π). As the openness is clear, it remains to prove the density. Let q ∈ MT(π). Then, by the choice of c, q MT(π)-forces c = x ξ [G]. Thus, we may assume that, for some n, the inequality c(n) = x ξ [G](n) is MT(π)-forced by q. By Lemma 13, there is a tuple s ∈ ω n+1 and a multitree p ∈ MT(π), p q, such that p directly forces the sentence s ⊆ c. It remains to be checked that s / ∈ p(ξ). Indeed, assume otherwise: s ∈ p(ξ). Then, the tree T = p(ξ) s belongs to MT(π). Define a multitree r by r(ξ) = T and r(ξ ) = p(ξ ) for all ξ = ξ . Then, r ∈ MT(π) and we have r p q. However, r directly forces both c(n) and x ξ [G](n) to be equal to the same number = s(n), which contradicts the choice of n.
(ii) Suppose towards the contrary that Q ∈ π(ξ) and c[G] ∈ [Q]. Lemma 15 (ii) implies that c avoids π(ξ) over ϙ as well. Lemma 10 implies that the set D c Q (π)⇑ϙ is open dense in MT(π ∪ cw ϙ). Therefore, the set D c Q (π) itself is pre-dense in MT(π ∪ cw ϙ). We conclude that G ∩ D c Q (π) = ∅ by the genericity, so that some multitree r ∈ G directly forces c / ∈ [Q]. It follows that c[G] / ∈ [Q], which is a contradiction.

Part III: The Forcing and the Model
Here, we present the key forcing constructions of the proof of Theorem 1. We consider the constructible universe L as the ground model. Fix a natural number n ≥ 3 as in Theorem 1. Theorem 6 in Section 19 introduces a ω 1 -long < <-increasing sequence #" Π ∈ L of special multiforcings, whose properties include: first, sealing many dense sets during the course of the construction; second, a sort of definable genericity in L; and third, a definability requirement-as in Definition 23. The subsequent key forcing notion P P ∈ L (which depends on #" Π ) is defined in Section 20. Its properties include CCC by Theorem 7. Then, we consider P P-generic extensions of L, called key models. The main results about key models are Theorem 8, which characterizes generic reals, and Theorem 9, which provides a ∆ 1 n -good well-ordering, with (i) of Theorem 1 as a consequence. Along with Theorem 7, they are the main results of this Part.
We begin with routine stuff on < <-increasing sequences of special multiforcings.
The following is a related form of < <-type definitions.

Definability Lemma
Recall that HC is the set of all hereditarily countable sets. Thus, X ∈ HC if the transitive closure TC (X) is at most countable. Note that HC = L ω 1 under V = L. We use the standard notation Σ HC n , Π HC n , ∆ HC n (slanted lightface Σ, Π, ∆ ) for classes of parameter-free definability in HC (no parameters allowed), and Σ n (HC), Π n (HC), ∆ n (HC) for full definability in HC (parameters from HC allowed). We will make use of the following known result, see e.g., Lemma 25.25 in Jech [24]: if X ⊆ 2 ω and n ≥ 1 then and similar equivalences for the classes Π , Π , ∆ , ∆ instead of Σ , Σ ,.

Lemma 22 (in L). The following ternary relation belongs to the class
Proof. Note first of all that # " MF ⊆ HC, so that the claim makes sense. The proof goes on by routine verification that all sets and relations involved are definable by ∆ 0 formulas, i.e., those with only bounded quantifiers over suitable countable sets such as ω or 2 <ω , despite the fact that their prima facie definitions may include quantifiers over uncountable sets such as 2 ω . Consider for instance the relation

C(S, T) := S, T ∈ PT ∧ [S] ∩ [T] is clopen in [S]
, that participates in several definitions, e.g., in the definition of regular arboreal forcing (Definition 2), in the definition of refinements in Section 4, etc. We observe that, because of the compactness of 2 ω , if S, T ∈ PT then for [S] ∩ [T] to be clopen in [S] it is necessary and sufficient that there exists a finite set U ⊆ S such that t∈U S t = {r ∈ S ∩ T : (S ∩ T) r is infinite}, and this condition is obviously ∆ 0 . Thus, this implies that the refinement relations < and < D between arboreal forcings (Sections 4 and 5) are definable by ∆ 0 formulas.
To check that π < < D ϙ as a ternary relation (Definition 11) is definable by ∆ 0 formula, it suffices to prove the ∆ 0 definability of the relation [u] ⊆ v∈D [v] (see the beginning of Section 8), where it is assumed that u ∈ MT, D ⊆ MT is finite, and |v| = |u| for all v ∈ D . Then, the relation [u] ⊆ v∈D [v] is equivalent to the following: if s ξ ∈ u(ξ) for all ξ ∈ |u| then there is v ∈ D such that s ξ ∈ v(ξ) for all ξ ∈ |u|.
However, this condition is ∆ 0 as required.

Auxiliary Diamond Sequence
We argue in L. Let us recall the technique of diamond sequences in L.
We obtain the following lemma as an easy corollary.

The Key Sequence
The next theorem (Theorem 6) is a crucial step towards the construction of the forcing notion that will prove Theorem 1. The construction employs some ideas related to definable generic transfinite constructions, and it will go on by a transfinite inductive definition of a sequence #" Π ∈ # " MF ω 1 in L from countable subsequences. The result can be viewed as a maximal branch in # " MF, generic with respect to all sets of a given complexity.

Definition 23 (in L)
. From now on a number n ≥ 3 as in Theorem 1 is fixed.
A sequence #" π ∈ # " MF blocks a set W if either #" π belongs to W (a positive block) or no sequence #" ϙ ∈ W ∩ # " MF extends #" π (a negative block). Any sequence #" MF ω 1 ∩ L, satisfying (in L) the following four conditions (A)-(D) for this n, will be called a key sequence: Π in the sense of Definition 22.
(C) If in fact n ≥ 4 and W ⊆ # " MF is a boldface Σ n−3 (HC) set (a definition with parameters), then there exists an ordinal γ < ω 1 such that the subsequence #" Π γ blocks W -so that either #" Π γ ∈ W , or there is no sequence ϙ ∈ W extending #" Π γ . Theorem 6 (in L). There exists a key sequence #" Proof. We argue under V = L, with n ≥ 3 fixed. In case n ≥ 4, let un n (p, x) be a universal Σ n−3 formula. In other words, the collection of all boldface Σ n−3 (HC) sets X ⊆ HC is equal to the family of all sets of the form Υ n (a) = {x ∈ HC : HC |= un n (a, x)}, a ∈ HC.

Proof (Claim). We skip a routine verification that
MF and a ∈ HC then for #" π to block Υ n (a) it is necessary and sufficient that This is a disjunction of Σ HC n−3 and Π HC n−3 , hence, ∆ HC n−2 , and we are finished.

Definition 24 (in
MF ω 1 , given by Theorem 6 for the number n ≥ 3 fixed by Definition 23. It satisfies (A)-(D) of Definition 23. We call this fixed #" Π ∈ L the key sequence.

#"
Π γ blocks W for some ordinal γ < ω 1 . The negative block is rejected because W is dense. Therefore,

The Key Forcing Notion
Based on Definition 24, we introduce some derived notions.
Proof. The next lemma claims that P P satisfies CCC.
Theorem 7 (in L). The forcing notion P P satisfies CCC. Therefore P P-generic extensions of L preserve cardinals.
Corollary 13 (in L). Let a set D ⊆ P P be pre-dense in P P. There is an ordinal γ < ω 1 such that D ∩ P P <γ is already pre-dense in P P.
Proof. We can w. l. o. g. assume that D is even dense. Let A ⊆ D be a maximal antichain in D. Then, A is a maximal antichain in P P since D is dense. Then, A ⊆ P P <γ for some ordinal γ < ω 1 by Theorem 7. However, A is pre-dense in P P.

The Key Model
We aim to prove Theorem 1 using P P-generic extensions of L, which we call key models. We will mostly argue in L and in ω L 1 -preserving generic extensions, in particular, in P P-generic extensions of L (cardinal-preserving by Theorem 7). Therefore, we will always have ω L 1 = ω 1 . This allows us to view things so that |Π| = ω 1 (rather than ω L 1 ).

Definition 26.
Let a set G ⊆ P P be generic over the constructible set universe L. If ξ < ω 1 , then, following the remark in the end of Section 6, − We put G(ξ) = { p(ξ) : p ∈ G and ξ ∈ |p|} ⊆ Π(ξ); − We define x ξ = x ξ [G] ∈ 2 ω as the unique real which belongs to T∈G(ξ) [T]; To conclude, the forcing notion P P adjoins an array X[G] of reals x ξ [G] to L, where every real Theorem 8. Assume that a set G ⊆ P P is P P-generic over L, ξ < ω 1 , and x ∈ L[G] ∩ 2 ω . Then, the following statements are equivalent: (2) the real x is Π(ξ)-generic over L ; Suppose that x ∈ L[G] ∩ 2 ω but x = x ξ [G], i.e., (1) fails. As P P = MT(Π) is CCC by Theorem 7, Theorem 5(i) implies the existence of a small Π-complete real name c ∈ L, such that x = c[G], c ⊆ P P × ω × 2, and c is non-principal at ξ over Π, meaning that is a set open dense in P P = MT(Π). By the smallness, c is a Π <γ 0 -complete real name for some ordinal γ 0 < ω 1 .

Corollary 14.
Let a set G ⊆ P P be P P-generic over L. Then, it holds in L[G] that X[G] belongs to the definability class Π HC n−2 , and hence, to class Π 1 n−1 by (1) of Section 20.

Proof. By Theorem 8, it is true in L[G] that ξ, x ∈ X[G] if and only if
which can be re-written as Note that the equalities µ = α(ξ) and Y = Π α (ξ) belong to the class ∆ HC n−2 by Corollary 26. This implies that the whole relation is Π HC n−2 , since the quantifier ∃ T ∈ Y is bounded.

Well-Orderings in the Key Model
According to the following theorem, the key model satisfies (i) of Theorem 1. The reals are treated as points of 2 ω , the Cantor space. The proof see Theorem 2 in [1].

Theorem 9.
Assume that a set G ⊆ P P is P P-generic over L. Then, in L[G], there is a ∆ 1 n -good well-ordering of 2 ω of length ω 1 , and hence (i) of Theorem 1 holds.
Our final step is to prove the result complementary to Theorem 9, that is, the key model also fulfills (ii) of Theorem 1. This will need some more effort. We will argue under the following assumption. Assumption 1. We assume that n ≥ 4 from now on.
This leaves aside the case n = 3 in (ii) of Theorem 1. Therefore, this case requires a separate consideration to justify the assumption. Assume that n = 3. We assert that (ii) of Theorem 1 holds in L[G] (which is the key model), where G is an arbitrary set P P-generic over L. Suppose towards the contrary that (ii) of Theorem 1 fails, so that there is a ∆ 1 2 well-ordering of the reals. (We even do not assume that the well-ordering is good.) Then, Theorem 25.39 in [24] by the product forcing theory. We conclude that y / ∈ L[x], which contradicts the choice of x.

Part IV: Non-existence of Simpler Well-orderings
Claim (ii) of Theorem 1 involves one more important technical tool related to the abovedefined key forcing notion P P. It turns out that the P-forcing relation of Σ 1 n−1 formulas is equivalent (up to level n − 1 of the projective hierarchy of formulas) to a certain auxiliary forcing relation forc defined and studied in Sections 23-30 below. Theorem 11 proves the equivalence. This auxiliary forcing is invariant with respect to permutations of indices ξ < ω 1 (Theorem 12), whereas the forcing P P itself is, generally speaking, not invariant in that sense. Such a hidden invariance plays a crucial role in the construction. Here, we make use of the invariance to prove, using Theorem 13, that the full version of (ii) holds in P Pgeneric extensions of L. Theorems 11-13 are the main results of this Part.

Auxiliary Forcing Relation
We argue in L. We make use of the second-order arithmetic language. It involves variables k, l, m, n, . . . (type 0) assumed to run over ω , and variables a, b, x, y, . . . (type 1) over 2 ω . The atomic formulas are only those of the form x(k) = n. Consider the extension L of this language, which allows us to substitute natural numbers for variables of type 0, and small real names (Definition 12) c ∈ L for variables of type 1. • We define natural classes LΣ 1 n , LΠ 1 n of L -formulas, as usual. • Given a formula ϕ in LΣ 1 n (resp., LΠ 1 n ), let ϕ − be the result of canonical transformation of ¬ ϕ to the LΠ 1 n (resp., LΣ 1 n ) form. Now, we introduce a relation p forc π ϕ between multitrees p, small multiforcings π ∈ # " MF, and closed L -formulas ϕ in LΣ 1 n ∪ LΠ 1 n , n ≥ 1, which will approximate the true P P-forcing relation. The definition proceeds by induction on the L -structure of ϕ. 1 • . Let π be a small regular multiforcing, p ∈ MT (not necessarily p ∈ MT(π)), and ϕ be a closed LΣ 1 1 formula. We assume that ϕ has the following canonical Σ 1 1 form: Consider a tree-name τ = τ(R) which consists of all pairs q, s ∈ MT × ω <ω such that there exist tuples t 1 , . . . , t k ∈ ω m , where m = lh(s), and multitrees r i j ∈ K c i j,t i (j) , 1 ≤ i ≤ k, j < m (see Definition 12), satisfying: (I) R(s n, t 1 n, . . . , t k n) for all n ≤ m; (II) q is equal to the multitree cw 1≤i≤k , j<m r i j -see Section 6 on cw , therefore q satisfies q r i j for all i, j, and hence q directly forces t i ⊂ c i for all 1 ≤ i ≤ k. We define p forc π ϕ if the following conditions (a)-(d) and (e1) hold: (a) Every q ∈ dom τ is sealed by π (see Example 2); (b) If q ∈ dom τ and p ⊥ π q then p ⊥ π q is sealed by π (see Example 3); (c) Every name c i in ϕ is sealed π-complete (see Definition 12); (d) τ = τ(R) is sealed in π (see Definition 19); (e1) p directly forces Λ ∈ τ ∞ π , i.e., there is a multitree p 0 p with p 0 , Λ ∈ τ ∞ π . 2 • . Let π be a small regular multiforcing, p ∈ MT (not necessarily p ∈ MT(π)), and ψ be a closed LΠ 1 1 formula. We assume that ψ has the following canonical , c 1 m, . . . , c k m), where R ⊆ (ω <ω ) k+1 is a recursive relation and every c i is a small real name.

Remark 3.
If p forc π ϕ holds then it is not necessary that p ∈ MT(π), and it is not necessary that every name c in ϕ satisfies c ⊆ MT(π) × (ω × 2).

Elementary Properties of the Auxiliary Forcing
We still argue in L.
Proof. If ϕ is a formula in LΣ 1 1 as in 1 • of Section 23, and p forc π ϕ, which is witnessed by (a)-(d) and (e1), then q forc ϙ ϕ also holds.
Recall that ϕ − is the canonical transformation of ¬ϕ to the prenex form.

Forcing the First Level Formulas
The following theorem shows that the auxiliary forcing relation is properly connected with the ordinary forcing at least for formulas in LΣ 1 1 ∪ LΠ 1 1 .
Proof. Case 1: ϕ is a formula in LΣ 1 1 , of the canonical form (f1), that is, where R ⊆ (ω <ω ) k+1 is a recursive relation and every c i ⊆ MT × (ω × 2) is a small real name, and p forc π α ϕ, so that properties (a)-(d) and (e1) of Section 23 hold for π = π α , τ = τ(R), and p. In particular, (*) p directly forces Λ ∈ τ ∞ π α by (e1). Now, consider any set G ⊆ Q, generic over the given universe V and containing p; the goal is to prove that ϕ[G] holds in V[G]. The following lemma simplifies the task . Lemma 32. The set G α = G ∩ MT(π α ) is π α -generic over τ in the sense of Definition 18, and is π α -generic over each name c i occurring in ϕ, in the sense of Definition 13.
In addition, c i [G] = c i [G α ] for any such name c i , as well as Proof (Lemma). First of all, we have to check (2) of Definition 13 for G α . Thus, let u, v ∈ MT(π α ) belong to G α . However, u, v are sealed by π α by (a) of Section 23, thus the sets D u (π α ) = {q ∈ MT(π) : |u| ⊆ |q| ∧ (q u ∨ q ⊥ u)} and D v (π α ) are sealed dense in MT(π α ). Then, D u (π α ) ∩ D v (π α ) is sealed dense in MT(π α ) as well by Lemma 11(i). Therefore, if α < β < κ = dom( #" π) then is a sealed dense, and therefore open dense, set in MT(π β ) by Lemma 10(iii). We conclude, by the genericity of G, that there is a multitree w ∈ G that belongs to (D u (π α ) ∩ D v (π α ))⇑π β for some β > α. Then, there is a multitree q ∈ D u (π α ) ∩ D v (π α ) satisfying w q. We have q ∈ G since w ∈ G. On the other hand, u, v ∈ G as well; therefore, q cannot be incompatible with u, v. It follows that q u and q v.
The proof that c i [G] = c i [G α ] for any name c i in ϕ is similar.
Let s ∈ τ[G], m = lh(s). We have q, s ∈ τ for some q ∈ G. By definition, there are tuples t 1 , . . . , t k ∈ ω m satisfying (I) and (II) of Section 23; in particular, q directly forces t i ⊂ c i by (II) for all i = 1, . . . , k, and hence, t i = y i m for all i. Thus, s ∈ T y 1 ,...,y k by (I).
Conversely let s ∈ T y 1 ,...,y k , that is, R(s n, t 1 n, . . . , t k n) holds for all n ≤ m = lh(s), where t i = y i m for all i. As y i = c i [G], there is a family of conditions r i j ∈ K c i j,t i (j) ∩ G, 1 ≤ i ≤ k, j < m. Then, the multitree q = cw 1≤i≤k , j<m r i j belongs to G as well as G is generic, and by definition we have q, c ∈ τ(R). It follows that s ∈ τ [G].

Forcing Inside the Key Sequence
It is implied by Theorem 11 below that the forcing relation forc π , considered with the terms π = Π α of the key sequence #" Π , really approximates the true P P-forcing relation at level n − 1 and below. Recall that n ≥ 4 is assumed (see Assumption 1).
Proof. Lemma 29 implies (i). To prove (ii), choose an ordinal β with α < β < ω 1 , satisfying q ∈ MT( #" Π β), and apply (i). To check (iii), we observe that p, q are incompatible in P P, as otherwise (i) leads to contradiction. On the other hand, multitrees incompatible in P P are ⊥ by Corollary 2.
Theorem 11. Assume that 1 ≤ n ≤ n − 2 and ψ is a closed formula in LΠ 1 n ∪ LΣ 1 n+1 , with all names small and Π-complete, and p ∈ P P. Then p P P-forces ψ[G] over L in the usual sense, if and only if p forc ψ.
Proof. Let − be the ordinary P P-forcing relation over L.

Permutation Invariance
The theory of forcing admits various invariance theorems. Theorem 12 is related to the invariance of the auxiliary forcing under permutations.
We argue in L. Let PERM be the set of all bijections h : ω 1 onto −→ ω 1 , satisfying h = h −1 and such that the non-identity domain NI(h) = {ξ : h(ξ) = ξ } is at most countable. Bijections in PERM will be called permutations.

The Non-Well-orderability Claim, Part I
Here, we begin the proof of Theorem 1 in part (ii). It will be completed in the end of Section 30. We are going to establish the following even somewhat stronger result.
Theorem 13. Assume that a set G ⊆ P P is P P-generic over L. Then, in L[G], there is no Σ 1 n−1 well-orderings of the reals, and moreover, no Σ 1 n−1 relation well-orders the set {x ξ [G] : ξ < ω L 1 }.
Our plan is to infer a contradiction from the next assumption contrary to Theorem 13.

Assumption 2.
Assume that Φ(x, y) is a Σ 1 n−1 parameter-free formula, α < ω 1 , p ∈ P P α = MT(Π α ), and p P P-forces, over L, that "the relation < Φ , defined by x < Φ y if Φ(x, y), strictly well-orders the set {x ξ [G] : We begin with the next lemma. See Example 4 regarding real names of the form .
The goal of the next lemma is to strengthen Lemma 35 to the effect that a whole dense set of conditions with the same property will be obtained.
Proof. (i) Using Lemma 34, we define π k by induction so that for each k there is a certain pair of ξ k ∈ |π k | and q k ∈ MT(π k ), satisfying: Moreover, the enumeration by ξ k and q k can be arranged so that for each ordinal ξ ∈ | #" π| and condition q ∈ MT(π) there exists k such that ξ k = ξ and q k = q. However, #" π is as required. Claim (ii) is just a reformulation of (i).

The Non-Well-orderability Claim, Part II
Still arguing under the conditions of Assumption 2, we proceed with the following construction.
Let − be the P P-forcing relation over L. It essentially coincides with forc by Proposition 11. Therefore, the lemma implies the following corollary. Corollary 16. If ξ ∈ | # " π * | then the following set is open dense in P P : , .
Proof of Theorem 13. It follows from Corollary 16 that if G if P P-generic over L, then the set {x ξ [G] : ξ ∈ |π * |} contains no < Φ -minimal element, which contradicts Assumption 2. The contradiction negates Assumption 2 and thereby proves Theorem 13.
Combining Theorems 13 and 9, we complete the proof of Theorem 1.

Part V: Final
This final Part contains Section 31 with a short proof of Theorem 2 and a brief discussion of its possible reduction to a theory weaker than ZFC − + 'P(ω) exists'. We finish in Section 32 with conclusions and problems.

Proof of Theorem 2 and Comments
First of all, we recall that ZFC − is a subtheory of ZFC obtained as follows: (a) We exclude the Power Set axiom PS; (b) The well-orderability axiom WA, which claims that every set can be well-ordered, is substituted for the usual set-theoretic Axiom of Choice AC of ZFC; (c) The Separation schema is preserved, but the Replacement schema (which happens to be not sufficiently strong in the absence of PS) is substituted with the Collection schema: ∀ X ∃ Y ∀ x ∈ X ∃ y Φ(x, y) =⇒ ∃ y ∈ Y Φ(x, y) . A comprehensive account of main features of ZFC − is given in e.g., [27][28][29].
Proof of Theorem 2. Arguing in ZFC − , let us drop to the subuniverse L − of all constructible sets in the ZFC − universe of discourse. Then, L − satisfies ZFC − too, and if P(ω) exists then P(ω) ∩ L − ∈ L − exists in L − . Thus, instead of ZFC − + 'P(ω) exists', we argue in the theory ZFC − + (V = L) + 'P(ω) exists', whose universe is L − . Now, the existence of the power set P(ω) = {X : X ⊆ ω} leads to the existence of sets such as ω 1 and HC = L ω 1 , and basically, the existence of all sets involved in the construction of the key forcing notion P P (including P P itself). After this remark, all arguments in the proof of Theorem 1 in Parts I, II, III, and IV above naturally go through, giving the proof of Theorem 2 by means of a P P-generic extension of L − .
It is really interesting to further reduce the assumptions of Theorem 2 down to PA 2 (see [20,30,31] and elsewhere on second-order Peano arithmetic PA 2 ) or ZFC − without the extra assumption of the existence of P(ω), or the associated class theory GBc − , which is formalized in a two-sorted language with separate variables and quantifiers for sets and classes, so that lower-case letters are used for set variables, whereas upper-case letters are used for class variables. The minus − still reflects the absence of the Power Set axiom. The axiomatization of GBc − (see e.g., [29]) includes axioms for sets (exactly those of ZFC − ) and those for classes. In particular, (1) extensionality for classes; (2) the class replacement axiom asserting that every class function restricted to a set is a set; and (3) a predicative comprehension schema asserting that every collection of sets, definable by a formula with no quantified class variables, is a class.
Such objects as ω 1 and HC are legitimate classes in GBc − , and such are all ZFCsets that play any role in the proof of Theorem 1 above, with one notable exception. The exceptional case being the ∆ HC 1 3 ω 1 -sequence used in Lemma 24. The ZFC construction of such a sequence (as e.g., in [24]) can be maintained as a proper class in GBc − + 'all sets are constructible' as well as in ZFC + (V = L). However, unfortunately, the proof of the 3 ω 1 -property of the resulting sequence does not go through in GBc − because the ZFC proof involves ordinals beyond ω 1 , and hence, does not directly translate to the GBc − level. This will be the subject of our forthcoming paper aimed at solving this technical obstacle by means of recently discovered methods as, e.g., in [35,36].

Conclusions and Problems
In this study, the method of finite-support products of Jensen's forcing was applied to the problem of obtaining a model of ZFC in which, for a given n ≥ 3, there is a ∆ 1 n -good well-ordering of the reals, but no well-orderings of the reals exist in the class ∆ 1 n−1 at the preceding level of the hierarchy. This is achieved by Theorem 1, our first main result. We also demonstrate that this theorem can be obtained on the basis of the consistency of ZFC − (i.e., ZFC sans the Power Set axiom) plus the claim that P(ω) exists, which is a much weaker assumption than the consistency of ZFC usually assumed in such independence results obtained by forcing method. This is achieved by Theorem 2, our second main result. Two principal technical achievements, related to getting rid of countable models of ZFC − as a technical tool and according treatment of the auxiliary forcing, were mentioned in Section 2. These are new results in such a generality (with n ≥ 3 arbitrary), and valuable improvements upon our earlier results in [1]. They may lead to further progress in studies of the projective hierarchy.
From our study, it is concluded that the technique of definable generic inductive construction of forcing notions in L that carry hidden automorphisms, developed for Jensen-type product forcing in our earlier papers [17,18,21], succeeds to solve other important descriptive set theoretic problems of the same kind, using Theorems 1 and 2.
These results (Theorems 1 and 2) continue the series of recent research such as a model [37] in which there is Π 1 n real singleton {a} that codes a cofinal map f : ω → ω L 1 , while every Σ 1 n set X ⊆ ω is constructible, and hence, cannot code a cofinal map ω → ω L 1 , and another model [38], in which there is a non-ROD-uniformizable Π 1 n set with countable cross-sections, while all Σ 1 n sets with countable cross-sections are ∆ 1 n+1 -uniformizable-in addition to the research already mentioned in Section 2 above.
This study may also be a contribution to the search for solutions of several similar and still open problems related to the projective hierarchy, such as separation of the countable AC at different levels of the projective hierarchy, a similar problem for the principle DC of dependent choices, and a critically significant problem posed by S. D. Friedman in ( [39], p. 209) and ( [40], p. 602): assuming the consistency of an inaccessible cardinal, find a model for a given n in which all Σ 1 n sets of reals are Lebesgue measurable and have the Baire and perfect set properties, while there is a ∆ 1 n+1 well-ordering of the reals. From the result of Theorem 1, the following more concrete problems arise. The boldface specification means that the real parameters are allowed in the definitions of pointsets, whereas they are not allowed in the lightface case. This is a principal difference. Problem 2. Prove a version of Theorem 1 with the additional requirement that the negation 2 ℵ 0 > ℵ 1 of the continuum hypothesis holds in the generic extension considered.
The model for Theorem 1 introduced in Section 20 definitely satisfies the continuum hypothesis 2 ℵ 0 = ℵ 1 . The problem of obtaining models of ZFC in which 2 ℵ 0 > ℵ 1 and there is a projective well-ordering of the real line, has been known since the beginning of modern set theory. See, e.g., problem 3214 in an early survey [41] by Mathias. Harrington [42] solved this problem using a generic model in which 2 ℵ 0 > ℵ 1 and there is a ∆ 1 3 wellordering of the continuum, using a combination of methods based on such coding forcing notions as the almost-disjoint forcing [43] and a forcing by Jensen and Johnsbråten [44]. Such a different forcing notion as the product/iterated Sacks forcing [45,46] may also be of interest here.  Data Availability Statement: Not applicable. The study did not report any data.