1 Introduction↩︎

The structural theory of the projective hierarchy exhibits a profound dichotomy between regularity properties—typically induced by large cardinals and determinacy hypotheses—and the existence of definable wellorders, which are traditionally derived from the fine structure of inner models. Under full Projective Determinacy (\(\mathsf{PD}\)), the projective sets exhibit complete regularity, precluding the existence of any projective wellorder of the reals. Conversely, in canonical inner models, definability and regularity are sharply demarcated.

The primary motivation for this article is rooted in a version of Woodin’s 12th Delfino problem, which in its original formulation asks whether \(\mathsf{PD}\) is strictly equivalent to all its structural consequences for the projective hierarchy.

A localized formulation of this question, first posed by S. Friedman and R. Schindler (see [1] and [2]) goes as follows: if we observe a behavior of the projective sets of reals which is the same as the one in some \(M_{n+1}\), the canonical inner model with \(n+1\) Woodin cardinals, is it true that the local hypothesis of projective determinacy, namely \(\boldsymbol{\Pi}^1_{n+1}\)-determinacy or even the weaker (if \(n+1\) is odd) \(\boldsymbol{\Delta}^1_{n+1}\)-determinacy, must be true? Specifically:

For \(n=1\) and \(n=2\), S. Friedman used his substantial refinements of Jensen’s coding the universe technique to answer it in the negative (see theorem 8.51 [3]). The methods stall however after the fourth level due to considerable technical difficulties one inevitably faces when trying to apply coding arguments over more complicated inner models. In [1] the authors were able to show that assuming \(n\) many strong cardinals, there are universes where all \(\boldsymbol{\Sigma}^1_{n+3}\)-sets are Lebesgue measurable, have the Baire property and which allow a \(\Delta^1_{n+5}\)-definable wellorder of the reals. Their arguments use Jensen coding combined with collapsing strong cardinals which implies that the associated levels of the projective hierarchy become universally Baire due to a result of H. Woodin.

The main result of this work demonstrates that the answer to this localized Delfino question is unequivocally no, even if we additionally require the \(\Sigma^1_{n+2+m}\)-uniformization property for every \(m\in \omega\).

This theorem should be contextualized against the properties of the projective sets of reals in \(M_{n+1}\). In \(M_{n+1}\), the presence of \(n+1\) Woodin cardinals yields \(\boldsymbol{\Pi}^1_{n+1}\)-determinacy by the Martin-Steel theorem ([4]), guaranteeing that all sets in \(\boldsymbol{\Sigma}^1_{n+2}\) are Lebesgue measurable and possess the Baire property via unfolding games. Simultaneously, the fine structure of \(M_{n+1}\) provides a good \(\Sigma^1_{n+3}\)-definable wellorder of the reals (see [5]), which in turn implies \(\Sigma^1_{n+3+m}\)-uniformization for every \(m \in \omega\) by an old result of J. Addison (see [6]). Thus, the universe we force over \(M_n\) mimics the behaviour of the reals in \(M_{n+1}\), but uses significantly less large cardinal strength. As in \(M_n\), \(\Delta^1_{n+1}\)-determinacy fails ² the negative answer to the Friedman-Schindler question follows.

Our proof takes advantage of a result of G. Hjorth ([8], Corollary 2.4) who observed that in the presence of “every real has a sharp”, \(\mathsf{MA_{\omega_1}}\) implies that every \(\boldsymbol{\Sigma}^1_3\)-set is Lebesgue measurable and has the Baire property. Hjorth’s result relies on the classical Martin-Solovay tree ([9]) denoted by \(T_2\). Using the generalized versions \(T_n\) of the Martin-Solovay tree, Hjorth’s result can be pushed up to higher projective pointclasses, thus establishing a path for a proof of the theorem: working over \(M_n\) as the ground model, the objective is to force \(\mathsf{MA}_{\omega_1}\) while simultaneously introducing a \(\Delta^1_{n+3}\)-definable wellorder and forcing global tail \(\Sigma\)-uniformization. Then apply a generalization of Hjorth’s argument to argue for Lebesgue measurability and the Baire property. To achieve the former, we adapt the coding techniques pioneered in [10], which possess the necessary flexibility to accommodate the forcing of \(\mathsf{MA}\).

There are several technical obstacles in the suggested path of our proof. We list two of them. First there is considerable tension between forcing a projectively definable wellorder of the reals and the \(\Sigma^1_{n+2+m}\)-uniformization on the one hand and forcing \(\mathsf{MA_{\omega_1}}\) on the other. Indeed forcing for the first objective is in need of a very tight control over certain projective predicates, whereas forcing for \(\mathsf{MA_{\omega_1}}\) necessarily creates a substantial amount of generically added chaos. A second substantial technical obstacle arises because the methods in [10] rely on \(L\) as the ground model, which behaves considerably nicer when applying coding forcings than \(M_n\), in particular in iterations of length \(\omega_2\) which we must use. To bridge this gap, we implement and refine the forcing machinery introduced in [11] and [12]. The main innovation here is to replace \(M_n\) with an intermediate generic extension that exhibits greater robustness under the subsequent coding forcings required for the definable wellorder and the uniformization. Notably, this intermediate extension is itself constructed via coding forcings, meaning the entire architecture functions as a “coding over codes” construction, effectively circumventing the shortcomings of \(M_n\).

We also record a byproduct of the \(M_1\) case which connects the construction with generalized notions of measurability for tree forcings. Fischer, Friedman and Khomskii study, among others, the random, Cohen, Sacks, Miller, Laver, Silver and Mathias notions of measurability, and Friedman–Schrittesser ask whether one can preserve Woodin cardinals while arranging prescribed combinations of projective \(P\)-measurability and failures of projective \(P\)-measurability. The \(M_1\) instance of our construction gives such a model for \(P\in\{B,C,S,M\}\), where \(B\) denotes random/Lebesgue measurability, \(C\) denotes Cohen/Baire measurability, \(S\) denotes Sacks/Marczewski measurability, and \(M\) denotes Miller measurability. This is not an additional forcing argument: it follows from Hjorth’s observation together with the Brendle–Löwe implications \[\Gamma(C)\Rightarrow \Gamma(M) \quad\text{and}\quad \Gamma(B)\Rightarrow \Gamma(S)\] for pointclasses \(\Gamma\) closed under continuous preimages.

We mention that this paper has a companion paper [13], which treats the complementary \(\Pi\)-side direction. There the same exact-placement background is combined with adjacent \(\Pi^1_{n+3}\)-uniformization, using a local coding predicate and a derivative hierarchy of allowable hybrid forcings. Thus the two papers should be viewed as complementary: the present paper develops the coding-over-codes construction needed for Martin’s Axiom and the global \(\Sigma\)-uniformization tail, while the companion paper develops a separate local mechanism for the adjacent \(\Pi\)-uniformization theorem.

We end with a brief description of the organization of this article. In the first sections we introduce the tools we shall use in our proof. This includes several properties of \(M_1\) and \(M_n\), and three coding forcings which we will weave together for a suitable coding machinery. The main proof will be shown with \(M_1\) as the ground model and we later rather extensively indicate how to implement the changes one has to make in order to lift the proof to \(M_n\).

2 Canonical inner models with Woodin cardinals and the trees \(T_n\)↩︎

2.1 The canonical inner model with one Woodin cardinal↩︎

We recall the fragment of inner model theory which will be used. The ground model for the main construction is the canonical proper class mouse \(M_1\), the minimal iterable proper class premouse with one Woodin cardinal. In this article, when we say \(M_1\), or more generally \(M_n\) for \(n>1\), exists, we implicitly assume that \(M_1^\#\) exists, the minimal active inner model with \(1\) Woodin cardinal that is not \(1\)-small (see below), and, in this case, \(M_1\) is the result of iterating the active measure of \(M_1^\#\) out of the universe. We use the notation and comparison conventions of Steel’s outline of inner model theory, and the projective definability analysis of Steel’s work on projectively well-ordered inner models; see [5], [14]. In particular, every proper initial segment of \(M_1\) is \(1\)-small and \(\omega\)-sound, and the order of construction gives a good \(\Delta^1_3\) well-order of the reals of \(M_1\).

A premouse \(\mathcal{M}\) is \(1\)-small above \(\eta\) if whenever \(E\) is an extender on the \(\mathcal{M}\)-sequence and \(\eta<\operatorname{crit}(E)\), the initial segment \(\mathcal{J}^{\mathcal{M}}_{\operatorname{crit}(E)}\) has no Woodin cardinal above \(\eta\). We say that \(\mathcal{M}\) is \(1\)-small if it is \(1\)-small above \(0\).

We also recall the weak iterability notion used by Steel at the first odd level. Let \(\mathcal{T}\) be an \(\omega\)-maximal putative iteration tree on \(\mathcal{M}\), let \(b\) be a maximal branch through \(\mathcal{T}\), and let \(\alpha\) be a countable ordinal. The branch \(b\) is \(\alpha\)-good if whenever \(\mathcal{N}\) is either \(\mathcal{M}_b^{\mathcal{T}}\) itself, or the \(\alpha\)-th linear iterate of an initial segment \(\mathcal{P}\trianglelefteq \mathcal{M}_b^{\mathcal{T}}\) by one extender on the \(\mathcal{P}\)-sequence and its images, then either \(\mathcal{N}\) is well-founded or \(\alpha\in\operatorname{wfp}(\mathcal{N})\). A countable premouse \(\mathcal{M}\) is \(\Pi^1_2\)-iterable if player II wins the corresponding one-round weak iteration game: player I plays a countable putative \(\omega\)-maximal tree together with a countable ordinal \(\alpha\), and player II either accepts a last well-founded model or plays a maximal \(\alpha\)-good branch. Steel proves that this iterability condition is \(\Pi^1_2\) in the codes. For \(1\)-small mice it is the \(n=1\) instance of the general \(\Pi_n\)-iterability analysis; see [5].

We now fix the local notation for limit length trees.

We shall use the following form of Steel’s branch uniqueness theorem, often called the zipper lemma.

The point of the preceding definition is that, in the \(1\)-small context, a \(\mathcal{Q}\)-structure determines at most one good cofinal branch. If two distinct branches had the same relevant well-founded \(\mathcal{Q}\)-structure, the zipper lemma would produce strength below \(\delta(\mathcal{T})\), contradicting the initial segment which witnesses that \(\delta(\mathcal{T})\) is not Woodin.

In the next few statements, an ordinary premouse means a premouse in Steel’s usual premouse language, with no real parameter, predicate parameter, or base set added to the structure. This is only a terminological convention distinguishing these premice from mice over a real or over another base; no additional fine-structural notion is being introduced.

2.1.0.1 Preservation convention for the outer models used below.

We shall use Lemma 1 only in the forcing extensions which occur in this paper. The preservation fact needed there is the following: if \(\mathcal{M}\triangleleft M_1\) is countable and belongs to one of these extensions, then \(\mathcal{M}\) remains \(\Pi^1_2\)-iterable there. This is the realizability preservation supplied by Steel’s weak iteration-game analysis for initial segments of \(M_1\), applied to the proper, \(\omega_1\)-preserving forcing extensions used below. We do not claim that the set of all real codes for \(\Pi^1_2\)-iterable premice is absolute between arbitrary outer models with the same \(\omega_1\).

2.1.0.2 Remark.

The notation in Definition 3 is a convention for outer models of \(M_1\) which preserve \(\omega_1\). The same first-order-looking assertion, if evaluated inside an arbitrary transitive model, need not imply that the object obtained is the true \(M_1|\omega_1\). Later, whenever this definition is used inside countable auxiliary models, the relevant countable premouse is also required externally to belong to the class \(\mathcal{I}\).

We shall need a canonical diamond sequence which is available from the same fine structure. The argument is Jensen’s proof of diamond in \(L\), with Steel’s condensation theorem for initial segments of \(M_1\) replacing condensation for \(L\).

We fix once and for all the \(<_{M_1}\)-least sequence \(\vec{D}\) satisfying Lemma 4. In later sections this sequence will be used to define a canonical \(\omega_1\)-sequence of independent Suslin trees over \(M_1\). The construction of those trees is postponed until the point where the coding machinery needs them.

2.2 The canonical inner model with \(n\)-many Woodin cardinals↩︎

We now record the higher-level analogue of Subsection 2.1. Throughout this subsection \(1\leq n<\omega\) is fixed, and \(M_n\) denotes the canonical minimal proper class premouse with \(n\) Woodin cardinals, in the sense of Steel’s construction of tame mice with full background extenders. Thus \(M_0=L\), and for \(n>0\) the model \(M_n\) is obtained from the Steel background construction by stopping at the first failure of \(n\)-smallness, or as the limit of the \(n\)-small construction if no such failure occurs. We shall use only the following standard consequences of Steel’s analysis.

First, if there are \(n\) Woodin cardinals, then \(M_n\) exists and satisfies that there are \(n\) Woodin cardinals. Secondly, every proper initial segment of \(M_n\) is \(n\)-small and \(\omega\)-sound. Thirdly, the order of construction of \(M_n\) gives a canonical good well-order of the reals of \(M_n\), and Steel’s projective analysis shows that this well-order is \(\Delta^1_{n+2}\) over the reals of \(M_n\); see [5], [14]. The case \(n=1\) is exactly the situation isolated in the previous subsection.

We shall also use Steel’s finite-level iterability condition. The exact definition depends on the parity of \(n\). For even \(n\), one uses the \(n\)-round weak iteration game in which the branches played by player II are required to be sufficiently definable over the trees played by player I. For odd \(n\), one adds the corresponding \(\alpha\)-goodness requirement at the last round. In both cases the definition is arranged so that realizable mice satisfy it, and so that it is projectively simple in the codes.

For later reference we isolate the three consequences of the definition which are used in this paper.

The next lemma is the direct analogue of Lemma 1. We state it separately because it is the form in which it will be used later.

2.2.0.1 Preservation convention for the outer models used below.

As in the \(M_1\) case, we shall only use the preceding comparison in forcing extensions which occur in this paper. The preservation fact needed is the following one: if \(\mathcal{M}\triangleleft M_n\) is countable and belongs to such an extension, then \(\mathcal{M}\) remains \(\Pi_n\)-iterable there. This is the higher-level version of the realizability preservation used in Paragraph 2.1.0.1. We do not claim that the class of all real codes for \(\Pi_n\)-iterable premice is absolute between arbitrary outer models with the same \(\omega_1\).

For the coding arguments we only need to recover the lower part \(M_n|\omega_1\). Thus we isolate the lower-part initial segments of \(M_n\), rather than the larger class of all \(n\)-small \(\Pi_n\)-iterable mice. This distinction is harmless for \(n=1\), but it is important at higher odd levels, where one has to avoid the familiar nonstandard \(\Pi_n\)-iterable mice. The lower-part restriction below excludes these objects from the definition used in the coding apparatus.

2.2.0.2 Remark.

The notation \(M_n|\omega_1\) in Definition 7 is a convention for the outer models of \(M_n\) used in this paper. It should not be read as saying that an arbitrary transitive model which internally satisfies the same first-order-looking definition has computed the true \(M_n|\omega_1\). Later, whenever countable auxiliary models are used, the relevant countable premouse is also required externally to belong to \(\mathcal{I}_n\).

The condensation and diamond arguments from the previous subsection lift without change once \(M_1\) is replaced by \(M_n\).

We shall use this fact only as a source of canonical choices inside \(M_n\). In particular, using \(<_{M_n}\) we fix once and for all the canonical almost disjoint family \[D=\langle d_\xi\mid \xi<\omega_1\rangle\in M_n\] used for almost disjoint coding, and the canonical \(\diamondsuit_{\omega_1}\)- sequence used in the bookkeeping. No later argument uses the well-order to choose elements from projective sections in the final model.

We fix once and for all the \(<_{M_n}\)-least sequence \(\vec{D}^{\,n}\) satisfying Lemma 8. In the higher-level version of the construction, this sequence plays exactly the role played by the \(M_1\)-diamond sequence in the detailed \(M_1\) case: it gives the canonical lower-part bookkeeping from which the corresponding sequence of independent Suslin trees and the localized coding apparatus are built.

2.3 The trees \(T_n\), weak homogeneity, and small generic absoluteness↩︎

We next fix the tree representations of the projective pointclasses which will be used in the regularity argument. The main application in this paper is the case \(n=2\): the Martin–Solovay tree \(T_2\in M_1\) represents the universal \(\boldsymbol{\Sigma}^1_3\) set. Since the higher-level notation is no harder, we record the construction uniformly. Thus, for \(n\geq 2\), the tree \(T_n\) will be a canonical tree belonging to \(M_{n-1}\) whose projection is the universal \(\boldsymbol{\Sigma}^1_{n+1}\) set of reals. In the final model we shall use only the following consequence of its construction: membership in \(p[T_n]\) is absolute to the small generic extensions in which the relevant real appears.

We recall the form of weak homogeneity used below. The notation follows Steel’s exposition of the Martin–Solovay theorem in terms of towers of measures [15]. If \(Z\) is a set and \(\kappa\) is a cardinal, let \(\operatorname{meas}_\kappa(Z)\) be the set of \(\kappa\)-additive measures on \(Z^{<\omega}\). If \(\mu\in\operatorname{meas}_\kappa(Z)\), its dimension is the unique \(m<\omega\) such that \(\mu\) concentrates on \(Z^m\). If \(\mu\) has dimension \(m\), \(\nu\) has dimension \(k\), and \(m\leq k\), we say that \(\nu\) projects to \(\mu\) if for every \(A\subseteq Z^m\), \[A\in\mu \quad\Longleftrightarrow\quad \{u\in Z^k\mid u\upharpoonright m\in A\}\in\nu .\] A tower \(\langle\mu_i\mid i<\omega\rangle\) is countably complete if, whenever \(A_i\in\mu_i\) for every \(i\), there is an \(f\in Z^\omega\) such that \(f\upharpoonright\operatorname{dim}(\mu_i)\in A_i\) for every \(i\).

The connection between the preceding two notions is the Martin–Solovay tree construction. If \(\bar\mu\) is a weak homogeneity system for \(T\) and \(\Theta\) is sufficiently large, the Martin–Solovay tree \[\operatorname{ms}(\bar\mu,\Theta)\] searches for a coherent descending sequence of ordinal ranks through the ultrapowers determined by the measures in \(\bar\mu\). A branch through this Martin–Solovay tree is precisely a continuous certificate that the corresponding tower is ill-founded. The following is the form of the Martin–Solovay theorem used in the rest of the paper [15].

We shall use Theorem 7 only in a localized form. Let \(M\) be one of the mice \(M_{n-1}\) and let \(\delta\) be its least Woodin cardinal. If \(T\in M\) is \(\kappa\)-weakly homogeneous in \(M\) for every \(\kappa<\delta\), then for every forcing \(\mathbb{P}\in M\) of \(M\)-cardinality \(<\delta\), and every \(M\)-generic \(G\subseteq\mathbb{P}\), the same tree \(T\) and its Martin–Solovay complement compute the same projective set in \(M[G]\). Equivalently, for reals \(x\in M[G]\), membership in \(p[T]\) can be checked by the ground-model tree \(T\) and is not changed by passing to further forcing extensions of size below the relevant completeness bound.

2.3.0.1 Remark.

Later forcing notions need not be regarded globally as small over the relevant mouse. What is used is the local consequence: every real and every name appearing in the coding construction is read in a bounded regular subforcing, and the relevant subforcing has size below the completeness bound of the homogeneity system. The assertion above is therefore applied in the intermediate model generated by that local support.

2.4 Defining the canonical trees \(T_n\) inside \(M_{n-1}\)↩︎

We now choose the particular trees to which the preceding subsection will be applied. Fix \(n\geq 2\). Work in \(M_{n-1}\), and let \[\delta^n_0<\delta^n_1<\cdots<\delta^n_{n-2}\] be the Woodin cardinals of \(M_{n-1}\). We write simply \(\delta_0\) for the least of them when \(n\) is fixed.

Let \(U_{n+1}\subseteq\omega^\omega\) be the standard universal \(\boldsymbol{\Sigma}^1_{n+1}\) set, with the first real coordinate coding the index and the remaining coordinates coding the parameter and the argument. We fix this universal set once and for all using the usual Moschovakis parametrization conventions for projective pointclasses [16]. Thus every boldface \(\boldsymbol{\Sigma}^1_{n+1}\) set is a section of \(U_{n+1}\).

The following theorem is the precise object needed later. It is a standard consequence of the Martin–Steel analysis of projective scales, the Martin–Solovay construction, and Steel’s tree-production argument for mice with finitely many Woodin cardinals [5], [15], [17].

For definiteness, we make the choice canonical.

2.4.0.1 Remark (how \(T_n\) will be used).

The tree \(T_n\) is not an additional coding device. It is a fixed descriptive-set-theoretic witness, belonging to the relevant mouse, which makes the projective truth predicate robust in small generic extensions. In the proof of Lebesgue measurability and the Baire property, the only instance needed is \(T_2\): after a real parameter \(a\) appears in a local intermediate model, cofinally many later Random and Cohen stages ensure that almost every, respectively comeagerly many, reals are generic over the corresponding \(L[T_2,a]\)-type model. The weak homogeneity and Martin–Solovay complementing pair are what allows Hjorth’s Solovay-style argument to read the \(\boldsymbol{\Sigma}^1_3\) definition correctly in those generic extensions.

3 The coding forcings which are used↩︎

3.1 Coding Reals by Triples of Ordinals↩︎

We utilize the coding technique developed by A. Caicedo and B. Veličković (see [18]). Sets coded via their method serve as a suitable ground-model \(H(\omega_2)\). Over this \(H(\omega_2)\), we apply further coding utilizing independent Suslin trees; in essence, this amounts to coding on top of already coded sets. A key property of the Caicedo-Veličković coding is that ccc forcings cannot introduce new codes. Consequently, the ground-model \(H(\omega_2)\) remains definable in ccc generic extensions. This allows us to identify objects from the ground-model \(H(\omega_2)\)—in our case, Suslin trees—and modify them generically by adding cofinal branches or specializing trees via ccc forcings. This ultimately yields a coding mechanism whose codes are easily decoded.

Although several proofs are omitted in the following discussion (details can be found in [11]), it is necessary to describe the Caicedo-Veličković coding in detail, as its features are crucial for the present work.

For the rest of this article, we will fix easily definable ladder systems, which are the only ones we will use throughout. Working over \(M_n\), we will use the \(\Delta^1_{n+2}\)-definable wellorder \(<_n\) of the reals to define \(\vec{C}= ( C_{\alpha} \mid \alpha < \omega_1 \cap Lim)\) inductively via picking at each stage \(\alpha < \omega_1 \cap Lim\) the \(<_n\) least real coding a ladder \(C_{\alpha} \subset \alpha\). A standard computations yields that this ladder system is \(\Sigma_1 (\{ \omega_1 \} )\)-definable over \(M_n | \omega_2\). From now on whenever a ladder system is mentioned when working over some \(M_n\), it is always implied that we talk about this particular ladder system \(\vec{C}\).

Given three subsets of natural numbers, \(x, y, z \subseteq \omega\), we can define an oscillation function. First, the set \(x\) is transformed into an equivalence relation \(\sim_x\) defined on \(\omega \setminus x\). For natural numbers \(n, m\) in the complement of \(x\) where \(n \leq m\), we define \(n \sim_x m\) if and only if the interval \([n,m]\) has no elements in common with \(x\) (i.e., \([n,m] \cap x = \emptyset\)). This allows us to define the following:

Next, we will define how appropriate countable subsets of ordinals can be used to code real numbers. For the remainder of this section, we fix a ladder system \(\vec{C}\). Suppose \(\omega_1 < \beta < \gamma < \delta\) are fixed limit ordinals, each with uncountable cofinality. Let \(N \subseteq M\) be countable subsets of \(\delta\). Furthermore, assume that \(\{ \omega_1, \beta, \gamma\} \subseteq N\), and for each \(\eta \in \{ \omega_1, \beta, \gamma\}\), \(M \cap \eta\) is a limit ordinal and \(N \cap \eta < M \cap \eta\).

We can use the pair \((N,M)\) to code a finite binary string. Let \(\bar{M}\) be the transitive collapse of \(M\), and let \(\pi : M \rightarrow \bar{M}\) be the collapsing map. Define \(\alpha_M := \pi(\omega_1)\), \(\beta_M := \pi(\beta)\), \(\gamma_M := \pi(\gamma)\), and \(\delta_M := \bar{M}\). These are all countable limit ordinals. Also, set \(\alpha_N := \sup(\pi[\omega_1 \cap N])\). The height \(n(N,M)\) of \(\alpha_N\) in \(\alpha_M\) is the natural number defined by:

\[n(N,M) := |\alpha_N \cap C_{\alpha_M}|\] where \(C_{\alpha_M}\) is an element of our pre-fixed ladder system. Since \(n(N,M)\) will appear frequently, we will abbreviate it as \(n\). Note that because the order type of each \(C_{\alpha}\) is \(\omega\), and \(N \cap \omega_1\) is bounded below \(M \cap \omega_1\), \(n(N,M)\) is indeed a natural number.

Now, we can assign a triple \((x,y,z)\) of finite subsets of natural numbers to the pair \((N,M)\) as follows: \[x := \{ |\pi(\xi) \cap C_{\beta_M}| : \xi \in \beta \cap N \}\] Note that \(x\) is finite because \(\beta \cap N\) is bounded in the set \(C_{\beta_M}\) (which is cofinal in \(\beta_M\) and has order type \(\omega\)). Similarly, we define: \[y := \{ |\pi(\xi) \cap C_{\gamma_M}| : \xi \in \gamma \cap N \}\] and \[z := \{ |\pi(\xi) \cap C_{\delta_M}| : \xi \in \delta \cap N \}\] Again, it is straightforward to see that these are finite subsets of natural numbers.

We can then consider the oscillation \(o(x \setminus n, y \setminus n, z \setminus n)\) (recalling that \(n = n(N,M)\)). If the domain of the oscillation function at these points is greater than or equal to \(n\), we write: \[s_{\beta, \gamma, \delta} (N,M) := \begin{cases} o(x \setminus n, y \setminus n, z \setminus n) \restriction n & \text{if defined} \\ \ast & \text{otherwise} \end{cases}\] Here, \(\ast\) denotes an undefined state. Similarly, if \(l > n\), we let \(s_{\beta, \gamma, \delta} (N,M) \restriction l = \ast\).

Finally, we can define what it means for a triple of ordinals \((\beta, \gamma, \delta)\) to code a real number \(r\).

Witnesses for this coding can be added using a proper forcing. Conversely, for fixed triples of ordinals, there is a degree of control over the behavior of continuous, increasing sequences on them:

Both partial orders \(\mathbb{P}_{\beta \gamma \delta}\) and \(\mathbb{P}_{r}\) which force \((\dagger)\) and \((\ddagger)\) respectively are actually instances of a general class of notions of forcing which were investigated first by J. Moore’s in his work on the Set Mapping Reflection Principle (\(\mathsf{MRP}\)) (see [19]). We shall see later that these forcings never kill Suslin trees. For that reason we have to introduce a couple of notions from [19]. We need first the following local version of stationarity:

Moore has shown that for any open stationary map \(\Sigma\) it is possible to force a reflecting sequence \((N_{\xi} \, : \, \xi < \omega_1)\) with a proper forcing \(\mathbb{P}_{\Sigma}\) (see [19], Theorem 3.1.).

The forcing \(\mathbb{P}_{\Sigma}\) is defined as expected: for an open stationary map \(\Sigma\) let \(\mathbb{P}_{\Sigma}\) consist of conditions \(p\) which are functions \(p: \alpha +1 \rightarrow dom(\Sigma)\), \(\alpha\) countable, which are continuous and \(\in\)-increasing, and which additionally satisfy the \(\mathsf{MRP}\)-condition on its limit points, namely that for every \(0< \nu < \alpha\), \(\nu\) a limit ordinal, there is a \(\nu_0 < \nu\) such that \(p(\xi) \cap X \in \Sigma(p(\nu))\) for every \(\xi\) in the interval \((\nu_0, \nu)\). The order is by extension.

Now, as already mentioned above, both forcings \(\mathbb{P}_{\beta \gamma \delta}\) and \(\mathbb{P}_{r}\) which will produce \((\dagger)\) and \((\ddagger)\) respectively are of the form \(\mathbb{P}_{\Sigma}\) (see [18], Lemma 1, 4 and 5).

As a consequence, if we show that \(\mathbb{P}_{\Sigma}\) always preserves Suslin trees for \(\Sigma\) an arbitrary open stationary map, we will have proven that \(\mathbb{P}_r\) and \(\mathbb{P}_{\beta \gamma \delta}\) preserve Suslin trees. This is indeed the case as we will show now. We need a Lemma of T. Miyamoto ([20]) first.

3.2 Almost Disjoint Coding↩︎

The second coding method we utilize is almost disjoint coding, originally developed by R. B. Jensen and R. M. Solovay [21]. Throughout this section, we identify subsets of \(\omega\) with their characteristic functions, using the term “reals” interchangeably for both elements of \(2^{\omega}\) and subsets of \(\omega\).

Let \(F=\{f_{\alpha} : \alpha < 2^{\aleph_0}\}\) be a family of almost disjoint subsets of \(\omega\); that is, for any distinct \(r, s \in F\), the intersection \(r \cap s\) is finite. Given a set of ordinals \(X \subset \kappa\) (where \(\kappa \le 2^{\aleph_0}\)), there exists a ccc forcing notion, denoted \(\mathbb{A}_F(X)\), which generically adds a new real \(x\). This real \(x\) codes the set \(X\) relative to the family \(F\) via the following equivalence: \[\alpha \in X \iff x \cap f_{\alpha} \text{ is finite.}\]

Another variant of this forcing, presumably due to L. Harrington [22], codes sets of reals relative to a newly added real. To define this, we first fix a definable bijection between the set of finite sequences of integers and \(\omega\). For any real \(b \in \omega^{\omega}\), let \(\bar{b}(n)\) denote the natural number coding the initial segment \(b \upharpoonright n\). We can associate to each real \(b\) a corresponding set of integers \(S(b) \subset \omega\), defined as the set of codes of all its finite initial segments: \[S(b) := \{ \bar{b}(n) : n \in \omega \}.\]

It is a standard fact that \(\mathbb{A}(A)\) possesses the Knaster property, ensuring that arbitrary products of \(\mathbb{A}(A)\) satisfy the ccc. If \(A\) is a set of reals in the ground model \(V\), forcing with \(\mathbb{A}(A)\) yields a generic filter \(G\). The union of the first coordinates of the conditions in \(G\) forms a new real \(a = \bigcup \{ p(0) : p \in G \}\). This real \(a\) perfectly codes the set \(A\) relative to the ground-model reals. Specifically, in the generic extension \(V[G]\), membership in \(A\) is characterized by: \[x \in A \iff x \in V \land (S(x) \cap a \text{ is finite}).\]

This precise characterization plays a crucial role in our broader construction. As established earlier, our objective is to isolate a universe where the \(H(\omega_2)\) of the ground model remains definable across arbitrary ccc generic extensions. By giving definable access to \(H(\omega_2)^V\), we can use this forcing variant to condense complex structural information into a single generic real \(a\), guaranteeing that this information can be decoded in any subsequent ccc extension (see the upcoming Lemma 18).

3.3 Independent Suslin trees↩︎

Suslin trees form the foundation of the third coding technique used in our proof. Recall that a set-theoretic tree \((T, <)\) is a Suslin tree if it is a normal tree of height \(\omega_1\) with no uncountable antichains. Because every tree appearing in this paper is normal, we implicitly assume normality in all subsequent tree constructions.

For two trees \((T_0, <_{T_0})\) and \((T_1, <_{T_1})\), their tree product \(T_0 \times T_1\) consists of the nodes \(\{ (t_0, t_1) \in T_0 \times T_1 : \operatorname{height}(t_0) = \operatorname{height}(t_1) \}\), ordered by \((t_0, t_1) <_{T_0 \times T_1} (s_0, s_1)\) if and only if \(t_0 <_{T_0} s_0\) and \(t_1 <_{T_1} s_1\). Hereafter, whenever we refer to a product of trees, we specifically mean this tree product.

For our purposes, we must iteratively add sequences of Suslin trees \(\langle T_\alpha : \alpha < \kappa \rangle\) such that any finite subproduct of these trees remains Suslin. We construct such sequences using Jech’s forcing, which adds a Suslin tree via countable conditions.

It is well known that \(\mathbb{P}_J\) is \(\sigma\)-closed and generically adds a Suslin tree. Furthermore, the generically added tree \(T\) has the property that for any ground-model Suslin tree \(S\), the product \(S \times T\) remains a Suslin tree in the generic extension.

A similar argument demonstrates that we can add an \(\omega\)-sequence of such Suslin trees via a full-support iteration. While longer sequences are possible by extending the iteration, \(\omega\)-blocks suffice for our construction.

Because these sequences of Suslin trees play a central role in our subsequent coding arguments, we formalize their defining property.

Finally, we will utilize the following preservation theorem, due to T. Miyamoto [20] and independently to U. Abraham and S. Shelah [23].

4 Definition of \(W_0\)↩︎

We finally have all the ingredients to successfully define the suitable ground model \(W_0\) over which coding arguments will work in a nice way. To form \(W_0\) we start with \(M_1\) as our ground model and let \(\eta\) be an inaccessible limit of inaccessible cardinals in \(M_1\). Using a \(\eta\)-long iteration we shall produce a generic extension \(W_0\) of \(M_1\) which forces with forcings of the form

1. \(\mathbb{P}_{\beta \gamma \delta}\) for triples \(\omega_1 < \beta < \gamma < \delta <\omega_2\) of uncountable cofinality.

2. \(\mathbb{P}_r\) for reals \(r\).

3. \(\mathbb{A}_F (X)\) for \(X \subset \omega_1\).

4. \(\mathbb{P}_J\) which denotes Jech’s forcing for adding a Suslin tree.

We use countable support for the iteration which has the feature that “Suslin tree preservation\("\) is preserved by theorem 13. This preservation is key as we will use the added Suslin trees for a second iteration over \(W_0\) which will use these Suslin trees for coding purposes. The goal is to arrive after \(\eta\)-many steps at a universe \(W_0\) with the following properties

1. \(\aleph_1^{M_1} = \aleph_1^{W_0}\).

2. The inaccessible limit of inaccessibles \(\eta\) becomes \(\aleph_2\) in \(W_0\).

3. There is a \(\eta=\omega_2\)-length, independent sequence \(\vec{T}\) of \(\omega_1\)-Suslin trees.

4. For every subset \(X\) of \(\omega_1\), there is a real \(r_X\) which is an almost disjoint code of \(X\) with respect to the canonical \(M_1\)-definable, almost disjoint family of reals \(D \subset \omega^{\omega} \cap M_1\).

5. Every triple of limit ordinals \((\alpha, \beta, \gamma) < \omega_2\) of uncountable cofinality is stabilized in the sense of \((\dagger)\).

6. Every real is coded by a triple of limit ordinals of uncountable cofinality \((\alpha,\beta,\gamma)\).

The actual iteration to obtain \(W_0\) is standard and we are not interested in its specific properties as long as the model \(W_0\) it produces has the 6 properties above. There is a canonical, definable well-order of \(P(\omega_1)\) in \(W_0\).

The definable wellorder \(\unlhd\) of \(P(\omega_1)\) unlocks a definition for a canonical sequence of length \(\omega_2\) of independent Suslin trees. The first entry of that sequence is, for technical reasons which will become clear later defined differently.

Working over \(M_1\) and using \(\diamondsuit\), we define the canonical Suslin tree \(T\) on \(\omega_1\). Specifically, let \(\langle A_\alpha : \alpha < \omega_1 \rangle\) be a \(\diamondsuit\)-sequence. Then \(T\) is constructed inductively as follows:

• Initialize the tree with a single root node.

• For every node \(x \in T_\alpha\), assign it \(\aleph_0\) many immediate successors in \(T_{\alpha+1}\).

• To form \(T_\alpha\), we must select branches through the lower part of the tree, \(T_{<\alpha}\), to extend.

  • Check if the guessed set \(A_\alpha\) happens to be a maximal antichain in \(T_{<\alpha}\).

  • If \(A_\alpha\) is a maximal antichain, ensure that every branch chosen to be extended into \(T_\alpha\) intersects \(A_\alpha\). Because \(T_{<\alpha}\) is countable, it is always possible to find enough branches that hit \(A_\alpha\) to continue the tree.

  • If \(A_\alpha\) is not a maximal antichain, extend any set of branches that keeps the level countable.

It is known that the tree \(T\) obtained this way is a full Suslin tree (see [24], pp. 431), which means that for every antichain \((a_n \in T \mid n \in \omega)\), the sequence of derived trees \((T_{a_n} \mid n \in \omega)\) is an independent sequence of Suslin trees.

We start with our fixed independent \(\omega\)-sequence \(\vec{T}^0\) and let \(\vec{T}^{\alpha}\) be the \(\unlhd\)-least \(\omega_1\) sequence of Suslin trees such that \(\bigcup_{\beta < \alpha} \vec{T}^{\beta}\) concatenated with \(\vec{T}^{\alpha}\) remains an independent sequence of Suslin trees.

As the wellorder \(\unlhd\) in fact talks about the reals which are almost disjoint codes for the corresponding elements of \(P(\omega_1)\) it will be useful to give that sequence of reals a name as well. For every element \(X\) in \(P(\omega_1)\), the set of reals which are almost disjoint codes for \(X\) is infinite. In the following we nevertheless talk about the real \(r_X\) which codes \(X \in P(\omega_1)\) by which we mean the \(\unlhd\)-least such real coding \(X\).

What is very important is that this definable \(\omega_2\)-sequence of independent Suslin trees will be definable in certain outer models of \(W_0\).

5 Definition of \(W_1\) and Suitability↩︎

We begin to define thoroughly how the forcing over \(W_0\) which will yield \(W_1\), does look like. We already hinted that, in order to use the independent sequence of Suslin trees \(\vec{T}\) we need a new notion for suitable models which will be able to define the sequence of Suslin trees \(\vec{T}\) correctly. With the notion of suitability it will become possible to correctly define \(\vec{T}\) already in \(\aleph_1\)-sized \(\mathsf{ZF}^-\) models, as we shall see soon, which is non-trivial, as being an \(\omega_1\)-Suslin tree in some \(\aleph_1\)-sized \(\mathsf{ZF}^-\) model not necessarily implies that this tree is actually Suslin in the real world.

Note that the statement ’“\(M\) is a pre-suitable model\("\) is completely internal in \(M\) and hence a \(\Sigma_1(\vec{C}, \vec{T}^0)\) and in particular a \(\Sigma_1 (\{\omega_1 \} )\)-formula. Further note that by the proof of Lemma 13, if \(W^{\ast}\) is a ccc extension of \(W_0\) and \(M\) is a pre-suitable model in \(W^{\ast}\) then \(M \subset W_0\), as ccc extensions will not add new reflecting sequences.

We have already seen in Lemma 13 that ccc extensions of \(W_0\) will still be able to define our \(\omega_2\)-sequence of independent \(W_0\)-Suslin trees \(\vec{T}\). With the notion of suitability we can localize this property in the following sense:

So suitable models will compute \(\vec{T}\) correctly, and if we start to write information into \(\vec{T}\), a suitable model can be used to read it off. In \(W_0\) cofinally many suitable model below \(\omega_2\) exist.

We shall use forcing to obtain a universe in which the just defined set of suitable models

• \(:=\{ M \, : \, \exists \kappa < \eta (\kappa \text{ is inaccessible } \land M=H(\omega_2)^{M_1[G_{\kappa}]} \}\)

becomes easily definable. This will be our first forcing in order to obtain \(W_1\) over \(W_0\). We note that every \(M \in U'\) is itself coded by a real relative to the almost disjoint family \(F\). We want to use the variant of almost disjoint coding forcing to code up

• \(:= \{ r_M \, : \, r_M\) is the least almost disjoint code for a subset of \(\omega_1\) which codes \(M \in U' \}\)

into one real \(r_{U}\). Recall that \(\mathbb{A}(U)\) is a forcing of size \(|U|=\eta\) which is Knaster and which adds a real \(r_U\) such that in \(W_0^{\mathbb{A}(U)}\) the following holds:

• \(\forall x \in 2^{\omega} \cap W_0 \,( x \in U \leftrightarrow r_U \cap S(x)\) is finite\()\).

In a next step we will code the characteristic function of the real \(r_U\) into a pattern on \(\vec{T}^0\). We use the fact that a Suslin tree can generically be destroyed in two mutually exclusive ways. We can generically add a branch or generically add an antichain without adding a branch. We fix the first \(\omega\)-block of independent Suslin trees \(\vec{T}^0\) and we let \(\mathbb{D} (r_U) := \prod_{n \in \omega} \mathbb{P}_n\) with finite support, where \[\mathbb{P}_n = \begin{cases} T^{0}_n & \text{ if } r_U(n)=1 \\ Sp(T^{0}_n) & \text{ if } r_U(n)=0 \end{cases}\] and \(T^{0}_n\) denotes the forcing notion one obtains when forcing with nodes of \(T^{0}_n\) as conditions, and \(Sp(T^{0}_n)\) denotes Baumgartner’s forcing which specializes \(T^{0}_n\) with finite conditions and which is known to be ccc (see [25]). Iterations of the just described form always have the countable chain condition.

In particular this means that \(\mathbb{D} (r_U)\) as defined above is a forcing with the countable chain condition which writes the real \(r_U\) into a pattern of 0 and 1’s on the sequence \(\vec{T}^0\) of Suslin trees. We let \(H_0\) be a generic filter for \(\mathbb{A}_U\) over \(W_0\) and let \(H_1\) denote a \(W_0[H_0]\)-generic filter for \(\mathbb{D} (r_U)\). The resulting model \(W_1:=W_0[H_0][H_1]\) is the ground model for a second iteration we define later.

To summarize we arrived at a situation where the real \(r_U\), which captures all the information about a set of suitable models, is written into the sequence of Suslin trees \(\vec{T}^0\). Consequently, any transitive, \(\aleph_1\)-sized model \(M\) which contains \(\vec{C}\), \(\vec{T}^0\) and which sees that every tree in \(\vec{T}\) is destroyed can compute the real \(r_U\) and thus has access to a set of suitable models, which in turn can be used to compute \(\vec{T}\) inside \(M\) in a correct way. This line of reasoning remains sound in all outer ccc extensions of \(W_1\) as we shall see. Thus the ability of finding \(\vec{T}\) in suitable models gives rise to the possibility of using \(\vec{T}\) for additional coding arguments over \(W_1\).

6 Definition of \(W_2\)↩︎

We shall use the just created \(W_1\) to start a second iteration which will code a fresh sequence of Suslin trees \(\vec{S}\) into the \(W_1\)-definable sequence \(\vec{T}\) of Suslin trees. The definability of \(\vec{T}\) will therefore be passed over to \(\vec{S}\). The reason for this seemingly redundant choice, is that we later want to force \(\mathsf{MA_{\omega_1}}\), which of course will necessarily create a lot of noise on any definable sequence of independent Suslin trees. This noise is a threat to any intentional coding we aim to build in order to have a universe for the \(\Sigma^1_n\)-uniformization property.

The addition of the second sequence \(\vec{S}\) of independent Suslin trees enables us to later use coding with \(\vec{S}\) trees along the usual forcing which produces \(\mathsf{MA_{\omega_1}}\) and yet be in full control of the codes we create on the \(\vec{S}\)-sequence. This effect cannot be reproduced by forcing over \(W_1\) alone; this is why we need to pass from \(W_1\) to a more suitable generic extension.

The universe \(W_2=W_1^{\mathbb{P}_0 \ast \mathbb{P}_1}\) is a two step generic extension of \(W_1\). The first factor \(\mathbb{P}_0\) is just an \(\omega_2\)-length product of ordinary Cohen forcing with finite support \[\mathbb{P}_0:= \prod_{\alpha < \omega_2} \mathbb{C}.\] As is known due to S. Shelah (see [25], Theorem 28.12, Lemma 28.13 for a proof of this which is due to S. Todorčević) every generically added Cohen real \(r\) can be used to define a new Suslin tree \(S_r\). To be more precise, if \(V\) is our ground model and \[(e_{\alpha} \, : \, \alpha < \omega_1)\] is an \(\omega_1\) sequence of functions in \(V\) which satisfies

1. for every \(\alpha < \omega_1\), \(e_{\alpha}\) is an injection from \(\alpha\) to \(\omega\);

2. and for every \(\alpha < \beta < \omega_1\), \(e_{\alpha} (\xi)=e_{\beta} (\xi)\) for all but finitely many \(\xi < \alpha\),

then \[S_r :=\{ r \circ (e_{\alpha} \upharpoonright \beta) \, : \, \alpha,\beta < \omega_1\}\] is a Suslin tree, as long as \(r\) is a Cohen real over \(V\).

Consequently, if \((c_{\alpha} \, : \, \alpha < \omega_2)\) denotes the sequence of Cohen reals in \(W_1^{\mathbb{P}_0}\), the sequence of trees \[\vec{S}:=(S_{{\alpha}} \, : \, \alpha < \omega_2)\] (note that here we write \(S_{\alpha}\) where we actually mean \(S_{c_{\alpha}}\)) is an independent sequence of Suslin trees. Indeed, any finite product of \(S_{\alpha}\)’s must be a Suslin tree again, as otherwise, if say \(S_1\) and \(S_2\) are such that \(S_1 \times S_2\) is not Suslin, then \(\mathbb{C} \times \mathbb{C} \times S_1 \times S_2\) is not a ccc forcing. But rearranging the factors yields that \(S_2\) is a Suslin tree in \(W_1^{\mathbb{C} \times S_1 \times \mathbb{C}}\), hence \(\mathbb{C} \times S_1 \times \mathbb{C} \times S_2\) has the ccc, which is a contradiction.

Our second forcing \(\mathbb{P}_1\) uses our already created independent sequence \(\vec{T}\) to code up the trees from \(\vec{S}\) with a method completely analogous to our forcing \(\mathbb{Q}^1\) which we used in the definition of \(W_0\). That is if we fix an arbitrary \(\alpha < \omega_2\) with the associated tree \(S_{\alpha} \subset \omega_1\), and for \(\beta < \omega_1\), we let \[\mathbb{P}^{\beta}_{1,\alpha}= \begin{cases} T_{ \omega_1 \alpha + \beta} & \text{ if S_{\alpha} (\beta)=1 } \\ SP (T_{\omega_1 \alpha+\beta}) & \text{ if S_{\alpha} (\beta) =0 } \end{cases}\] where \(T_{ \omega_1 \alpha + \beta}\) just denotes the forcing which adds an \(\omega_1\)-branch through \(T_{ \omega_1 \alpha + \beta}\), and SP\((T_{ \omega_1 \alpha + \beta})\) is the forcing which specializes \(T_{ \omega_1 \alpha + \beta}\).

Then \[\mathbb{P}_{1,\alpha} = \prod_{\beta < \omega_1} \mathbb{P}^{\beta}_{1,\alpha}\] using finite support. Note that for every \(\alpha < \omega_2\), \(\mathbb{P}_{1,\alpha}\) has the ccc.

Finally we let \[\mathbb{P}^1 := \prod_{\alpha < \omega_2} \mathbb{P}_{1,\alpha}\] again using finite support. The upshot of these manipulations is, that \(\vec{S}\) is now a uniformly definable \(\omega_2\)-sequence of independent Suslin trees which has a second crucial feature which is not shared by \(\vec{T}\), and which we briefly want to sketch: Let \[W_2= W_1^{\mathbb{P}_0 \ast \mathbb{P}_1}\] and suppose that \(\mathbb{Q}\) is a partial order of size \(\aleph_1\) in \(W_2\). There will be an \(\gamma < \omega_2\) such that \(\mathbb{Q}\) is already in the intermediate model \(W^{(\prod_{\alpha < \gamma} \mathbb{C} ) \ast (\prod_{\alpha < \gamma} \mathbb{P}_{1,\alpha} )}\) which is an inner model of \(W_2\). If we decide to force with \(\mathbb{Q}\) over \(W_2\) then the resulting generic extension \(W_2^{\mathbb{Q}}\) can be written as \[W^{(\prod_{\alpha < \gamma} \mathbb{C} ) \ast (\prod_{\alpha < \gamma} \mathbb{P}_{1,\alpha}) \ast ( \mathbb{Q}) \ast (\prod_{\alpha \in [\gamma, \omega_2) } \mathbb{C} ) \ast ( \prod_{\alpha \in [\gamma, \omega_2)} \mathbb{P}_{1,\alpha} )}.\] As a result the Suslin trees which \(\prod_{\alpha \in [\gamma,\omega_2)} \mathbb{C}\) creates are still Suslin in \(W_2^{\mathbb{Q}}\). This feature will be very useful when we try to force \(\mathsf{MA_{\omega_1}}\) and simultaneously a \(\Delta^1_4\)-definable wellorder of the reals and the \(\Sigma^1_n\)-uniformization property.

7 Coding forcings to alter a \(\Sigma^1_4\)-predicate↩︎

We work now over \(W_2\). Our goal is to define a \(\Sigma^1_4\)-predicate \(\sigma(x)\) such that

1. \(W_2 \models \lnot \exists x \sigma (x)\).

2. For an arbitrary real \(x\), there is a forcing \(\operatorname{Code} (x)\) which has size \(\aleph_1\) and has the ccc and which has the effect that \(W_2^{\operatorname{Code} (x)} \models \sigma (x)\), yet for every real \(y \ne x\), \(W_2^{\operatorname{Code} (x)} \models \lnot \sigma (y)\)

We shall define the coding forcing \(\operatorname{Code} (x)\) along our desired \(\Sigma^1_4\)-formula \(\sigma\). Assume that \(x \in 2^{\omega}\) is an arbitrary real in \(W_2\). In a first step we fix the first \(\omega\)-block \(\vec{S}^1\) from \(\vec{S}\) and define the forcing \(\operatorname{Code} (x)\) relative to this first \(\omega\)-block of trees. This choice should decrease notation as we spare an additional ordinal in the definition of \(\operatorname{Code}\). We add however that the coding forcing can be defined in exactly the same way for other \(\omega\)-blocks of Suslin trees from \(\vec{S}.\)

We use the same technique as in Lemma 16 to write \(x\) into \(\vec{S}^1\), i.e. we first let the first factor of \(\operatorname{Code}(x)\) be the finitely supported product of factors defined as follows: \[\mathbb{P}_{n } = \begin{cases} S^1_{n} & \text{ if } x(n)=1 \\ Sp(S^1_{n}) & \text{ if } x(n)=0 \end{cases}\]

To form the second forcing we need to find a “nice\("\) subset of \(\omega_1\) which codes all the branches and specializing functions along with some suitable models. Let \(A\) be a suitable model which computes \(\vec{T}^1\) correctly. We collect the set \(\mathcal{J}^{M_1}_{\omega_1}\), the relevant branches and specializing functions through elements of \(\vec{T}^0\) which create the pattern which codes up \(r_U\), the relevant reflecting sequences to define the suitable \(A\) and write everything into one set \(X \subset \omega_1\). Note that if \(L_{\zeta}[X]\) is a \(\mathsf{ZF}^-\)-model which contains \(X \subset \omega_1\), we obtain that \[\begin{align} L_{\zeta} [X] \models &n \in x \rightarrow S^1_{n} \text{ has an \omega_1-branch and } \\& n \notin x \rightarrow S^1_{n} \text{ has a specializing function} \end{align}\] where we define \(\vec{T}^1\) and \(\vec{S}^1\) inside \(A \in L_{\zeta} [X]\) and \(A\)’s suitability is confirmed via \(M_1 | \omega_1\), the \(M_1 | \omega_1\)-definable ladder system \(\vec{C}\) and the \(M_1 | \omega_1\)-definable trees \(\vec{T}^0\).

We first note that any transitive, \(\aleph_1\)-sized \(\mathsf{ZF}^-\) model \(M\) which contains \(X\) will satisfy \[\begin{align} (M, \in, \mathcal{J}^{M_1}_{\omega_1}) \models &``\text{Decoding X yields several objects namely} \\& \text{ a model m and m=\mathcal{J}^{ M_1}_{\omega_1}}=\bigcup_{\mathcal{J}^{M_1}_{\eta} \in \mathcal{I}} \mathcal{J}^{M_1}_{\eta}, \\ & \text{ a suitable model a which in turn computes trees \vec{s^1} } \\& \text{some branches \vec{b} and specializing functions \vec{f} through \vec{s^1}},\\& \text{ such that for any \eta< \omega_2 such that L_{\eta}[a][\vec{b}] [\vec{f}] \models \mathsf{ZF}^-} \\& L_{\eta}[a][\vec{b}] [\vec{f}] \models \forall n ( n \in x \rightarrow S^1_{n} \text{ has an \omega_1-branch} \\& \qquad \qquad n \notin x \rightarrow S^1_{n} \text{ has a specializing function}) \end{align}\] In particular, this will be true for a \(\mathsf{ZF}^-+`` \aleph_1\) exists\("\) model of the form \((L_{\xi}[X],\in, \mathcal{J}^{M_1}_{\omega_1})\), \(\xi < \aleph_2\) as we can always assume that the decoding function is absolute.

We consider the club of countable structures \[C:= \{ \eta < \omega_1 \, : \, \exists (M, \in, P) \prec (L_{\xi}[X],\in,\mathcal{J}^{M_1}_{\omega_1})( |M|=\aleph_0 \land \eta= \omega_1 \cap M) \}\] If we assume that \((M,\in,P) \in C\), \(M \cap \omega_1 = \eta\), then the model \(m\in M\) which satisfies the formula \(m=\mathcal{J}^{M_1}_{\omega_1}\) locally in \(M\), will also satisfy \(m \in \mathcal{I}\) as seen from the outside, as it is the limit of reals which are elements of \(\mathcal{I}\) and taking unions is absolute. So \(m= \mathcal{J}^{M_1}_{\eta} \in \mathcal{I}\).

Consequently if \((N, \in)\) is an arbitrary countable transitive model of \(\mathsf{ZFC}^-\) and “\(\aleph_1\) exists”, such that \(X\cap \omega_1^N \in N\) and \(\omega_1^N \in C\) then also \(\mathcal{J}^{M_1}_{\eta} \in N\) and \(N\) will decode out of \(X \cap \omega_1^N\) with the help of the ladder system and the almost disjoint family computed from \(\mathcal{J}^{M_1}_{\eta}\) exactly what \((\bar{M},\in,\mathcal{J}^{M_1}_{\eta})\) decodes out of \(X \cap \omega_1^N\), where the latter is the transitive collapse of \((M,\in, P) \prec (L_{\xi}[X],\in, \mathcal{J}^{M_1}_{\omega_1})\). In particular, if we denote the \(\Delta_1\)-definable decoding functions with \(dec_1,dec_2\), \(dec_3\) and \(dec_4\) respectively, then we obtain \[\begin{align} N \models \exists m_1 \, \exists \vec{c} \, \exists \vec{b}, \vec{f} (& dec_1(X\cap \omega_1^N)=m_1 \land dec_2(X \cap \omega_1^N)= a \\& \land dec_3(X \cap \omega_1^N)=\vec{b} \land dec_4(X\cap \omega_1^N)=\vec{f} \\& \text{ and for any \mathsf{ZF}^- model of the form L_{\eta}[X \cap \omega_1^N] we have that} \\& L_{\eta} [X \cap \omega_1^N] \models \forall n ( n \in x\rightarrow S^1_{n} \text{ has an \omega_1-branch} \\& \qquad \qquad \qquad \quad \quad n \notin x\rightarrow S^1_{n} \text{ is special})) ), \end{align}\] where again the trees \(S^1_n\) are computed inside \(a\) with the help of the formula which defines \(\vec{S}^1\) in \(W_2\). Further, as \(dec_1(X \cap \omega_1^N)=m_1=\mathcal{J}^{M_1}_{\eta}\), we get that (from the outside, not from \(N\)’s perspective) \[dec_1(X \cap \omega_1^N) \in \mathcal{I}.\]

Now let the set \(Y\subset \omega_1\) code the pair \((C, X)\) such that the odd entries of \(Y\) should code \(X\) and if \(Y_0:=E(Y)\) where the latter is the set of even entries of \(Y\) and \(\{c_{\alpha} \, : \, \alpha < \omega_1\}\) is the enumeration of \(C\) then

1. \(E(Y) \cap \omega\) codes a well-ordering of type \(c_0\).

2. \(E(Y) \cap [\omega, c_0) = \emptyset\).

3. For all \(\beta\), \(E(Y) \cap [c_{\beta}, c_{\beta} + \omega)\) codes a well-ordering of type \(c_{\beta+1}\).

4. For all \(\beta\), \(E(Y) \cap [c_{\beta}+\omega, c_{\beta+1})= \emptyset\).

The set \(Y\subset \omega_1\) codes \(X\) but has the following additional property:

• Let \(M\) be an arbitrary countable transitive model of \(\mathsf{ZF}^-+ `` \aleph_1\) exists\("\) for which there is a \(\mathcal{J}^{M_1}_{\eta} \in \mathcal{I}\) (again from the outside, not from \(M\)’s perspective) such that \(\omega_1^M=\omega_1^{\mathcal{J}^{M_1}_{\eta}}\) and \(\mathcal{J}^{M_1}_{\eta} \in M\). Assume that \(Y \cap \omega_1^M \in M\) then \(M\) can decode out of \(Y \cap \omega_1\) with the help of the almost disjoint family of \(\mathcal{J}^{M_1}_{\omega_1}\) the following objects:

  • a transitive model \(m\) which \(M\) believes to be of size \(\aleph_1\),

  • a model \(a\) which, \(M\) believes, is suitable relative to the ladder system and the almost disjoint family of reals defined in \(\mathcal{J}^{M_1}_{\eta}\) and which computes \(\vec{s}^1\) using the same formula which computes the trees in \(W_2\),

  • a set of branches \(\vec{b}\) through \(\vec{s}^1\).

  • and specializing functions \(\vec{f}\) through elements of \(\vec{s}^1\) such that for any \(\mathsf{ZF}^-+ ``\aleph_1\) exists\("\)-model of the form \(L_{\zeta} [a,\vec{b},\vec{f}]\): \[\begin{align} L_{\zeta} [a,\vec{b},\vec{f} \models &\forall n ( n \in x \rightarrow s^1_{n} \text{ has an \omega_1-branch} \\& \qquad n \notin x \rightarrow s^1_{n} \text{ is special})) ). \end{align}\]

  Moreover \(m\) is an \(M_1\) initial segment as seen from the outside, i.e. \(m = \mathcal{J}^{M_1}_{\eta} \in \mathcal{I}\).

In the last step, we use almost disjoint coding forcing relative to the \(\mathcal{J}^{M_1}_{\omega_1}\)-definable almost disjoint family of reals to obtain a real \(r_Y\) which codes our set \(Y \subset \omega_1\). Thus we obtain that the following formula \(\psi(x,r_Y)\) holds, where \(\psi(x,r_Y)\) is defined as:

• Assume \(M\) is a countable transitive model of \(\mathsf{ZF}^-+ ``\aleph_1\) exists\("\), and \(r_Y \in M\). Assume that for \(M\) there is a \(\mathcal{J}^{M_1}_{\eta} \in \mathcal{I}\) such that \(\omega_1^M=\omega_1^{\mathcal{J}^{M_1}_{\eta}}\) and \(\mathcal{J}^{M_1}_{\eta} \in M\). Assume further that \(r_Y \in M\) then \(M\), relative to the almost disjoint family of reals from \(\mathcal{J}^{M_1}_{\eta}\), can decode out of \(r_Y\) the following

  • a transitive model \(m\) of \(\mathsf{ZFC}^-\) and “\(\aleph_1\) exists\("\),

  • a model \(a\) which, \(M\) believes, is suitable, and which computes \(\vec{s}^1\),

  • a set of branches \(\vec{b}\) through \(\vec{s}^1\).

  • and specializing functions \(\vec{f}\) through elements of \(\vec{s}^1\) such that for any \(\mathsf{ZF}^-+ ``\aleph_1\) exists\("\)-model of the form \(L_{\zeta} [a,\vec{b},\vec{f}]\) (computed in \(m\)): \[\begin{align} L_{\zeta} [a,\vec{b},\vec{f} \models &\forall n ( n \in x \rightarrow s^1_{n} \text{ has an \omega_1-branch} \\& \qquad n \notin x \rightarrow s^1_{n} \text{ is special})) ). \end{align}\]

  Moreover \(m\) is an \(M_1\) initial segment as seen from the outside, i.e. \(m = \mathcal{J}^{M_1}_{\eta} \in \mathcal{I}\).

A straightforward calculation shows that the statement \(\psi(x,r_Y)\) is of the form \((\Sigma^1_3 \rightarrow \Pi^1_3)\), thus it is a \(\Pi^1_3\)-formula, and stating the existence of such a real \(r_Y\) is \(\Sigma^1_4\) and results in our desired \(\Sigma^1_4\)-formula \(\sigma\): \[\sigma(x) \equiv \exists r \psi (x,r).\]

The existence of a real \(r\) witnessing \(\psi(x,r)\) is sufficient to conclude that \(L[r]\) contains branches through \(\aleph_1\)-many trees from \(\vec{S}\).

As an immediate consequence of the preceding lemma, the truth of the \(\Sigma^1_4\)-formula \(\sigma\) has absolute consequences for the ambient universe: whenever \(\sigma(x)\) holds, the characteristic function of \(x\) is genuinely written into \(\vec{S}^1\). This property ensures that our iteration does not inadvertently introduce spurious codes; that is, no unintended real \(x\) will satisfy \(\sigma(x)\).

Furthermore, we state again that while we initially defined \(\operatorname{Code}(x)\) with respect to \(\vec{S}^1\), this choice was arbitrary. We can generalize the definition of \(\operatorname{Code}(x)\) to utilize \(\vec{S}^{\eta}\) for any \(\eta < \omega_2\).

Moving forward, we adopt the convention that whenever a coding forcing \(\operatorname{Code}(x,\eta)\) is used as a factor in an iteration, \(\eta\) is always chosen to be the least ordinal such that no forcing of the form \(\operatorname{Code}(y,\eta)\) has appeared in any prior stage. In other words, the iteration uses the \(\omega\)-blocks of \(\vec{S}\) in consecutive order, leaving no unused gaps.

8 Appropriate \(\Sigma^1_n\)-predicates↩︎

We now fix the higher coding predicates used in the final iteration. Let \[\operatorname{Coded}(z)\] abbreviate the \(\Sigma^1_4\) assertion that the real \(z\) is coded into the sequence \(\vec{S}\) in the sense of the preceding section. We shall also write “\(z\) is not coded” for \(\neg\operatorname{Coded}(z)\). The point is that the sign of the final coding statement alternates with the parity of the desired projective level.

For \(k\geq 4\) put \(\ell=k-4\). If \(z\) is a real, define \(\Phi^k(z)\) by \[\Phi^k(z) \quad\Longleftrightarrow\quad \exists a_0\, Q_1 a_1\, Q_2 a_2\cdots Q_\ell a_\ell\; \Theta_\ell(z,a_0,\ldots,a_\ell),\] where \(Q_i\) is \(\forall\) for odd \(i\) and \(\exists\) for even \(i\), and where \[\Theta_\ell(z,a_0,\ldots,a_\ell)= \begin{cases} \operatorname{Coded}(z,a_0,\ldots,a_\ell), & \text{if \ell is even},\\ \neg\operatorname{Coded}(z,a_0,\ldots,a_\ell), & \text{if \ell is odd}. \end{cases}\] Here, as throughout, the expression \((z,a_0,\ldots,a_\ell)\) denotes the fixed recursive real coding the tuple. In applications \(z\) will itself often be a code for a tuple, for instance \(z=(x,y,m)\).

Thus the first few cases are \[\begin{align} \Phi^4(x,y,m) &\Longleftrightarrow \exists a_0\; \operatorname{Coded}(x,y,m,a_0),\\ \Phi^5(x,y,m) &\Longleftrightarrow \exists a_0\forall a_1\; \neg\operatorname{Coded}(x,y,m,a_0,a_1),\\ \Phi^6(x,y,m) &\Longleftrightarrow \exists a_0\forall a_1\exists a_2\; \operatorname{Coded}(x,y,m,a_0,a_1,a_2),\\ \Phi^7(x,y,m) &\Longleftrightarrow \exists a_0\forall a_1\exists a_2\forall a_3\; \neg\operatorname{Coded}(x,y,m,a_0,a_1,a_2,a_3). \end{align}\] Since \(\operatorname{Coded}\) is \(\Sigma^1_4\), the displayed definition gives exactly a \(\Sigma^1_k\) predicate. The cases \(k=2,3\) are dealt with by the earlier \(\Sigma^1_2\) and \(\Sigma^1_3\) uniformization arguments; the present predicates are the ones needed from level \(4\) upward.

The following observation is the only feature of these predicates used later.

9 Forcing \(\mathsf{MA}\) and global \(\Sigma\)-uniformization and a definable wellorder↩︎

We now perform the final forcing over \(W_2\). The forcing is a finite-support iteration of length \(\omega_2\) whose nontrivial stages have three roles: they add the diagonal ccc forcing needed for \(\mathsf{MA}\), they code the comparisons which yield the definable wellorder of the reals, and they carry out the global \(\Sigma\)-uniformization construction. The diagonal form of the \(\mathsf{MA}\) stages is used only to preserve the unused tail of the sequence \(\vec{S}\).

Before starting the final iteration we make one harmless normalization which will be used only for the compression of long lists of reals into single reals. Fix, in a part of the construction disjoint from the Suslin-tree coding apparatus, an almost disjoint family \[\mathcal{H}=\langle h_\xi:\xi<\omega_1\rangle\] of reals. If \(A\subseteq\omega_1\), let \(\mathbb{A}_{\mathcal{H}}(A)\) be the usual almost disjoint coding forcing which adds a real \(r_A\) such that, for all \(\xi<\omega_1\), \[\xi\in A \quad\Longleftrightarrow\quad |r_A\cap h_\xi|=\omega .\] Equivalently, one may use the dual convention with finite intersections for membership; the choice is immaterial. We use a finite-support bookkeeping iteration of such forcings, relative to \(\mathcal{H}\), to code all subsets of \(\omega_1\) which occur in the relevant intermediate extensions into reals. This auxiliary forcing is completely separate from the forcings which read or destroy the trees in \(\vec{S}\).

After this normalization we keep the notation \(W_2\) for the resulting model.

9.1 Definition of the final iteration↩︎

Fix a bookkeeping function \[F:\omega_2\longrightarrow H(\omega_2)\] such that every relevant object, name, and tuple of names occurs unboundedly often. We define by induction a finite-support iteration \[\bigl((\mathbb{P}_\beta,\dot{\mathbb{Q}}_\beta):\beta<\omega_2\bigr)\] of ccc forcings over \(W_2\). Let \(G_\beta\) denote the \(\mathbb{P}_\beta\)-generic filter. At stage \(\beta\) we use the least still unused \(\omega\)-block of \(\vec{S}\) whenever a coding forcing is required. Thus, after stage \(\beta\), the blocks already used for coding form an initial segment of the block sequence, while the remaining blocks still form an independent sequence of Suslin trees in the current extension.

The value of \(F(\beta)\) is interpreted in one of the following ways. If it is not of one of these forms, we let \(\dot{\mathbb{Q}}_\beta\) be trivial.

1. Uniformization request. The value \(F(\beta)\) codes an \(\omega\)-tuple of ordinals which, in the usual way, determines a tuple of \(\mathbb{P}_\beta\)-names \[(\dot{x},\dot{m},\dot{\alpha},\dot{b}_1,\dot{b}_2,\ldots).\] Here \(\dot{x}\) and the \(\dot{b}_i\) are names for reals, \(\dot{m}\) is a name for a natural number coding a projective formula, and \(\dot{\alpha}\) is a name for an ordinal below the continuum. The precise factor \(\dot{\mathbb{Q}}_\beta\) is then defined by the four uniformization cases in Subsection 9.3 below.

2. Wellorder coding request. The value \(F(\beta)\) is a pair \((\dot{x}_0, \dot{x}_1)\) of \(\mathbb{P}_\beta\)-names for reals. Let \(x_i=\dot{x}_i^{G_\beta}\), and let \(\sigma_i\) be the \(<_{M_1}\)-least \(\mathbb{P}_\beta\)-name whose \(G_\beta\)-value is \(x_i\). After interchanging the indices if necessary, assume \(\sigma_0<_{M_1}\sigma_1\). We then put \[\dot{\mathbb{Q}}_\beta^{G_\beta}=\operatorname{Code}((0,x_0,x_1),\eta_\beta),\] where \(\eta_\beta\) is the least unused block index. The initial coordinate \(0\) only marks this as a wellorder code.

3. Diagonal \(\mathsf{MA}\) request. The value \(F(\beta)\) is a pair \((\delta,\gamma)\). In \(W_2[G_\delta]\), let \(\mathbb{B}\) be the \(\gamma\)-th partial order of size \(\aleph_1\), provided that \(\mathbb{B}\) is still ccc in \(W_2[G_\beta]\). The forcing \(\mathbb{P}_\beta\ast\mathbb{B}\) has a dense subforcing of size \(\aleph_1\) consisting of fully decided conditions, and therefore can be coded as a subset of \(\omega_1\) in \(W_2\). Since \(W_2=W_1[H_{\omega_2}]\), there is some \(\nu<\omega_2\) such that a forcing equivalent to this dense subforcing belongs to \(W_1[H_\nu]\). If \(\nu\leq\beta\), we set \[\dot{\mathbb{Q}}_\beta^{G_\beta}=\mathbb{B};\] otherwise we force trivially at stage \(\beta\).

The point of the condition \(\nu\leq\beta\) in the \(\mathsf{MA}\) case is that the unused tail of the \(\vec{S}\)-sequence remains genuinely generic over the model in which the side forcing \(\mathbb{B}\) has already appeared. This prevents the \(\mathsf{MA}\) stages from creating unintended coding patterns.

The same freshness convention handles the wellorder and uniformization coding stages: such stages intentionally consume the next unused block, and Lemma 24 handles the only possible interference from the diagonal \(\mathsf{MA}\) side forcing.

9.2 Proof that \(\mathsf{MA}\) holds true↩︎

Let \(G_{\omega_2}\) be \(\mathbb{P}_{\omega_2}\)-generic over \(W_2\).

The combination of \(\mathsf{MA}\) with the \(\Sigma^1_3\) generic absoluteness available in our background model gives the third-level regularity consequences needed for the theorem.

9.3 Global \(\Sigma\)-uniformization: the coding rules↩︎

We now make the uniformization part of the iteration explicit. The point of the construction is to mark, at the same projective level, the first successful triple in the canonical list of possible witnesses. We keep the notation from the previous subsection and write simply \[\operatorname{Code}(z)\] for coding \(z\) into the next unused block of \(\vec{S}\).

We shall use the following compression lemma throughout the argument.

If \(b\) is a real and \(\alpha\) is understood, we write \[\pi_\alpha^{-1}(b)=(b^\eta:\eta<\alpha).\] Thus one real \(b\) codes the whole sequence of reals \((b^\eta)_{\eta<\alpha}\). The bookkeeping is understood at the level of names; in the verification below we suppress the dots and write the corresponding final reals.

Let \(<\) be the final canonical wellorder of the reals. For each real \(x\) let \[(x,y^0,a_0^0),(x,y^1,a_0^1),\ldots,(x,y^\xi,a_0^\xi),\ldots\qquad(\xi<\omega_2)\] be the \(<\)-increasing enumeration of all triples with first coordinate \(x\), without repetitions, and with \((x,y^0,a_0^0)=(x,0,0)\).

For the even levels write a formula coded by \(m\) in the form \[\varphi_m(x,y)\equiv \exists a_0\,\forall a_1\exists a_2\cdots\exists a_{2n-4}\, \psi(x,y,a_0,a_1,\ldots,a_{2n-4}),\] where \(n\geq 2\) and \(\psi\in\Pi^1_3\). Let \[S^{\mathrm{ev}}_m(x,y,a_0)\] abbreviate the displayed tail formula \[\forall a_1\exists a_2\cdots\exists a_{2n-4}\, \psi(x,y,a_0,a_1,\ldots,a_{2n-4}).\] Here and below, if \(2n-4=0\), the string of quantifiers is empty and the formula is just \(\psi(x,y,a_0)\).

The even marker is \[M^{\mathrm{ev}}_m(x,y,a_0)\] and means \[\exists b_1\forall b_2\exists b_3\cdots\forall b_{2n-4}\, \operatorname{Coded}(\#\psi,x,y,a_0,b_1,\ldots,b_{2n-4}).\] Again, for \(2n-4=0\) this means simply \[\operatorname{Coded}(\#\psi,x,y,a_0).\] The associated uniformizing relation is \[\begin{align} \sigma_{\mathrm{even}}(x,y,m)\quad\Longleftrightarrow\quad &\bigl(y=0\wedge S^{\mathrm{ev}}_m(x,0,0)\bigr)\;\lor\; \\ &\bigl(\neg S^{\mathrm{ev}}_m(x,0,0)\wedge \exists a_0\bigl(S^{\mathrm{ev}}_m(x,y,a_0)\wedge M^{\mathrm{ev}}_m(x,y,a_0)\bigr)\bigr). \end{align}\] This is a \(\Sigma^1_{2n}\) formula. Indeed, the first disjunct is \(\Pi^1_{2n-1}\), hence belongs to \(\Sigma^1_{2n}\), while the second disjunct is a conjunction of a \(\Sigma^1_{2n-1}\) statement with a \(\Sigma^1_{2n}\) statement.

For the odd levels write \[\varphi_m(x,y)\equiv \exists a_0\,\forall a_1\exists a_2\cdots\forall a_{2n-3}\, \psi(x,y,a_0,a_1,\ldots,a_{2n-3}),\] where \(n\geq 2\) and \(\psi\in\Sigma^1_3\). Let \[S^{\mathrm{odd}}_m(x,y,a_0)\] abbreviate \[\forall a_1\exists a_2\cdots\forall a_{2n-3}\, \psi(x,y,a_0,a_1,\ldots,a_{2n-3}).\] The odd marker is the dual non-coding marker \[M^{\mathrm{odd}}_m(x,y,a_0)\] meaning \[\exists b_1\forall b_2\exists b_3\cdots\exists b_{2n-3}\, \neg\operatorname{Coded}(\#\psi,x,y,a_0,b_1,\ldots,b_{2n-3}).\] The final quantifier in (MO) is existential, since \(2n-3\) is odd. The associated relation is \[\begin{align} \sigma_{\mathrm{odd}}(x,y,m)\quad\Longleftrightarrow\quad &\bigl(y=0\wedge S^{\mathrm{odd}}_m(x,0,0)\bigr)\;\lor\; \\ &\bigl(\neg S^{\mathrm{odd}}_m(x,0,0)\wedge \exists a_0\bigl(S^{\mathrm{odd}}_m(x,y,a_0)\wedge M^{\mathrm{odd}}_m(x,y,a_0)\bigr)\bigr). \end{align}\] This is a \(\Sigma^1_{2n+1}\) formula.

We now describe the forcing factor used at a uniformization stage. Work in \(W_2[G_\beta]\) and suppose that the bookkeeping value at stage \(\beta\) gives \[(x,m,\alpha,b_1,b_2,\ldots),\] where \(0<\alpha<\omega_2\). Only the finitely many \(b_i\) needed by the formula coded by \(m\) are used. If \[\pi_\alpha^{-1}(b_i)=(b_i^\eta:\eta<\alpha),\] then the stage is defined as follows.

If \(m\) codes an even formula as in (E), put \[\dot{\mathbb{Q}}_\beta^{G_\beta}= \operatorname{Code}(\#\psi,x,y^\alpha,a_0^\alpha,b_1,\ldots,b_{2n-4})\] exactly in the case that \[\neg\psi(x,y^\eta,a_0^\eta,b_1^\eta,\ldots,b_{2n-4}^\eta)\] holds for every \(\eta<\alpha\). If this condition fails, we force trivially.

If \(m\) codes an odd formula as in (O), put \[\dot{\mathbb{Q}}_\beta^{G_\beta}= \operatorname{Code}(\#\psi,x,y^\alpha,a_0^\alpha,b_1,\ldots,b_{2n-3})\] exactly in the case that \[\psi(x,y^\eta,a_0^\eta,b_1^\eta,\ldots,b_{2n-3}^\eta)\] holds for some \(\eta<\alpha\). If no such \(\eta\) exists, we force trivially.

If the bookkeeping value is ill formed, or if \(m\) does not code one of the displayed normal forms, the stage is trivial. Notice that the coding marker is always \(\#\psi\), never \(\#\varphi_m\). This is important: the marker records the matrix computation which is being compressed.

The truth of the \(\Sigma^1_3\) or \(\Pi^1_3\) matrix \(\psi\) is permanent between the intermediate model in which the stage is evaluated and the final model. This is the base-level generic absoluteness supplied by the \(M_1\) analysis used in Section 11. Thus the above decisions are not later reversed.

10 Proof that \(\Sigma^1_n\)-uniformization holds in the final model↩︎

Let \[V^*=W_2[G_{\omega_2}].\] We verify that the relations \(\sigma_{\mathrm{even}}\) and \(\sigma_{\mathrm{odd}}\) defined above uniformize the corresponding projective relations in \(V^*\).

We first record the no-accidental-coding consequence used below. If a tuple \(z\) is not intentionally coded at a stage using the block assigned to that stage, then no later stage can make \(z\) coded in that block. Different coding stages use different blocks, and the predicate \(\operatorname{Coded}(z)\) is read from the unique block assigned to the relevant marker. Hence all marker statements decided by the rules above remain fixed in the final extension.

Combining Theorem 14 with the previously established \(\Sigma^1_2\) case and the \(\Sigma^1_3\) base case proved in Section 11, we obtain the announced global tail uniformization over the final model.

11 The \(\Sigma^1_3\)-uniformization in \(W_{2}[G_{\omega_2}]\)↩︎

The higher uniformization argument leaves the \(\Sigma^1_3\) base case to be checked separately. At this first nontrivial level the coding iteration is not needed. What is used is the standard Steel analysis of \(M_1\) and of its relativizations. We only need this analysis for the small generic extensions which occur in the construction. Thus, throughout this section, a small generic extension of \(M_1\) means an extension \(M_1[G]\) by a set forcing whose size in \(M_1\) is below the Woodin cardinal of \(M_1\) and to which the canonical iteration strategy of \(M_1\) lifts. All intermediate models used in the construction, and in particular the final model \(W_2[G_{\omega_2}]\), are of this form.

For a real \(s\) in such an extension, let \(M_1(s)\) denote the canonical proper class mouse over \(s\) with one Woodin cardinal. The relevant mice exist and are iterable by the lifted \(M_1\) strategy. The argument below uses a capture theorem for this canonical mouse over \(s\); it does not identify \(M_1(s)\) with the forcing extension generated by \(s\).

The assertion that a real \(c\) codes a sound simple \(s\)-mouse is a \(\Pi^1_2(s,c)\) condition. Any two such mice compare by initial segment. Hence the reals of \(M_1(s)\) carry the usual good \(\Sigma^1_3(s)\) wellorder \(<^1_s\): first compare the least simple \(s\)-mouse containing the real, and then use the canonical wellorder of that mouse.

We shall use the following standard form of Steel’s relativized capture theorem at the first mouse level; see the projective definability and comparison analysis of \(M_1\) in [5], [14].

For completeness, let us spell out why this is the right theorem to apply. The mouse \(M_1(s)\) is the canonical mouse over the parameter \(s\). The comparison theorem for \(\Pi^1_2\)-iterable \(s\)-mice identifies its countable initial segments with the segments captured by the \(\Pi^1_2\) mouse condition. Therefore a small generic extension of \(M_1\) cannot create a new \(\Sigma^1_3(s)\) witness without some countable initial segment of \(M_1(s)\) already capturing the corresponding branch. The same comparison analysis gives the \(\Pi^1_2\) correctness used in item (2), and the usual definition of the wellorder by least simple \(s\)-mouse gives item (3).

12 A byproduct on generalized regularity properties↩︎

In this section we record a consequence of the \(M_1\) case for generalized notions of measurability associated with tree forcings. We use the notation of Fischer–Friedman–Khomskii (see [26]) and Friedman–Schrittesser ([27]). Thus \(B\) denotes the random algebra, equivalently Lebesgue measurability; \(C\) denotes Cohen forcing, equivalently the Baire property; \(S\) denotes Sacks measurability, equivalently Marczewski measurability; and \(M\) denotes Miller measurability. If \(P\) is one of these tree forcings and \(\Gamma\) is a pointclass, we write \(\Gamma(P)\) for the assertion that every set in \(\Gamma\) is \(P\)-measurable.

We shall use the following theorem of Brendle and Löwe, as quoted in [26]. If \(\Gamma\) is closed under continuous preimages, then \[\Gamma(C)\Rightarrow \Gamma(M) \quad\text{and}\quad \Gamma(B)\Rightarrow \Gamma(S).\] They also show the implication chain \[\Gamma(L)\Rightarrow \Gamma(M)\Rightarrow \Gamma(S),\] where \(L\) denotes Laver measurability. We shall only use the first two implications above.

The preceding corollary should not be read as treating all tree forcings in the Friedman–Schrittesser diagram. In particular, the Brendle–Löwe implications used above do not yield \(\Sigma^1_3(L)\), \(\Sigma^1_3(V)\) or \(\Sigma^1_3(R)\), where \(L,V,R\) denote Laver, Silver and Mathias measurability, respectively. The implication involving Laver goes in the direction \(\Gamma(L)\Rightarrow\Gamma(M)\), not conversely. Thus the corollary is limited to \(P\in\{B,C,S,M\}\).

13 Forcing over \(M_n\)↩︎

In this final section we explain how the preceding construction is lifted from \(M_1\) to the canonical inner model \(M_n\) with \(n\) Woodin cardinals. The point of this section is not to repeat the coding computations of Section 8. Those computations are entirely formal once the relevant initial segments of the ground model are available in the decoding models. The only change is that, over \(M_n\), the assertion that a decoded premouse is an initial segment of the ground model has higher projective complexity.

We therefore isolate the precise modification. In the construction over \(M_1\) we used the fact that, in any \(\omega_1\)-preserving generic extension, the relevant countable initial segments of \(M_1\) are captured by the \(\Pi^1_2\)-definable class \(\mathcal{I}\). Over \(M_n\) the corresponding class is more complex.

We now repeat the definition of the coding predicate in the form in which it will be used over \(M_n\). Let \(x\in 2^\omega\) be a real. As before, we first force a pattern into an unused \(\omega\)-block \[\vec{S}^\xi=(S^\xi_k:k<\omega)\] of the independent sequence of Suslin trees: for each \(k<\omega\) we add an \(\omega_1\)-branch through \(S^\xi_k\) if \(x(k)=1\), and specialize \(S^\xi_k\) if \(x(k)=0\). We then collect the resulting branches and specializing functions, together with the auxiliary objects needed to reconstruct the suitable model which computes the block \(\vec{S}^\xi\), into a set \(X\subseteq\omega_1\). Finally, exactly as in Section 8, we replace \(X\) by a set \(Y\subseteq\omega_1\) which also codes a club of countable elementary submodels, and then almost-disjoint-code \(Y\) by a real \(r_Y\) using the almost disjoint family computed from the relevant initial segment of \(M_n\).

The resulting decoding formula is the following.

Thus \(\psi_n(x,r)\) is literally the formula \(\psi(x,r)\) of Section 8 with \(\mathcal{I}\) replaced by \(\mathcal{I}_n\) and \(M_1\) replaced by \(M_n\).

The crucial absoluteness consequence of the coding predicate is unchanged.

We define the \(M_n\)-version of the coding forcing as follows.

Thus the full computation of Section 8 transfers to \(M_n\) after the single replacement \[\mathcal{I}\quad\rightsquigarrow\quad \mathcal{I}_n.\] The price paid for this replacement is purely descriptive-set-theoretic: the basic coding predicate becomes \(\Sigma^1_{n+3}\) rather than \(\Sigma^1_4\).

The only level not covered by this shifted coding predicate is the first one, namely \(\boldsymbol{\Sigma}^1_{n+2}\). We now record the exact higher analogue of the \(M_1(s)\) capture theorem used in Section 11. Throughout the statement, a small generic extension of \(M_n\) means an extension of the kind fixed in Paragraph 2.2.0.1; equivalently, it is one of the extensions in which the canonical \(M_n\) strategy and the relevant comparison arguments used in this paper are preserved.

We now define the predicates used for the uniformization part of the final iteration. Let “\(z\) is coded into \(\vec{S}\)” mean \(\sigma_n(z)\). Since this is a \(\Sigma^1_{n+3}\) predicate, we define, for each \(m<\omega\), the corresponding predicate \(\Phi^{n+3+m}\) by adding the usual alternating block of real quantifiers in front of the coding predicate. For example, \[\Phi^{n+3}(x,y,k) \quad\Longleftrightarrow\quad \exists a_0\, \bigl((x,y,k,a_0)\text{ is coded into }\vec{S}\bigr),\] and in general

\[\begin{align} \Phi^{n+3+m}(x,y,k) \Longleftrightarrow \exists a_0\,\forall a_1\,\cdots\,Q_m a_m\, & ((x,y,k,a_0,\ldots,a_m) \text{ is coded} \\& \text{ into } \vec{S} ) \end{align}\] where the final quantifier \(Q\) depends on the arity of \(m\) alternate. The complexity of \(\Phi^{n+3+m}\) is therefore \(\Sigma^1_{n+3+m}\).

The final \(\omega_2\)-length iteration is now defined exactly as before. At successor stages we again distinguish the three cases: the stages devoted to forcing Martin’s Axiom, the stages devoted to the definable wellorder of the reals, and the stages devoted to the uniformization predicates \(\Phi^{n+3+m}\). Whenever the earlier construction used \(\operatorname{Code}(x,\xi)\), we now use \(\operatorname{Code}_n(x,\xi)\). The bookkeeping again ensures that every relevant real, every relevant forcing for \(\mathsf{MA_{\omega_1}}\), and every relevant projective relation to be uniformized is considered cofinally often, and that unused \(\omega\)-blocks of \(\vec{S}\) are always available.

It remains only to record the regularity conclusion. We spell out the higher-level version of the Hjorth argument, since this is the point at which the tree \(T_{n+1}\) replaces the classical Martin–Solovay tree \(T_2\).

Combining Propositions 18 and 19 gives the desired theorem.

14 Open Questions↩︎

We end with a couple of natural questions whose answers would need new ideas. A first and natural question is whether the large cardinals used in our argument can be improved.

Note that by R. Solovay ([28])and S. Shelah ([29]) we only need an inaccessible to force all boldface projective sets become Lebesgue measurable and have the Baire property. The other two properties have no large cardinal strength at all. The question of whether already an inaccessible suffices to get a universe with the combination of the four properties was already asked in [1].

The following strengthening of the original 12th Delfino problem is still wide open (see [30]):

There are other local versions of the 12th Delfino problem which are mostly open as well:

For \(n=N=1\) the above has a positive answer due to J. Steel (see [2]) but for the higher levels this is open.

References↩︎

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1. The first author’s research was funded in whole by the Austrian Science Fund (FWF) Grant-DOI 10.55776/P37228. The second author’s research was funded in part by the Austrian Science Fund (FWF) [10.55776/Y1498, 10.55776/I6087]. For the purpose of open access, the authors have applied a CC BY public copyright license to any Author Accepted Manuscript version arising from this submission.↩︎

2. The failure of \(\boldsymbol{\Delta}^1_{n+1}\)-determinacy in \(M_n\) can be seen via comparison. Woodin’s inner model characterization establishes that \(\boldsymbol{\Delta}^1_{n+1}\)-determinacy is equivalent to the statement that for every real \(x\), there exists a countable, iterable \(x\)-mouse with \(n\) Woodin cardinals. If \(\boldsymbol{\Delta}^1_{n+1}\)-determinacy were to hold in \(M_n\), then \(M_n\) would contain a set sized, iterable mouse \(N\) with \(n\) Woodin cardinals (see [7]). Coiterating \(N\) with \(M_n\) yields a proper initial segment of an iterate of \(M_n\) containing \(n\) Woodin cardinals. However, by the minimality of \(M_n\), no proper initial segment of \(M_n\) (or its iterates) can contain \(n\) Woodin cardinals. This contradiction demonstrates that no such mouse \(N\) exists in \(M_n\), and thus \(\boldsymbol{\Delta}^1_{n+1}\)-determinacy must fail in \(M_n\).↩︎