A topological completeness theorem for transfinite provability logic

We prove a topological completeness theorem for the modal logic GLP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsf{GLP}$$\end{document} containing operators {⟨ξ⟩:ξ∈Ord}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\langle \xi \rangle :\xi \in \textsf{Ord}\}$$\end{document} intended to capture a wellordered sequence of consistency operators increasing in strength. More specifically, we prove that, given a tall-enough scattered space X, any sentence ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document} consistent with GLP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsf{GLP}$$\end{document} can be satisfied on a polytopological space based on finitely many Icard topologies constructed over X and corresponding to the finitely many modalities that occur in ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document}.


Introduction
The purpose of this article is to prove a topological completeness theorem for the transfinite extension of Japaridze's logic G L P. G L P is a provability logic in a propositional language augmented with a possibly transfinite sequence of modal operators; our case of interest is that in which the sequence is wellfounded. Each of these operators can be interpreted arithmetically as asserting provability within a given theory, and the logic G L P relates these notions of provability to one another. For arithmetical interpretations of G L P, see Beklemishev [6], Fernández-Duque and Joosten [18], Cordón-Franco et al. [15], and others. As a modal logic, G L P has some unusual properties. For example, it is not complete with respect to any class of relational frames; a natural question is whether it is complete with respect to its neighborhood (i.e., topological) semantics. Beklemishev and Gabelaia [11] showed that G L P ω , the restriction of G L P to ω-many modalities, is complete with respect to a natural topological space on the ordinal ε 0 . Because sentences in the language of G L P contain instances of only finitely many modalities, the spaces constructed by Beklemishev and Gabelaia serve as models also for formulas in the language of transfinite G L P; hence it is also topologically complete. However, it is an open problem whether transfinite G L P is complete with respect to a single Beklemishev-Gabelaia space. 1 Another open problem is that of completeness with respect to what are known as the canonical topological semantics for G L P. The question of completeness with respect to these spaces has very interesting connections with stationary reflection and indescribable cardinals (see Bagaria [4], Bagaria-Magidor-Sakai [5] and Brickhill [14]). Completeness for the two-modality fragment with respect to these topologies was proved by Beklemishev [8]. It is not hard to see that G L P is not strongly complete with respect to its canonical topological semantics; we shall prove this below.
Completeness with respect to Beklemishev-Gabelaia spaces was proved by Fernández-Duque [16] for restrictions of G L P to any countable amount of modalities. It is not known whether this result can be extended to arbitrarily long sequences of modalities, but the results from [3] show that the techniques would need to be very different.
The topological completeness theorem we shall prove here goes in this direction. Roughly, given a sentence consistent with G L P, we construct a topological model for it with finitely many topologies. The new feature is that these topologies in a way correspond to the modalities appearing in the sentence; we call these ϑ-polytopologies, for ϑ a finite sequence of ordinals. In particular, one can extend the space with intermediate increasing topologies corresponding to modalities not appearing in the sentence in such a way that each topology results in a model of the unimodal G L. Unfortunately, this extension (with the intermediate topologies) will not be a model of G L P, but we hope that a similar construction can yield completeness for Beklemishev-Gabelaia spaces with any amount of topologies. This hope is the main motivation for carrying out the work reported in this article.
Our main tool is a technical "product lemma." Essentially, given two ordinals κ and λ, we find an ordinal and natural projections π 0 : → κ and π 1 : → λ which preserve satisfiability of polymodal formulas, if the ordinals are equipped with the right topologies. This is a generalization of a technical lemma of Beklemishev-Gabelaia [10], which corresponds to the case in which the first element of the sequence ϑ is 1. The proof is largely arithmetical and relies heavily on the theory of hyperexponentials and hyperlogarithms of Fernández-Duque and Joosten [17].

The polymodal logic of provability
For any ordinal number we consider a language L consisting of a countable set of propositional variables P together with the constants , ⊥; Boolean connectives ∧, ∨, ¬, →; and a modality [ξ ] for each ordinal ξ < . As usual, we write ξ as a shorthand for ¬[ξ ]¬.

Definition 2.1
The logic G L P is then defined to be the least logic containing all propositional tautologies and the following axiom schemata: ϕ for all ξ < ζ < , (iv) ξ ϕ → [ζ ] ξ ϕ for all ξ < ζ < , and closed under the rules modus ponens and necessitation for each [ξ ]: We will often write simply G L P for G L P when we do not want to specify a . Note that G L P when restricted to any one modality is simply the well-known logic G L. Modal logics are usually studied by means of relational semantics. A Kripke -frame is a structure (W , {R ξ }) ξ< , where each R ξ is a binary relation on W . We define a valuation · to be a function assigning subsets of W to each L -formula such that · respects boolean connectives and such that Proposition 2.2 (Segerberg [19]) G L P 1 is complete with respect to the class of finite relational frames (W , R) that are conversely wellfounded trees.
The preceding proposition provides a convenient way to study G L P 1 . However, as is well known, G L P is incomplete with respect to any class of relational frames whenever 1 < . This motivates the search for topological models of the G L P.
Recall that x is a limit point of A if A intersects every punctured neighborhood of x. We call the set of limit points of A the derived set of A and denote it by d A. We may also denote it by d τ A to emphasize the topology we are considering. The derived set operator is iterated transfinitely by setting Since d α X ⊃ d β X whenever α < β, there exists a minimal ordinal ht(X )-the height or rank of X -such that d ht(X ) X = d ht(X )+1 X . For any x ∈ X , we let ρ τ x, the rank of x, be the least ordinal ξ such that x / ∈ d ξ +1 X , if it exists. Throughout this paper, we will speak about rank-preserving extensions of topologies. (A topology σ on a set X is a rank-preserving extension of a topology τ on X if τ ⊂ σ and ρ σ x = ρ τ x for all x ∈ X .) Lemma 2.3 ). A topology σ is a rank-preserving extension of a scattered topology τ if, and only if, ρ τ [U ] is an ordinal for each U ∈ σ .
A point in A that is not a limit point is isolated. Thus a point is isolated if and only if it has rank 0. We denote by iso(A) the set of isolated points in Not all scattered spaces are T 1 (e.g., X = {0, 1} with open sets ∅, {0} and X ), however, the examples in which we will focus are.
We study polytopological spaces-structures (X , {T ι } ι< ), where X is a set and {T ι } ι< is a sequence of topologies of length . Topological semantics for modal logics may be defined by interpreting diamonds as topological derivatives. Definition 2.4 (Topological semantics). Let X = (X , {T ι } ι< ) be a polytopological space. A valuation is a function · : L → P(X ) such that for any L -formulae ϕ, ψ: is a polytopological space together with a valuation. We say that ϕ is satisfied in M if ϕ is nonempty and we say ϕ is valid in a space X and write X | ϕ if ϕ = X for any model based on X.
Observe that x ∈ [ξ ]ϕ if, and only if, there is a T ξ -neighborhood of x all of whose points belong to ϕ , except possibly for x. In order that a space validate the axioms of G L P, we need to impose some regularity conditions (see ). A space (X , {T ι } ι< ) is a G L P -space if {T ι } ι< is nondecreasing, scattered, and d ξ A ∈ T ζ for all ξ < ζ and all A ∈ P(X ) (2.1) In the situation above, we refer to {T ι } ι< as a G L P -polytopology. Clearly, we have:

Lemma 2.5 Any G L P -space validates all theorems of G L P .
A natural way of constructing G L P -polytopologies appears to be to start with any scattered topology and simply add all derived sets at each stage. This results in what has come to be known as the canonical G L P-space generated by X . In doing so, the topologies quickly become extremely fine. In fact, for the most natural examples, their non-discreteness becomes undecidable within ZFC after two or three iterations.
One way out of this, explored in Beklemishev-Gabelaia [10], is to extend the topology at each stage before adding derived sets. Extending the topology reduces the amount of derived sets attainable and makes subsequent topologies coarser. A different approach, introduced in Fernández-Duque [16], is to fix increasing topologies from the beginning and restrict the algebra of possible valuations. We will consider the first approach here.
Let us finish this section with the remark that G L P is not strongly complete with respect to its canonical semantics. By strong completeness (with respect to a class of models X ), we mean the following assertion: whenever is a set of L -sentences consistent with G L P , then there is some model X ∈ X where is satisfied.

Proposition 2.6 Suppose X is a scattered space in which every G δ set is open. Then G L is not strongly complete with respect to X .
Proof This is a generalization of the usual proof that G L is not strongly complete with respect to trees. Let Suppose is satisfied at some x. Then, for each i, there is a neighborhood U i of x such that every y ∈ U i with x = y, if y satisfies p i , then y is a limit of points satisfying p i+1 . Since U := i<ω U i is G δ , it is open. Since x satisfies ♦ p 0 and U is a neighborhood of x, U contains some x 0 = x of some rank α 0 < ρ(x) satisfying p 0 . Inductively, for each i < ω, there is some x i ∈ U satisfying p i with x i = x, and, since x i ∈ U i , x i is a limit of points satisfying p i+1 ; in particular, there is one such point x i+1 in U , say of rank α i+1 < α i (so in particular x i+1 = x). This gives an infinite decreasing sequence of ordinals.
Recall that if κ is an ordinal of uncountable cofinality, then the intersection of countably many sets which are closed and cofinal in κ is also closed and cofinal in κ. Hence, Proposition 2.6 applies to the closed-unbounded topology from Blass [13]. More generally: Corollary 2.7 G L is not strongly complete with respect to topologies on ordinals given by countably complete filters, such as the closed-unbounded topology.
The spaces we will consider will instead be built around the Generalized Icard topologies.
These are called the generalized Icard topologies.

Arithmetic, I
Definition 2.9 We fix some notation related to ordinal arithmetic.

Whenever
A is a set of ordinals, we denote by α + A the set {α + β : β ∈ A}. Expressions such as −α + A are defined analogously, if they make sense. 3. For all nonzero ξ , there exist ordinals α and β such that ξ = α + ω β . Such a β is unique. We denote it by ξ and call it the end-logarithm of ξ . 4. For all nonzero ξ , there exists a unique ordinal η such that ξ can be written as ω η + γ , with γ < ξ. We denote this ordinal by Lξ and call it the initial logarithm of ξ .
The operations and L should be regarded as functions on (a sufficiently large subset of) Ord. Nonetheless, in its use and in general whenever we deem it convenient, we will omit the symbol '•' for function composition, as well as perhaps parentheses.
Our completeness proof will rely heavily upon an analysis of generalized Icard topologies and their structure induced by the arithmetical properties of ordinals. Hence, developing a thorough intuition about them will be crucial. A most useful remark in this direction is the fact that they are to arbitrary topological spaces as the usual order topology is to ordinal numbers. Indeed, define the initial segment topology I 0 on an ordinal (or on Ord) to be generated by all initial segments [0, α], for α < . Then ( , I 0 ) is a scattered space: a rather trivial scattered space-it carries no further information than the usual ordering on Ord. For instance, we have ρ I 0 α = α for all α and ht( , I 0 ) = .
Proof It is not hard to see that I 1 is the order topology, and that the rank function is is established by a simple induction. Finally, let H be the class of additively indecomposable ordinals. It follows that Since α → Lα is non-decreasing, this last supremum is equal to L( )+1, as claimed.
In what follows, we write simply I λ instead of I 0↑λ . These topologies are important because, as we will see, the completeness theorem can quickly be reduced to the case when the underlying space is an ordinal equipped with a topology of the form I λ .

d-maps and J-maps
There is an appropriate notion of structure-preserving mappings between scattered spaces. We say that a function between topological spaces is pointwise discrete if the preimage of any singleton is a discrete subspace. Clearly, any homeomorphism is a d-map. In particular, ordinal addition and substraction, i.e., functions of the form are d-maps. The rank function is also a d-map. A more interesting example is given by end-logarithms of the form: A proof of this, and the more general Lemma 2.21 below can be found in Fernández-Duque [16].
Since the composition of d-maps is a d-map, they can be thought of as morphisms in the category of scattered spaces. We will now state various properties of d-maps.
1. If Y is an ordinal with the initial segment topology, then f is the rank function on X .

For any
4. If f is surjective, then for any L 1 -formula ϕ, X | ϕ implies Y | ϕ.
Proof Items 1 and 2 appear in Beklemishev-Gabelaia [10]; item 4 appears in Bezhanishvili-Mines-Morandi [12] in the current formulation. Item 3 is proved in [3], but therein a different definition of τ ↑λ is used, and we still have not shown that they are equivalent. Nonetheless, the claim can be proved by an easy induction.
As mentioned in the proof of 2.12.3, Lemma 2.12.2 implies that d-maps are rankpreserving. Also, it follows from 2.12.3 that if the rank of (X , τ ) is , then is a d-map. The main feature of d-maps is as follows: Proof That completeness follows from the existence of d-maps is independently due to Abashidze [1] and Blass [13]. Note that it immediately follows from Proposition 2.2 and Lemma 2.12.4.
The converse is probably folklore and will not be needed below, but we prove it anyway. Suppose G L P 1 is complete with respect to (X , τ ), where τ is scattered. Let (T , <) be a finite, converse wellfounded tree. We define from T a modal formula ϕ consistent with G L P 1 . Let { p t : t ∈ T } be a set of distinct propositional variables and r be the root of T . Set Clearly, there is a Kripke model based on T where ϕ is true in r ; namely, any one where each p t holds only in t. Hence, ϕ is consistent with G L P 1 , whereby it is satisfiable in X . Fix a valuation over X and a point x r ∈ X such that x r | ϕ. Thus, x r satisfies p r and s,t∈T ; s =r ¬ p s and x r is a limit point of points satisfying each of p t , for t = r . Moreover, by each of the conjuncts above: 1. there is a punctured neighborhood of x r where each point satisfies p t for some t ∈ T ; 2. there is a punctured neighborhood of x r where no point satisfies p r ; 3. for each pair of distinct s, t ∈ T , there is a punctured neighborhood of x r of points satisfying at most one of p s and p t ; 4. for each pair of distinct s, t ∈ T with s < t, there is a punctured neighborhood of x r where all points satisfying p s are limits of points satisfying p t ; 5. for each pair of distinct s, t ∈ T with s < t, there is a punctured neighborhood of x r where all points satisfying p s are not limits of points satisfying p t ; and 6. there is a punctured neighborhood of x r where whenever a point x satisfies p t , then there is a punctured neighborhood of x where each points satisfies one of p s , with t < s.
Let S be the intersection of all those finitely many open neighborhoods of x r . Clearly, We claim f is a d-map. Let A t be an open subset of T of the form This clearly equals S if t = r . Otherwise, for each x ∈ S with x | p s and t ≤ s, there is an open neighborhood U of x where each point satisfies p u for some u > s.
and t < s. Then x is a limit of points satisfying p s , so that is the image of points satisfying p t and no point in S can satisfy p s ∧ ♦ p s for any s. Therefore, f is a d-map.
Hence, the need to check whether a given space X satisfies a formula is replaced by the definition of a suitable mapping between X and some other space which is known to do so. In practice, polymodal analogs of Lemma 2.13 do not even require us to use full d-maps, but rather a weaker form of embeddings, as shown by Beklemishev [7]: is called a J-frame if each relation is transitive and conversely wellfounded and it satisfies the following two conditions: (I) For all x, y ∈ W and all m < n: x < n y implies that for all z ∈ W : x < m z if, and only if y < m z. (J) For all x, y, z ∈ W and all m < n: if x < m y and y < n z, then x < m z.
We call a J-frame a J n -frame if all binary relations past the nth one are empty.
Let (T , < 0 , . . . , < N ) be a frame. Denote by E n the reflexive, symmetric, and transitive closure of n≤k<ω < k . The equivalence classes under E n are called n-planes. A natural order is defined on the set of (n + 1)-planes: We say that a J-frame is a J-tree if for all n, the (n + 1)-planes contained in each n-plane form a tree under < n and if whenever α < β for two (n + 1)-planes α, β, we have x < n y for all x ∈ α and y ∈ β. This means that each J n -tree can be thought of as a tree each of whose nodes is itself a J n−1 -tree. Below, a node t ∈ T is a hereditary k-root if for no j ≥ k and no s ∈ T do we have s < j t. We also write x k y if x < j y for some j ≥ k and Definition 2.15 (J-map) Let (T , σ 0 , . . . , σ n ) be a J n -tree and (X , τ 0 , . . . , τ n ) be a space with n + 1 topologies. We say that a function f : Lemma 2.16 (Beklemishev [7]).
Lemma 2.17 ). For each L n+1 -formula ϕ consistent with G L P, there exists a J n -tree T such that if X is a G L P n+1 -space and f : X → T is a surjective J n -map, then X | ¬ϕ, i.e., ϕ is satisfiable in X.
We call the tree obtained in Lemma 2.17 the canonical tree for ϕ.

If ξ and δ are nonzero, then
. From this follows that if γ < δ, then Finally, it is proved by induction that (γ δ) = Lγ + δ, so that if 1 < ξ, then as desired.
We now give an alternative characterization of topologies τ ↑λ and their rank functions: Lemma 2.20 Let (X , τ ) be a scattered space of rank .
Proof The second claim follows from Lemmas 2.12.1 and 2.12.3. We use this to prove the first claim by induction. Suppose τ ↑λ is generated by τ and all sets of the form for ξ < λ. By definition, τ ↑(λ+1) = (τ ↑λ ) ↑1 is generated by τ ↑λ and all sets of the form but ρ τ ↑λ = λ • ρ τ by induction hypothesis. So (τ ↑λ ) ↑1 is generated by all sets of the form for ξ < λ + 1. The limit case is immediate.
The following lemma provides the key relationship between arithmetic and topology for ordinals: are d-maps.
We will make use of the following two lemmata from [3]:

Lemma 2.22
Let (X , τ ) be a scattered space and λ be an ordinal. Any x in (X , τ ↑λ ) has a λ-neighborhood U such that whenever x = y ∈ U , λ ρ 0 y < λ ρ 0 x. Lemma 2.23 Let 1 < λ be an additively indecomposable ordinal and x ∈ X be such that ρ τ x = e λ > 0. Then for any τ ↑λ -neighborhood V of x, there exist • a set U ∈ τ , and • ordinals η < e λ and ζ < λ, For ranks not of the form e λ , we have a more general result, also from [3]: Then for any τ ↑λ -neighborhood V of x, there exist • a set U ∈ τ , and • a finite partial function r : λ → Or d such that letting We conclude this section with a final observation on logarithms:

Lemma 2.25
Suppose that λ is additively indecomposable, ζ is of the form e λ ζ 0 , and λ ξ < ζ 0 . Let Then η is a successor ordinal or zero.

#-polytopologies
In this section, we state our completeness theorem and prove it modulo the product lemma, which will be proved in the next section. Let us begin with some motivation by recalling the constructions from [10] and [16]. Let X = (X , τ ) be a scattered space; by [3], G L is complete with respect to each topology τ ↑λ , with 0 < λ, provided X is tall enough. Thus, one would attempt to prove completeness of G L P with respect to the polytopology However, this is not a G L P-space and thus does not validate the axioms of G L P. The idea is then to replace each topology τ ↑λ by a rank-preserving extension and prove completeness for that space. It is not known whether this is possible for arbitrary . What we will do here is, given a formula φ consistent with G L P, say, with occurrences of modalities λ 0 , . . . , λ n , and a (tall enough) scattered space X = (X , τ ), we produce a sequence of topologies τ 0 , . . . , τ n such that 1. (X , τ 0 , . . . , τ n ) satisfies φ, and 2. each τ i is a rank-preserving extension of τ ↑1+λ i . Definition 3.1 (ϑ-maximal topology) Let ϑ be a nonzero ordinal and (X , τ ) be a scattered topological space. We say that τ * is a ϑ-extension of τ if 1. τ ⊂ τ * , 2. ρ τ * = ρ τ , and 3. the identity function id : (X , τ ) → (X , τ * ) is continuous at all points x such that ϑ ρ τ (x) = 0.
We say that τ * is an ϑ-maximal topology if there are no proper ϑ-extensions of τ * .

Lemma 3.3 Let
Let us refer to the polytopologies considered in [10] and [16] as BG-polytopologies. We will not need that notion below, so we do not define them. ϑ-polytopologies are weak versions of BG-polytopologies. For example, suppose X is a {ω 1 }-polytopology over the interval topology on an ordinal. Then τ 1 is a rank-preserving extension of I ω 1 obtained just by adding sets that would be already included in the corresponding rank-preserving extension of I ω 1 in any corresponding BG-space of length ≥ ω 1 over I 1 . We now state the completeness theorem we shall prove: Theorem 3.4 Let ϑ be an increasing sequence of nonzero ordinals. Denote by G L P ϑ the fragment of G L P whose only modalities appear in ϑ. The result also holds also for successor ordinals by replacing e 1 with e 1+ ω. In fact, this general version is what we will prove; the smaller bound in the statement of the theorem follows from the fact that, for limit , Notice that in Theorem 3.4.3.4, we satisfy the consistent formula on a ϑpolytopology over a subspace (X , τ ↑1 ). We cannot in general replace this with (X , τ ↑0 )-consider an ordinal with the initial-segment topology. It is still useful to consider polytopologies of this sort. We will call ϑ-polytopologies over the initialsegment topology improper.
In the remainder of this article, we prove Theorem 3.4. Soundness follows from Lemma 3.6 below, which in turn follows from Lemma 3.5. The proofs are the same as in the case ϑ = 1 from [10]. They can also be found in [2]. Lemma 3.5 (X , τ ) is a ϑ-maximal space if, and only if, for all x ∈ X whose rank ρ is such that ϑ ρ > 0 and all V ∈ τ with V ⊂ [0, ρ) τ 0 , one of the following holds:

Lemma 3.6 Suppose (X , τ ) is ϑ-maximal and λ ≥ ϑ. Then {d
It follows from Lemma 3.6 that all ϑ-polytopologies are G L P n -spaces, which implies soundness.

is a d-map for each i ≤ n.
Proof This is essentially the same proof as for the case ϑ = 1 (see [10,Lemma 8.5]).
The key point is the following claim: To see this suffices, suppose the claim holds. Then, letting T 0 be any ϑ 1 -maximal extension of the topology given by T and f −1 S 0 , we obtain that (3.1) holds for i = 0. Inductively, suppose (3.1) holds for some i. By Lemma 2.12.3, is a d-map. If i + 1 = n, then we are done; otherwise, by definition, S i+1 is a ∂ϑ i+2maximal extension of S i ↑∂ϑ i+1 , whereby the claim yields that (3.1) holds for i + 1 if we set T i+1 to be some ∂ϑ i+2 -maximal extension of the topology given by T i ↑∂ϑ i+1 and f −1 S i+1 . Hence, it suffices to prove the claim. it is not hard to see that f : (X , R) → (Y , S ) is also a d-map. By definition, T ⊂ R, whence R is a rank-preserving extension of T . Let x ∈ X be such that ϑ ρ T x = 0. We need to show that id : (X , T ) → (X , R) is continuous at x. This follows from the fact that S is a ϑ-extension of S: for any R-neighborhood of x of the form

Proof of the claim
Note that we must have 0 < ϑ ρ. Without loss of generality, we may assume ρ is the least possible rank of a counterexample and W contains no other point of rank ≥ ρ, so that W = W 0 ∪ {x}, for some W 0 ∈ T with W 0 ⊂ [0, ρ) T 0 . We will arrive at a contradiction using Lemma 3.5: since f is rank-preserving, we have that ϑ ρ > 0, f (W 0 ) ∈ S , and f (W 0 ) ⊂ [0, ρ) S 0 . Hence, by Lemma 3.5, one of the following holds: It must be the second one that holds, for f is not S -open by hypothesis. Observe that f −1 U ∩ W is a T -neighborhood of x and thus contains points with rank of every ordinal up to, and including, ρ. Because W contains only one point of rank ρ and f is rank-preserving, It follows that the set contains points with rank of every ordinal up to, but not including, ρ. However, this is impossible by 3 above, because ρ is a limit ordinal. This finishes the proof of the claim and the lemma. Lemma 3.7 is still true in the degenerate case σ = I 0 . In this case, notice that I 0 is already ϑ-maximal for every ϑ, for there is only one point of each rank. The proof of Theorem 3.8 below is postponed to the next section. It is stated in a way that directly gives all information we will need when applying it below.
• the polytopology (R 0 , . . . , R n ), when restricted to X ↑ , is the one obtained from Lemma 3.7 by pulling back via π 1 ; • π −1 1 ({λ}) = { }. Theorem 3.8 is the main new ingredient of our proof. With it, we can adapt the usual proofs to obtain completeness. First, we need an embedding lemma: Lemma 3.9 Let (T , < 0 , < 1 , . . . , < n ) be a finite J n -tree with root r and ϑ be an increasing n-sequence of nonzero ordinals. Then, for any ς > 0, there exist • a ϑ-polytopology ([0, ], T 0 , . . . , T n ) over ([0, ], I ς ) such that < e ς+ϑ n ω; and • a surjective J n -map f : Proof The proof is by induction on n. The base case follows from [3, Theorem 6.11], so we assume that the result holds for all m < n and proceed by a subsidiary double induction on 1. ς , which we decompose as ς 0 + ω ρ ; and 2. hgt 0 (T ), the height of < 0 , in that order. Let hgt i (T ) be the height of < i . We need to consider various cases: Case I: ω ρ < ς. By the induction hypothesis (for ς ) applied to ω ρ , there are: e ω ρ +ϑ n ω; and • a surjective J n -map g :
Case III: ς is additively indecomposable and 0 < hgt 0 (T ). Let r 1 , . . . , r m be all < 0successors of r that are hereditary 1-roots and (following earlier notation) let 0 (r i ) denote the generated subtrees. Also let 1 (r ) denote the subtree consisting of all nodes that are < 0 -incomparable with r (i.e., the < 0 -roots). By induction hypothesis (applied to hgt 0 (T )), there exist ϑ-polytopologies over ([0, κ i ], I ς ) and surjective J n -maps be the topological sum. We also denote by over I 0 as in Case II in such a way that there is a J n−1 -map such that, letting f 0 = g • ς and Similarly, suppose ξ is any ordinal and let Since −ξ + ς : 0, e ς (ξ + λ) ∩ ξ, ∞ ς , I ς → [0, λ], I 0 is a d-map (it is the rank function), by applying Lemma 3.7 we obtain a polytopologŷ S 0 , . . . ,Ŝ n on 0, e ς (ξ + λ) ∩ ξ, ∞ ς such that is a d-map for each i. Arguing as in Case II, we see that is also a J n -map. We now fix for any α ≤ κ, and in particular when polytopology (R 0 , . . . , R n ), when restricted to X ↑ , is the one obtained from Lemma 3.7 by pulling back via π 1 ; Thus, when restricted to X ↑ , the topologies R i andŜ i coincide (directly by definition), so is a J n -map. We define a function given by: Since X ↑ and X ↓ are I ς+1 -clopen and 1 ≤ ∂ϑ 1 , it follows that X ↑ and X ↓ are R iclopen for all 0 < i. The facts that f ξ and f * are J n -maps and that the projection π 0 is a d-map yield condition ( j 1 ), as well as conditions ( j 2 )-( j 4 ) for 0 < i. We verify the remaining ones: for any 0 < i ≤ m, then there are ordinals ξ 0 , . . . , ξ m ∈ U such that π 0 (ξ i ) = κ i for each i. But then π 0 (U ∩ X ↓ ) contains a neighborhood U i of each κ i and by choice of f * , f * (U i ) = 0 (r i ). ( j 3 ) Any hereditary 1-root x is either r or in some T i . In the former case, f −1 ( 0 (r )) and f −1 ({r } ∪ 0 (r )) ∈ τ 0 equal [1, ) and [1, ], respectively. In the latter case, the result follows from the continuity of π 0 and the fact that f * is a J n -map.
is discrete and π 0 is pointwise discrete.
Therefore, f is a indeed a surjective J n -map. Since we have considered all cases, the lemma follows.
We can now finish the proof of Theorem 3.4. Let ≥ ϑ n and (X , τ ) be any scattered space of height ≥ e 1+ ω. Suppose ϕ only contains modalities in ϑ and is consistent with G L P ϑ. We need to show that ϕ is satisfied on a ϑ-polytopology over (X , τ ↑1 ).
Let U be the neighborhood of x consisting of all points of rank ≤ , so (U , τ ) is a scattered space of height + 1. By Lemma 3.7, there is a ϑ-polytopolgy is a d-map for each i ≤ n. By Lemmata 2.17 and 2.16, ϕ is then satisfiable in U. This completes the proof of Theorem 3.4.

Proof of the product lemma
For convenience, we restate the lemma:  polytopology (R 0 , . . . , R n ), when restricted to X ↑ , is the one obtained from Lemma 3.7 by pulling back via π 1 ; Let ς , κ, and λ be as in the statement. Since logarithms map initial segments of Ord to initial segments of Ord (by definition), ξ is an ordinal. In fact, it is a successor ordinal. To see this, suppose towards a contradiction that it were a limit ordinal. Thus, for each ζ < ξ, we have ζ ∈ ς [0, κ], so e ς ζ < κ. By continuity, e ς ξ ≤ κ, so ξ ∈ ς [0, κ]-a contradiction.
Thus ξ is a successor ordinal. Write ζ for its predecessor. We have for otherwise e ς ξ belongs to the interval [0, κ] and so ξ ∈ ς [0, κ], contrary to its definition. The case ς > 1 will require most of our efforts and will be considered first, over the next few sections. The case ς = 1 will be covered in Sect. 4.4.

The partition
Definition 4.2 As in the statement of the theorem, we define: Observe that if λ < e ϑ n ω and κ < e ς+ϑ n ω, then e ς (ξ + λ) < e ς+ϑ n ω, as desired. We also set: See the following picture: The purpose of this section is to partition X ↓ into smaller clopen cells. The idea is that the projection π 0 will be defined differently on different cells. We begin by defining a partition of the interval [0, e ς ξ) into smaller intervals [α ι , β ι ] and then we lift it to all of X ↓ . What α ι and β ι are, as well as the set over which ι ranges, will depend on various parameters. Condition (2.4) states that e ς (ζ + 1) = lim α→ς e α (e ς ζ + 1).
Let us look at the definition of π 0 a bit more closely. Consider a typical element α * of X ↓ and generate the sequence Let η be least such that α := η α * < e ς ξ . Then α belongs to some cell Let us suppose first that e ς ζ ≤ κ < e(e ς ζ + 1). Then, with the maximum being attained at the value e ι+2 κ ∈ [α ι , β ι ].
Proof Surjectivity we verified in the discussion preceding this lemma; we verify that it is a d-map. Consider cells of the form x < e ς ξ and e ς ξ ≤ η x and η+1 x ∈ X ι = x ∈ X ↓ : η+1 x ≤ e ς ξ and e ς ξ < η x and η+1 (The second equality follows from the definition of X ↓ .) Observe that [α ι , β ι ] η+1 is an I ς -clopen interval if η < ς, even when α ι is a limit ordinal; this is because α ι is always an isolated point in I ς by its definition. The definition of π 0 is the same within each X η ι and in each of those sets, π 0 is defined as a logarithm and is thus a d-map (recall that ς is additively indecomposable). Additionally, by Lemma 2.25, if x ∈ X ↓ , then the least η such that η x < e ς ξ is a successor ordinal or zero. If we additionally define X −1 ι := X ι , then the collection of all X η ι forms a clopen partition of X ↓ . Since π 0 is a d-map on each cell, it is a d-map on all of X ↓ . Lemma 4.10 π −1 0 α is I ς -dense in X ↑ for any α ≤ κ.
Since U was arbitrary, this finishes the proof in this case.
Since U was arbitrary, this finishes the proof in this case.
Observe that η < ς, for otherwise we would have contradicting the choice of β. Thus, if one writes out the normal form expansion of β, one obtains an expression of the form e β 0 β 1 + e β 2 . . . (β + nf(e ς ξ)) . . . , (4.5) where e ς ξ < β + e ς ξ and all the exponents to the left of β + nf(e ς ξ) add up to η. Consider the sequence {β(ι) : ι < e ς ξ }, where β(ι) is the ordinal one obtains if one substitutes ι for the rightmost occurrence of nf(e ς ξ) in the normal form expansion (4.5) of β. If U is a I ς -neighborhood of β, then, by Lemma 2.24, U contains a set of the form where r : ς → β + 1 is a finite partial function. It follows from this and from the continuity of exponentials that every such set U , if nonempty, contains cofinally many ordinals of the form β(ι). Since e ς ξ < β +e ς ξ and e ς ξ is additively indecomposable, we must have e ς ξ < β , so it follows that for each β(ι), An argument as in Case I or Case II (according as κ < e(e ς ζ + 1) or e(e ς ζ + 1) ≤ κ) shows that there is some ι < e ς ξ such that β(ι) ∈ U and π 0 β(ι) = α.

The polytopology
It remains to define a ϑ-polytopology (R 0 , . . . , R n ) on ([0, ], I ς ) such that the projection mappings is a d-map by Lemma 4.9, we may apply Lemma 3.7 to obtain a ϑ-polytopology (X ↓ ,T 0 , . . . ,T n ) over I ς such that is a d-map for each 0 ≤ i ≤ n. This ϑ-polytopology (T 0 , . . . ,T n ) is not, however, a topology on [0, ], so we need to extend it. For each i, letR i be the smallest topology on [0, ] extending I ς+ϑ i and containing all sets inT i . Since X ↑ is I ς+1 -clopen,R i is simply equal to I ς+ϑ i when restricted to X ↑ , for 0 < i. We are closer to our goal, but not done yet, since the space might not be a ϑ-polytopology, asR 0 might not be ϑ 1 -maximal around points in X ↑ .
The projection π 1 , which was defined as the function α → −ξ + ς α is the rank function of (X ↑ , I ς ) (viewed as a subspace of ([0, ], I ς )), so is a d-map. Having only one point of each rank, the space ([0, λ], I 0 ) has no proper rank-preserving extensions, and in particular is ϑ 1 -maximal. By the claim within the proof of Lemma 3.7, if (X ↑ , R) is any ϑ 1 -extension of (X ↑ , I ς ), then remains a d-map. Let ([0, ], R 0 ) be a ϑ 1 -maximal extension of ([0, ],R 0 ). Then, R 0 only adds neighborhoods around points of rank some ρ such that 0 < ϑ 1 ρ and, moreover, only neighborhoods around points in X ↑ , since (X ↓ ,T 0 ) was already ϑ 1maximal. Given a point x ∈ X ↑ , and recalling that ξ , the minimum I ς -rank of points in X ↑ , is a successor ordinal, we see that if, and only if, 0 < ϑ 1 ρ (X ↑ ,I ς ) x.
Thus, the space (X ↑ , R 0 ) is a ϑ 1 -extension of (X ↑ , I ς ). It follows that remains a d-map. We may now apply Lemma 3.7 to obtain a ϑ-polytopology (X ↑ , R 0 ,Ŝ 1 , . . . ,Ŝ n ) over (X ↑ , R 0 ) such that is a d-map for each 1 ≤ i ≤ n. For each 1 ≤ i ≤ n, we let R i be the disjoint union The sets X ↑ and X ↓ are R 0↑ϑ 1 -clopen and so it follows that is a ϑ-polytopology. Moreover, we have seen that is a d-map for each 0 ≤ i ≤ n and that is also a d-map for each 0 ≤ i ≤ n. The other conditions in the statement of the Product Lemma we have shown already, so its proof is complete.
If η is a limit ordinal, then, β is a limit of ordinals of the form ω ξ ·η, for any I 1neighborhood U of β, the previous argument applied to any sufficiently large enough η shows that U contains someα such that π 0α = α. This proves the lemma.