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

: The problem of the existence of analytically deﬁnable well-orderings at a given level of the projective hierarchy is considered. This problem is important as a part of the general problem of the study of the projective hierarchy in the ongoing development of descriptive set theory. We make use of a ﬁnite support product of the Jensen-type forcing notions to deﬁne a model of set theory ZFC in which, for a given n > 2, there exists a good ∆ 1 n well-ordering of the reals but there are no such well-orderings in the class ∆ 1 n − 1 . Therefore the existence of a well-ordering of the reals at a certain level n > 2 of the projective hierarchy does not imply the existence of such a well-ordering at the previous level n − 1. This is a new result in such a generality (with n > 2 arbitrary), and it may lead to further progress in studies of the projective hierarchy.


Introduction
The problem of the well-orderability of the continuum of real numbers R has been known in set theory since the time of Cantor and Hilbert. Zermelo's axiom of choice AC directly postulates the existence of a well-ordering of R (and of any other set of course), but this is far from an effective construction of a concrete, "nameable" well-ordering of R. We refer to the famous "Sinq Lettres" [1] in matters of the discussion on these issues in early set theory.
Somewhat later, using the methods of the descriptive set theory that just emerged, it was established that no well-ordering of R belong to the first-level projective classes Σ 1 1 , Π 1 1 -and then to ∆ 1 1 since x y iff x = y or y x. This is an easy consequence of Luzin's theorem [2] that sets in Σ 1 1 ∪ Π 1 1 are Lebesgue measurable; see, for example, Sierpinski [3]. (We use the modern notation Σ 1 n , Π 1 n , ∆ 1 n for projective classes and Σ 1 n , Π 1 n , ∆ 1 n for their effective subclasses, sometimes also called "lightface"; see, for example, monographs [4,5].) The next key result was obtained by Gödel [6]: it is true that in the Gödel constructible universe L, there exists a ∆ 1 2 well-ordering L of the reals, or saying it differently, the existence of a ∆ 1 2 well-ordering of the reals is a consequence of the axiom of constructibility V = L. It follows that the existence of a ∆ 1 2 well-ordering of the reals is consistent with the axioms of the Zermelo-Fraenkel set theory ZFC (containing the axiom of choice AC) because the axiom of constructibility V = L itself is consistent by [6].
Addison [7] singled out an important additional property of the Gödel well-ordering L . Namely, let a ∆ 1 n -good well-ordering is any ∆ 1 n well-ordering such that for every binary ∆ 1 n relation P(x, y) on the reals, the relations Q(x, y) := ∃ x x P(x , y) and R(x, y) := ∀ x x P(x , y) belong to ∆ 1 n as well, so that the class ∆ 1 n is closed under -bounded quantification (see Moschovakis [5]). In these terms, the Gödel-Addison result says that L is a ∆ 1 2 -good well-ordering of the reals in L, and hence the existence of such a well-ordering follows from V = L and is consistent with ZFC. The property of ∆ 1 n -goodness of L is behind many key results on projective sets in Gödel's universe L, see Section 5A in [5].
In the opposite direction, it was established in the early years of modern set theory (see, for example, Levy [8] and Solovay [9]) that the statement of the non-existence of a well-ordering of the reals of any projective class is consistent as well.
Recent studies on projective well-orderings explore various topics concentrated around the general problem formulated by Moschovakis [5] (Introduction) as follows: [T]he central problem of descriptive set theory and definability theory in general [is] to find and study the characteristic properties of definable objects.
For instance, it is established in [10] that the bounded proper forcing axiom BPFA combined with ω 1 = ω L 1 implies the existence of a ∆ 1 3 well-ordering of the reals. Studies in [11][12][13] presented different constructions of countable support-iterated generic models which, first, admit controlled cardinal characteristics of the continuum, and second, admit a ∆ 1 3 well-ordering of the reals. A model of ZFC in which the nonstationary ideal on ω 1 is ℵ 1 -saturated and whose reals admit a ∆ 1 4 well-ordering, is defined in [14] under a large-cardinal hypothesis. A finite support product of clones of Jensen's minimal singleton forcing [15] is used in [16] to define a model in which any non-empty analytically definable set of reals contains an analytically definable real (the full basis theorem), but there is no analytically definable well-ordering of the reals of any class ∆ 1 n . However one of principal questions related to projective well-orderings remained unsolved by those studies. We let WO(∆ 1 n ), respectively, WO(∆ 1 n ) be the following statement: there is a well-ordering of the reals which, as a set of pairs, belongs to, respectively, ∆ 1 n , ∆ 1 n , for the sake of brevity. As the strict inclusions hold for all n ≥ 2, we have accordingly and the ensuing principal problem is as follows.
In other words, for a given n ≥ 3, are there models of ZFC in which WO(∆ 1 n ) holds but WO(∆ 1 n−1 ) fails, as well as those in which WO(∆ 1 n ) holds but WO(∆ 1 n−1 ) fails?
This problem is a version of a well-known problem posed by S. D. Friedman, one of the leading experts in set theory, in [17] (Problem 11 on page 209) and [18] (Problem 9 in Section 9). Friedman's problem asks for a model for WO(∆ 1 n ) plus the Lebesgue measurability and the Baire property of all Σ 1 n−1 sets of reals, which is somewhat stronger than the related requirements of the failure of WO(∆ 1 n−1 ) and WO(∆ 1 n−1 ) in Problem 1. The following theorem (our main result here) contributes to the studies of these problems. It gives a partial positive solution of the "lightface" part of Problem 1 that uniformly works for all values of the index. No such result has ever been obtained before. Theorem 1. Let n ≥ 3. There exists a generic extension of L, in which it is true that (i) there is a ∆ 1 n -good well-ordering of the reals, of length ω 1 ; (ii) there are no ∆ 1 n−1 -good well-orderings of the reals.

Outline of the Proof
Given n ≥ 3, our plan is to make use of a generic extension of L defined in [19] in order to get a model where the separation principle fails for both classes Σ 1 n and Π 1 n . This extension utilizes a sequence of forcing notions P(ξ), ξ < ω 1 , defined in L so that the finite-support product P = ∏ ξ P(ξ) satisfies CCC and adjoins a sequence of generic reals x ξ ∈ 2 ω , satisfying the following crucial definability property: the binary relation "x ∈ 2 ω is a real P ξ -generic over L" (with arguments x, ξ ) is 1 ]. This will suffice to define a well-ordering satisfying Theorem 1(i).
On the other hand, Claim (ii) of Theorem 1 involves another crucial property: the Pforcing relation of Σ 1 n formulas is equivalent to an auxiliary forcing relation forc invariant w. r. t. permutations of indices ξ < ω 1 .
Each factor forcing P(ξ) consists of perfect trees in 2 <ω and is a clone of Jensen's minimal forcing defined in [15]; see also [20] (28A) on this forcing. The technique of finitesupport products of Jensen's forcing, which we owe to Enayat [21], was exploited recently to obtain generic models with counterexamples to the separation theorem for both Σ 1 3 and Π 1 3 [22], and some counterexamples to the axiom of choice [23], to name a few applications. Section 3 introduces perfect trees in 2 <ω , arboreal forcing notions, multitrees (finite products of trees), multiforcings (countable products of arboreal forcing notions).
Section 4 defines the refinement relation and presents the principal properties of refinements. We define the set # " MF of all countable sequences #" π of small multiforcings, increasing in the sense of the refinement relation.
Then, following our earlier paper [19], we introduce the key forcing notion P P P = P P P n for Theorem 1 with a fixed n ≥ 3, and study the main properties of P P P-generic models in Section 5. Theorem 2 in Section 6 shows that condition (i) of Theorem 1 holds in P P P-generic extensions of L.
Sections 7 and 8 introduce an auxiliary forcing relation forc, which approximates the truth in P P P-generic extensions for Σ 1 n−1 -formulas and below, so that the relation forc restricted to any class Σ 1 m or Π 1 m , m ≥ 2, is Σ 1 m , respectively, Π 1 m itself. The tail invariance and permutation invariance of the relation forc is established in Section 9. (We may note in brackets that the product forcing notion P P P itself is not permutation invariant.) Using these results, we finally prove that condition (ii) of Theorem 1 holds in P P P-generic extensions of L in Section 10. This completes the proof of Theorem 1.
This paper is a sequel of [19] in many technical details, and hence some intermediate results involved in the proof of Theorem 1 are taken from [19] without proof.

Arboreal Forcing Notations and Multiforcings
Let 2 <ω be the set of all tuples (finite sequences) of numbers 0, 1. If s, t ∈ 2 <ω , then s ⊆ t means that t extends s, while s ⊂ t means proper extension. If t ∈ 2 <ω then lh(t) is the length of t, and 2 n = {t ∈ 2 <ω : lh(t) = n} (tuples of length n).
PT is the set of all perfect trees ∅ = T ⊆ 2 <ω . Thus a tree ∅ = T ⊆ 2 <ω belongs to PT if it has no endpoints and no isolated branches. In this case, is a perfect set. If s ∈ T ∈ PT then put T s = {t ∈ T : s ⊆ t ∨ t ⊆ s}; then T s ∈ PT.
Let an arboreal forcing be any set P ⊆ PT such that if u ∈ T ∈ P then T u ∈ P. Let AF be the set of all arboreal forcings P.
A forcing P ∈ AF is special, if there is a finite or countable antichain A ⊆ P such that P = {T s : s ∈ T ∈ A}-the antichain A is unique and P is countable in this case.
Accordingly, let a multitree be any function p : |p| → PT, with a finite support |p| = dom p; MT will be the collection of all multitrees. Every p ∈ MT can be seen as an indexed set p = T p ξ ξ∈|p| , where T p ξ = p(ξ) ∈ PT for all ξ ∈ |p|. We order MT componentwise: q p (q is stronger than p) if |p| ⊆ |q| and T q ξ ⊆ T p ξ for all ξ ∈ |p|. Assume that π = P ξ ξ∈|π| is a multiforcing. Let a π-multitree be any multitree p ∈ MT such that |p| ⊆ |π|, and if ξ ∈ |p|, then the tree p(ξ) = T p ξ belongs to P ξ . The set MT(π) of all π-multitrees can be identified with the finite support product ∏ ξ∈|π| P ξ of the arboreal forcings P ξ involved.
Any arboreal forcing P ∈ AF is considered a forcing notion (if T ⊆ T , then T is a stronger condition); such a forcing P adjoins a real in 2 ω .

The Key Sequence, Key Forcing Notation, and Key Model
In this section, we introduce the forcing notion to prove Theorem 1, defined in our earlier paper [19]. It has the form MT(Π), for a certain multiforcing Π with |Π| = ω 1 . The multiforcing Π is equal to the componentwise union of terms of a certain sequence #" Π ∈ # " MF ω 1 which we present in Definition 1. Yet we need to recall one more concept. Let HC be the set of all hereditarily countable sets. Thus X ∈ HC if the transitive closure TC (X) is at most countable.
We use standard notation Σ HC n , Π HC n , ∆ HC n (slanted Σ, Π, ∆ ) for classes of lightface definability over HC (no parameters allowed), and Σ n (HC), Π n (HC), ∆ n (HC) for boldface definability over HC (parameters in HC allowed). The following useful result connects projective hierarchy with the definability classes over HC.

Definition 1 (in L)
. From now on, we fix a number n ≥ 3 as in Theorem 1. We also fix a sequence #" MF ω 1 satisfying Theorem 15.3 in [19] for this n. This includes the equality α |Π α | = ω 1 and the following conditions (in L): We call this fixed #" Π ∈ L the key sequence. The construction of #" Π in [19] is rather long and too technical, so we do not reproduce it here. It employs some ideas related to diamondstyle constructions, as well as to some sort of definable generic inductive constructions. This method is realized by a special transfinite construction of the sequence #" Π in L from countable subsequences. The construction can be viewed as a maximal branch in a certain mega-tree, say P , whose nodes are such countable subsequences. A suitable character of extension in the mega-tree allows to define a maximal branch in P that blocks all sets in P as in (B) of Definition 1, and still satisfies (A).
The following definition introduces some derived notions.

Lemma 3
Our final goal is to prove Theorem 1 by means of P P P-generic extensions of L. We call these extensions key models.
From now on, we will typically argue in L and in ω L 1 -, preserving generic extensions of L, in particular, in P P P-generic extensions (see above). Thus it will always be the case that ω L 1 = ω 1 . This allows us to think that |Π| = ω 1 (rather than ω L 1 ).

Definition 3.
Let a set G ⊆ P P P be generic over the constructable universe L. If ξ < ω 1 then following the remark in the end of Section 3, we define and let x ξ [G] ∈ 2 ω be the only real in T∈G(ξ) [T]. Then we put Thus the forcing notion P P P adjoins an array Lemma 4 (Corollary 18.2 in [19]). Assume that G ⊆ P P P is P P P-generic over L. Then it is true in L[G] that X[G] is a set of definability class Π HC n−2 , hence, of class Π 1 n−1 by Lemma 2 above.

∆ 1 n -Good Well-Ordering in the Key Model
The next theorem proves that the key model L[G] satisfies condition (i) of Theorem 1. The reals are treated here as points of the Cantor space 2 ω . Theorem 2. If G ⊆ P P P is P P P-generic over L then it holds in L[G] that there is a ∆ 1 n -good wellordering of 2 ω of length ω 1 , hence (i) of Theorem 1 holds.

Proof.
We argue in L[G]. Lemma 4 will be the principal ingredient of the proof. Let ) then x ∈ L[X γ] for some γ < ω 1 by Lemma 4, hence we let γ(x) be the least γ < ω 1 such that x ∈ L[X γ], and ν(x) < ω 1 be the index of x in the canonical ∆ HC 1 ({X γ}) well-ordering ≤ X γ of 2 ω in L[X γ] by Gödel. We claim that the maps x → γ(x) and x → ν(x) are ∆ HC n−1 . Indeed, This easily yields the result for the map x → γ(x). The result for the other map follows by a similar rather routine estimation. Now let be the well-ordering of the set 2 ω ∩ L[G], according to the lexicographical well-ordering of the triples max{γ(x), ν(x)}, γ(x), ν(x) . It easily follows from the results for maps x → γ(x) and x → ν(x) that is ∆ HC n−1 , and hence ∆ 1 n by Lemma 2 of Section 5. Finally to check the ∆ 1 n -goodness, by definition it remains to prove that, given a ∆ 1 n set P ⊆ 2 ω × 2 ω , the set Q = { z, x : ∀ y x ¬ P(z, y)} is ∆ 1 n as well. The class Π 1 n is obvious, as is already shown to be ∆ 1 n . Thus we have to verify the definability class Σ 1 n , or equivalently, class Σ HC n−1 , for Q. However, this is true, as Q(z, x) is equivalent to the following: for all γ , ν ≤ max{γ(x), ν(x)}, if the triple max{γ , ν }, γ , ν non-strictly precedes max{γ(x), ν(x)}, γ(x), ν(x) lexicographically, then there is a real y ∈ 2 ω such that γ = γ(y), ν = ν(y), and ¬ P(z, y).
It remains to note that the quoted formula is essentially Σ HC n−1 since the bounded quantifiers ∀ γ , ν ≤ max{γ(x), ν(x)} do not destroy Σ -classes over HC.
Our final step is to prove that the key model also satisfies condition (ii) of Theorem 1. However, this will involve much more work and will be carried out under the following assumption. Assumption 1. We shall assume that n ≥ 4 henceforth.
This leaves aside the case n = 3 in (ii) of Theorem 1 which thus needs a separate consideration to justify the assumption. Thus suppose for a moment that n = 3. We claim that (ii) of Theorem 1 holds in the key model L[G], where G is P P P-generic over L. Suppose to the contrary that (ii) of Theorem 1 fails, so that there is a ∆ 1 2 well-ordering of the reals (even not necessarily good). Then by Theorem 25.39 in [20],

Real Names
We begin with a technical concept. The goal of the following definitions is to give a suitable notation for names of reals in 2 ω in the context of forcings of the form MT(π).
Let a real name be any set c ⊆ MT × (ω × 2) such that the sets satisfy the following: if n < ω and p ∈ K c n0 , q ∈ K c n1 , then p, q are somewhere almost disjoint, in the sense that there is an index ξ ∈ |p| ∩ |q| such that T n is at most countable-then the set |c| = n p∈K c n |p|, and c itself as a set are countable, too. Now let π be a multiforcing. A real name c is π-complete if the set is dense in MT(π) for each n. In this case, if a set (a filter) G ⊆ MT(π) is MT(π)-generic over the family of all sets K c n ↑ π , n < ω, then we define a real c[G] ∈ 2 ω so that c[G](n) = i if G ∩ (K c ni ↑ π) = ∅, where K c ni ↑ π is defined from K c ni similarly to (3). We do not require here that c ⊆ P P P × (ω × 2), or equivalently, K c n ⊆ P P P for all n. Finally, if #" π is a sequence in # " MF ∪ # " MF ω 1 , then a #" π-complete real name will mean a πcomplete real name, where π = cw #" π (the componentwise union).
As an elementary example, we let ξ < ω 1 and define a real name .
x ξ such that each set K .
We leave it as a routine exercise to prove that .
x ξ is a small real name, π-complete for any multiforcing π , and if a set G ⊆ MT(π) is MT(π)-generic over L, then the real .
is identical to x ξ [G] defined by Definition 3. Thus, .
x ξ is a canonical name for the real x ξ [G].

An Auxiliary Forcing Relation
We begin a lengthy proof of the non-existence of ∆ 1 n−1 -good well-orderings of the reals in the generic models considered. The proof involves an auxiliary forcing relation, not explicitly connected with any particular forcing notion, in particular, not explicitly connected with the key forcing P P P.
We argue in L. Consider the 2nd order arithmetic language, with variables k, l, m, n, . . . of type 0 over ω and variables a, b, x, y, . . . of type 1 over 2 ω , whose atomic formulas are those of the form x(k) = n. Let L be the extension of this language, which allows to substitute variables of type 0 with natural numbers and variables of type 1 with small real names (see Section 7) c ∈ L.
Proposition 3 just below demonstrates that the forcing relation forc #" π , considered with countable initial segments #" π = #" Π α of the key sequence #" Π (introduced by Definition 1), coincides with the true P P P-forcing relation (see Definition 2) up to the level n − 1.
Recall that n ≥ 4 by Assumption 1. We write p forc α ϕ instead of p forc #" Π α ϕ, for the sake of brevity. Let p forc ϕ mean: p forc α ϕ for some α < ω 1 . The next result makes use of (B) of Definition 1.

Invariance
Invariance theorems are very typical for all kinds of forcing. We present here two major invariance theorems on the auxiliary forcing forc , established in [19]. The first one shows tail invariance, while the other one explores the permutational invariance.
The Let h ∈ PERM. The action of h is extended as follows. (See [19], Section 24.) 1.
If c ⊆ MT × (ω × ω) is a real name, then put hc = { hp, n, i : p, n, i ∈ c}, thus easily hc is a real name as well.

No ∆ 1 n−1 -Good Well-Orderings in the Key Model
In this section, we accomplish the proof of Theorem 1 by verifying that the key model L[G] of Section 5 satisfies (ii) of Theorem 1. That the key model satisfies (i) of Theorem 1 was already established by Theorem 2. The following lemma is the principal step.
Lemma 5. If G ⊆ P P P is P P P-generic over L then it holds in L[G] that every Σ 1 n−1 set S ⊆ ω is constructable.
Proof. There is a parameter-free Σ 1 n−1 formula ϕ(j), such that S = { j < ω : ϕ(j)} in L[G]. We claim that (A) if j < ω then j ∈ X if and only if it is true in L that there exists a sequence #" σ ∈ # " MF and a multitree s ∈ MT( #" σ ) such that s forc #" σ ϕ(j). In the easy direction, assume that j ∈ X. There is a condition s ∈ P P P which P P P-forces ϕ(j) over L. Then s forc ϕ(j) by Proposition 3, that is, s forc #" Π α ϕ for some α < ω 1 . We can increase α if necessary to guarantee that s ∈ MT( #" Π α). It remains to take #" σ = #" Π α. In the difficult direction, suppose that s forc #" σ ϕ(j), where #" σ ∈ # " MF and s ∈ MT( #" σ ); we have to prove that j ∈ X. Suppose toward the contrary that j / ∈ X. Then there is a multitree p 0 ∈ G such that (B) p 0 P P P-forces ¬ ϕ(j) over L.
By routine estimation, U is a Σ 1 (HC) set (with δ, σ as the only parameters of a Σ 1 definition in HC), hence a Σ n−3 (HC) set as n ≥ 4 by Assumption 1. Therefore by Definition 1(B) there is an ordinal λ < ω 1 such that #" Π λ blocks U .
The contradiction obtained closes Case 1.
Combining Theorem 3 with the result of Theorem 2, we conclude that L[G] is a model for Theorem 1.

Conclusions and Problems
In this study, the method of finite-support products of Jensen's forcing was employed to the problem of obtaining a model of ZFC in which, for a given n ≥ 3, good wellorderings of the reals exist in the class ∆ 1 n but do not exist in ∆ 1 n−1 . This result (Theorem 1 of this paper) continues our series of recent research, such as a model defined in [24] for a given n, in which there is a Π 1 n Vitali equivalence class containing no ordinal-definable elements, whereas every countable Σ 1 n -set of reals contains only ordinal-definable reals, or a model defined in [25] in which there is a Π 1 n real singleton {a} such that a codes a cofinal map f : ω → ω L 1 , whereas every Σ 1 n set X ⊆ ω is constructible and hence cannot code a cofinal map ω → ω L 1 , or a very recent model defined in [26] in which the separation principle holds for a given class Σ 1 n for sets of integers. Theorem 1 may also be a step towards the solution of the all-important problem by S. D. Friedman mentioned in the introduction above (Section 1).
From our study, it is concluded that the technique of definable generic inductive constructions of forcing notions in L, developed for Jensen-type product forcing in our earlier papers [16,19], leads to a new result (Theorem 1), which is a significant advance toward solving an important set theoretic problem formulated in the introduction as Problem 1.
From the result of Theorem 1, we immediately come to the following problems.

Problem 2.
Prove that it is true in the key model L[G] of Section 5 that there is no "boldface" ∆ 1 n−1 well-ordering of the reals of any kind (that is, not necessarily ∆ 1 n−1 -good).
Such a strengthening of Theorem 1 would solve Problem 1 of Section 1 completely.

Problem 3.
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.
To comment upon Problem 3, note that the model for Theorem 1 introduced in Section 5 (the key model) 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 continuum, has been known since the early years of modern set theory. See, for example, problem 3214 in an early survey [27] by Mathias. Harrington [28] solved this problem by a generic model in which 2 ℵ 0 > ℵ 1 and there is a ∆ 1 3 well-ordering of the continuum, by a combination of methods based on different forcing notions, such as the almost-disjoint forcing [29] and the forcing notion by Jensen and Johnsbråten [30]. See [13] for further remarkable progress in forcing constructions of models with long projective well-orderings of low projective classes. Solving Problem 3 would be a further significant step in this direction.  Data Availability Statement: Not applicable. The study does not report any data.