aa r X i v : . [ m a t h . L O ] F e b HIGHER INDESCRIBABILITY AND DERIVED TOPOLOGIES
BRENT CODY
Abstract.
We introduce reflection properties of cardinals in which the at-tributes that reflect are expressible by infinitary formulas whose lengths canbe strictly larger than the cardinal under consideration. This kind of general-ized reflection principle leads to the definitions of L κ + ,κ + -indescribability andΠ ξ -indescribability of a cardinal κ for all ξ < κ + . In this context, universal Π ξ formulas exist, there is a normal ideal associated to Π ξ -indescribability andthe notions of Π ξ -indescribability yield a strict hierarchy below a measurablecardinal. Additionally, given a regular cardinal µ , we introduce a diagonalversion of Cantor’s derivative operator and use it to extend Bagaria’s [Bag19]sequence h τ ξ : ξ < µ i of derived topologies on µ to h τ ξ : ξ < µ + i . Finally, weprove that for all ξ < µ + , if there is a stationary set of α < µ that have a highenough degree of indescribability, then there are stationarily-many α < µ thatare nonisolated points in the space ( µ, τ ξ +1 ). Contents
1. Introduction 12. Canonical reflection functions 73. Restricting Π ξ formulas and consistency of higher Π ξ -indescribability 144. Restricting L κ + ,κ + formulas and consistency of L κ + ,κ + -indescribability 205. Higher Π ξ -indescribability ideals 225.1. Universal Π ξ formulas and normal ideals 235.2. A hierarchy result 275.3. Higher Π ξ -clubs 336. Higher ξ -stationarity, ξ -s-stationarity and derived topologies 37References 441. Introduction
When working with certain large cardinals, set theorists often use reflectionarguments. For example, if κ is a measurable cardinal then it is inaccessible, andfurthermore, there are normal measure one many α < κ which are inaccessible; wesay that the inaccessibility of a measurable cardinal κ reflects below κ . In this articlewe consider generalizations of this kind of reflection so that we may reflect attributesof large cardinals that are expressible by formulas whose lengths can be strictlylonger than the large cardinal under consideration. We will see that in many cases, Date : February 22, 2021.2010
Mathematics Subject Classification.
Primary 03E55, 54A35; Secondary 03E05.
Key words and phrases.
Derived topology, diagonal Cantor derivative, indescribable cardinals,stationary reflection. if κ is a measurable cardinal and κ has some property, which is expressible by aformula ϕ whose length is less than κ + , then the set of α < κ such that a canonicallydefined restricted version of this formula ϕ | κα is true of α , is normal measure one.We use this kind of generalized reflection to define the L κ + ,κ + -indescribability andΠ ξ -indescribability of a cardinal κ for all ξ < κ + , thus generalizing the notionsof indescribability previously considered in [Bag19]. Let us note that a precursorto this type of reflection principle was studied by Sharpe and Welch (see [SW11,Definition 3.21]). We then use our notion of Π ξ -indescribability to establish thenondiscreteness of certain topological spaces which are generalizations of the derivedtopologies considered in [Bag19], and which are defined by using a diagonal versionof the Cantor derivative operator (see the definition of τ ξ and d ξ at the end of thecurrent section).We believe the results presented below will open up new avenues for future workin many directions. For example, in order to define the restriction of formulas(Definition 3.2 and Definition 4.1) and then to establish basic properties of Π ξ -indescribability, we introduce the canonical reflection functions (see Definition 1.1and Section 2), which are interesting in their own right and will likely have appli-cations in areas far removed from this paper. We also expect that the notion ofrestriction of formulas defined below, will have applications in the study of infini-tary logics and model theory. Note that [Kue77] and [Bar75] both contain resultsinvolving a notion of restriction of L ∞ ,ω formulas to countable sets; we suspect thatthese results, as well as other results in this area [Kue77], will have analogues involv-ing our notion of restriction. Furthermore, the notion of higher Π ξ -indescribabilityshould allow for a finer analysis of the large cardinal hierarchy as in [Cod]. Finally,the notions and results contained herein, particularly those on higher ξ -stationarityand higher derived topologies (see Section 6), should also allow for generalizationsof many results concerning iterated stationary reflection properties (see [Bag19],[BMS15] and [BMM20]).Before we discuss the restriction of formulas in general, let us give some examples.Recall that for cardinals κ and µ , we use L κ,µ to denote the infinitary logic whichallows for conjunctions of < κ -many formulas that together contain < κ -many freevariables and quantification (universal and existential) over < µ -many variables atonce. If κ is a measurable cardinal and ϕ is any sentence in the L κ,κ language ofset theory such that V κ | = ϕ , then the set of α < κ such that V α | = ϕ is normalmeasure one in κ . On the other hand, for any cardinal κ there are L κ + ,κ + sentenceswhich are true in V κ and false in V α for all α < κ . For example, for each η < κ there is a natural L κ + ,κ + formula χ η ( x ) such that for all α ≤ κ and all a ∈ V α wehave V α | = χ η ( a ) if and only if a is an ordinal and a has order type at least η . Now χ = V η<κ ∃ xχ η ( x ) is an L κ + ,κ + sentence such that V κ | = χ , and yet there is no α < κ such that V α | = χ . However, the restriction χ | κα := V η<α ∃ xχ η ( x ) of χ to α holds in V α for all α < κ . In what follows we will define the restriction of L κ + ,κ + formulas in generality, which will allow for similar reflection results. However, themain focus of this article is on a different kind of infinitary formula.Generalizing the notions of Π n and Σ n formulas (see [L´ev71] or [Kan03, Section0]), Bagaria [Bag19] defined the classes of Π ξ and Σ ξ formulas for all ordinals ξ .For example, if ξ is a limit ordinal, a formula is Π ξ if it is of the form V ζ<ξ ϕ ζ where each ϕ ζ is Π ζ . A formula is Σ ξ +1 if it is of the form ∃ Xψ where ψ is Π ξ .For more on this definition see Section 3. Given a cardinal κ , Bagaria defined a set IGHER INDESCRIBABILITY AND DERIVED TOPOLOGIES 3 S ⊆ κ to be Π ξ -indescribable in κ if and only if for all A ⊆ V κ and all Π ξ sentences ϕ , if ( V κ , ∈ , A ) | = ϕ then there is an α ∈ S such that ( V α , ∈ , A ∩ V α ) | = ϕ . Bagariapointed out that, using his definition, no cardinal κ can be Π κ -indescribable becausethe Π κ sentence χ defined above is true in V κ but false in V α for all α < κ . Weintroduce a modification of Bagaria’s notion of Π ξ -indescribability which allows fora cardinal κ to be Π ξ -indescribable for all ξ < κ + . Given a cardinal κ and anordinal ξ < κ + , we say that a set S ⊆ κ is Π ξ -indescribable in κ if and only if forall Π ξ sentences ϕ (with first and second-order parameters from V κ ), if V κ | = ϕ then there is some α ∈ S such that a canonically defined restriction of ϕ is true in V α , which we express by writing V α | = ϕ | κα (see Definition 3.4 for details).In order to define the notions of restriction of L κ + ,κ + formulas and restrictionof Π ξ formulas, we use a sequence of functions h F κξ : ξ < κ + i we call the sequenceof canonical reflection functions at κ , which is part of the set theoretic folkloreand which is closely related to the sequence h f κξ : ξ < κ + i of canonical functionsat κ . Before defining the canonical reflection functions, let us recall some basicproperties of canonical functions. Given a regular cardinal κ , the ordering definedon κ ORD by letting f < g if and only if { α < κ : f ( α ) < g ( α ) } contains a club,is a well-founded partial ordering. The Galvin-Hajnal [GH75] norm k f k of such afunction is defined to be the rank f in the relation < . For each ξ < κ + , there is a canonical function f κξ : κ → κ of norm ξ , in the sense that k f κξ k = ξ and whenever k h k = ξ the set { α < κ : f κξ ( α ) ≤ h ( α ) } contains a club (see [Jec10, Page 99]). Forconcreteness, we will use the following definition of f κξ for ξ < κ + . If ξ < κ we let f κξ : κ → κ be the function with constant value ξ and if ξ = κ we let f κξ = id be theidentity function. If κ ≤ ξ < κ + we fix a bijection b κ,ξ : κ → ξ and define f κξ byletting f κξ ( α ) = ot( b κ,ξ [ α ]) for all α < κ . It is easy to see that for all ζ < ξ < κ + we have f κζ < f κξ and that f κξ is a canonical function of norm ξ . The sequence ~f = h f κξ : ξ < κ + i is sometimes referred to as the sequence of canonical functionsat κ , although, as pointed out by Foreman [For10, Section 2.6], this terminologyis slightly misleading as the canonical functions are only well-defined modulo thenonstationary ideal. Definition 1.1.
Suppose κ is a regular cardinal and ~b = h b κ,ξ : κ ≤ ξ < κ + i isa sequence of bijections b κ,ξ : κ → ξ . We define the corresponding sequence of canonical reflection functions ~F = h F κξ : ξ < κ + i at κ where F κξ : κ → P κ κ + foreach ξ < κ + .(1) For ξ < κ we let F κξ ( α ) = ξ for all α < κ .(2) For κ ≤ ξ < κ + we let F κξ ( α ) = b κ,ξ [ α ] for all α < κ .It is not difficult to see that for κ ≤ ξ < κ + , the ξ th canonical reflection function F κξ is independent, modulo the nonstationary ideal, of which bijection b κ,ξ : κ → ξ is used in its definition. That is, if b κ,ξ : κ → ξ and b κ,ξ : κ → ξ are two bijectionsthen the set { α < κ : b κ,ξ [ α ] = b κ,ξ [ α ] } contains a club.In Section 2, we establish many basic structural properties of the canonicalreflection functions which will be used later in the paper. For example, if κ ≤ ξ < κ +1 Note that ϕ may involve finitely-many second-order parameters A , . . . , A n ⊆ V κ , and whenwe write V κ | = ϕ we mean ( V κ , ∈ , A , . . . , A n ) | = ϕ . Since this abbreviated notion will not causeconfusion and greatly simplifies notation, we will use it throughout the paper without furthercomment. BRENT CODY and ξ is a limit ordinal, then the function F κξ is canonical in the sense that whenever F : κ → P κ κ + is a function such that for all ζ < ξ the set { α < κ : F κζ ( α ) ( F ( α ) } contains a club, then the set { α < κ : F κξ ( α ) ⊆ F ( α ) } contains a club. However,this canonicity property does not hold when ξ < κ or if ξ < κ + is a successor ordinal(see Remark 2.4). A particularly useful application of canonical functions [For10,Proposition 2.34] is that that the ξ th canonical function at a regular cardinal κ represents the ordinal ξ in any generic ultrapower by any normal ideal on κ . Asshown in Section 2 below, whenever I is a normal ideal on κ , G ⊆ P ( κ ) /I is genericand j : V → V κ /G ⊆ V [ G ] is the corresponding generic ultrapower embedding,the ξ th canonical reflection function F κξ represents j ” ξ in the generic ultrapower,that is, j ( F κξ )( κ ) = j ” ξ . Also in Section 2, we prove that the sequence of canonicalreflection functions at κ satisfies a natural coherence property: for all limit ordinals ξ < κ + the set { α < κ : ( ∀ ζ ∈ F κξ ( α )) F κξ ( α ) ∩ ζ = F κζ ( α ) } is a closed unbounded subset of κ . In Section 2, we prove a highly useful technicalresult which, no matter how bijections are chosen, draws a connection betweencanonical reflection functions at κ and canonical reflection functions at regular α < κ : if κ is a regular cardinal then for all ξ < κ + there is a club C ⊆ κ such thatfor all regular α ∈ C there are club-many β < α such that π ξ,α [ F κξ ( β )] = F αf κξ ( α ) ( β ),where π ξ,α : F κξ ( α ) → f κξ ( α ) is the transitive collapse of F κξ ( α ).In Section 3, given a regular cardinal κ , we review the definitions of Π ξ andΣ ξ formulas over V κ ; when we say that ϕ is Π ξ over V κ we mean that ϕ is Π ξ inBagaria’s sense, but ϕ is also allowed to have any number of first-order parametersfrom V κ and finitely-many second-order parameters from V κ (see Definition 3.1).In Definition 3.2, we use canonical reflection functions to define the notion of re-striction of Π ξ and Σ ξ formulas by transfinite induction on ξ < κ + . For example, if ϕ = ϕ ( X , . . . , X m , A , . . . , A n ) is a Π ξ formula over V κ and ξ < κ , then we define ϕ | κα = ϕ ( X , . . . , X m , A ∩ V α , . . . , A n ∩ V α ) . As another example, suppose κ ≤ ξ < κ + and ξ is a limit ordinal. If ϕ = ^ ζ<ξ ϕ ζ is a Π ξ formula and α < κ , then we define ϕ | κα = ^ ζ ∈ F κξ ( α ) ϕ ζ | κα . Let us point out that this definition of ϕ | κα depends on which sequence of canonicalreflection functions is chosen. However, the definition of ϕ | κα is independent of thischoice modulo the nonstationary ideal (see Lemma 3.3).For a given cardinal κ and ordinal ξ < κ + , we say that S ⊆ κ is Π ξ -indescribable if and only if for all Π ξ sentences ϕ over V κ , whenever V κ | = ϕ there must be an IGHER INDESCRIBABILITY AND DERIVED TOPOLOGIES 5 α ∈ S such that V α | = ϕ | κα . Notice that since ϕ can have any number of first-orderparameters from V κ , one might be concerned that for some α < κ the formula ϕ | κα could contain parameters which are not in V α ∪ P ( V α ), in which case V α | = ϕ | κα does not make sense. However, Lemma 3.6 states that for any ξ < κ + , if ϕ is anyΠ ξ formula over V κ then the set of α < κ for which ϕ | κα is Π f κξ ( α ) over V α containsa club subset of κ . We also show (see Lemma 3.7) that even though the definitionof ϕ | κα depends on our choice of canonical reflection functions, our definition of Π ξ -indescribability does not. Our last result in Section 3 states that if κ is a measurablecardinal then κ is Π ξ -indescribable for all ξ < κ + , and furthermore, the set of α < κ such that α is Π ζ -indescribable for all ζ < α + is normal measure one in κ .In Section 4, given a regular cardinal κ and an L κ + ,κ + formula ϕ in the languageof set theory, we use the canonical reflection functions at κ to define a notion ofrestriction ϕ | κα by induction on subformulas, for all α ≤ κ . For a regular cardinal κ , we say that a set S ⊆ κ is L κ + ,κ + -indescribable if and only if for all L κ + ,κ + sentences in the language of set theory with V κ | = ϕ there is an α < κ such that V α | = ϕ | κα . Proposition 4.4 states that if κ is a measurable cardinal, then κ is L κ + ,κ + -indescribable and furthermore, the set of regular cardinals α < κ that are L α + ,α + -indescribable is normal measure one in κ .Generalizing the results of L´evy [L´ev71] and Bagaria [Bag19] on universal for-mulas, in Section 5.1, we show that given a regular cardinal κ , for all ξ < κ + thereexist universal Π ξ and Σ ξ formulas over V κ in the following sense. For example, asa consequence of Theorem 5.3, given ξ < κ + , there is a Π ξ formula Ψ κξ, ( X, Y ) over V κ , where X and Y are second-order variables, such that for all Π ξ formulas ϕ ( X )over V κ there is a code K ϕ ⊆ κ and a club C ⊆ κ such that for all A ⊆ κ and all α ∈ C ∪ { κ } we have V α | = ϕ ( A ) | κα if and only if V α | = Ψ κξ, ( A, K ϕ ) | κα . Using universal formulas, we prove Theorem 5.5, which states that if κ is Π ξ -indescribable where ξ < κ + , then the collectionΠ ξ ( κ ) = { X ⊆ κ : X is not Π ξ -indescribable } is a nontrivial normal ideal on κ .In Section 5.2, again using the existence of universal Π ξ formulas discussed above,we prove Theorem 5.8, which states that given a regular cardinal κ and ξ < κ + , theΠ ξ -indescribability of a set S ⊆ κ is expressible by a Π ξ +1 formula in the followingsense. There is a Π ξ +1 formula Φ κξ ( Z ) over V κ such that for all S ⊆ κ we have S is Π ξ -indescribable in κ if and only if V κ | = Φ κξ ( S )and there is a club C in κ such that for all regular α ∈ C we have S ∩ α is Π f κξ ( α ) -indescribable in α if and only if V α | = Φ κξ ( S ) | κα . We then prove two hierarchy results for Π ξ -indescribability. For example, as aconsequence of these results, if κ is κ + n + 1-indescribable, where n < ω , then Sharpe and Welch [SW11, Definition 3.21] extended the notion of Π n -indescribability of acardinal κ where n < ω to that of Π ξ -indescribability where ξ < κ + by demanding that theexistence of a winning strategy for a particular player in a certain finite game played at κ impliesthat the same player has a winning strategy in the analogous game played at some cardinal lessthan κ . The relationship between their notion and the one defined here is not known. BRENT CODY the set of α < κ which are α + n -indescribable is in the filter Π κ + n +1 ( κ ) ∗ . Moregenerally, our first hierarchy result, Corollary 5.9, states that if κ is Π ξ -indescribablewhere ξ < κ + and ζ < ξ , then the set of α < κ which are Π f κζ ( α ) -indescribable isin the filter Π ξ ( κ ) ∗ . Our second hierarchy result, Corollary 5.12, states that if κ isΠ ξ -indescribable where ξ < κ + , then for all ζ < ξ we have Π ζ ( κ ) ( Π ξ ( κ ). Theproofs of these two hierarchy results require several lemmas which are interesting intheir own right. For example, Lemma 5.6 states that if Ψ κξ, ( X, Y ) is a universal Π ξ formula at κ , which is defined by transfinite recursion on ξ in the proof of Theorem5.3, then the set of α < κ such that Ψ κξ, ( X, Y ) | κα is a universal Π f κξ ( α ) formula at α contains a club subset of κ . Moreover, we also require Proposition 5.7, which statesthat for any regular cardinal κ and ordinal ξ < κ + , if ϕ is any Π ξ or Σ ξ formulathen there is a club C ⊆ κ such that for all regular α ∈ C , the set of β < α forwhich ( ϕ | κα ) | αβ = ϕ | κβ is club in α .Recall that one can characterize Π n -indescribable subsets of a Π n -indescribablecardinal κ by using a natural base for the filter Π n ( κ ) ∗ dual to Π n ( κ ). For a regularcardinal κ , a set C ⊆ κ is a Π -club in κ if it is club in κ . We say that C ⊆ κ isΠ n +1 -club in κ , where n < ω , if it is Π n -indescribable in κ and whenever C ∩ α isΠ n -indescribable in α we have α ∈ C . Then, if κ is Π n -indescribable, a set S ⊆ κ is Π n -indescribable if and only if S ∩ C = ∅ for all Π n -clubs C ⊆ κ . This resultis due to Sun [Sun93] for n = 1 and to Hellsten [Hel03] for n < ω . In Section 5.3,we generalize this to Π ξ -indescribable subsets of Π ξ -indescribable cardinals for all ξ < κ + . That is, for all ξ < κ + , we introduce a notion of Π ξ -club subset of κ suchthat if κ is Π ξ -indescribable then a set S ⊆ κ is Π ξ -indescribable if and only if S ∩ C = ∅ for all Π ξ -clubs C ⊆ κ . For more results involving Π ξ -clubs, one shouldconsult [Cod19], [CS20], [CGLH] and [Cod20].Finally, in Section 6, we generalize some of the results of Bagaria [Bag19] onderived topologies on ordinals. Given a nonzero ordinal δ , Bagaria defined a trans-finite sequence of topologies h τ ξ : ξ ∈ ORD i on δ , called the derived topologies on δ , and proved—using the definitions of [Bag19]—that if there is an α < δ whichis Π ξ -indescribable then the τ ξ +1 topology on δ is non-discrete. However, usingthe definitions of [Bag19], α can be Π ξ -indescribable only if ξ < α . Thus, Bagariaobtained the non-discreteness of the τ ξ topologies on δ only for ξ < δ .Given a regular cardinal µ , we now present a natural extension of Bagaria’snotion of derived topologies on µ by defining a transfinite sequence of topologies h τ ξ : ξ ∈ ORD i on µ such that for ξ < µ our τ ξ is the same as that of [Bag19] andBagaria’s condition for the nondiscreteness of τ ξ +1 has a generalization, outlinedhere, to all ξ < µ + .We let τ be the interval topology on µ , that is, τ is the topology on µ generatedby the collection B of intervals contained in µ together with { } . Given τ ξ , where ξ < µ + , let d ξ : P ( µ ) → P ( µ ) be a diagonal version of Cantor’s derivative operatordefined by letting d ξ ( A ) = { α < µ : α is a limit point of A in the τ f µξ ( α ) topology } . IGHER INDESCRIBABILITY AND DERIVED TOPOLOGIES 7
Notice that if ξ < µ then, since f µξ is constantly equal to ξ , our definition of d ξ agrees with that of [Bag19]. We define τ ξ +1 to be the topology on µ generated by τ ξ ∪ { d ξ ( A ) : A ⊆ µ } . If ξ < µ + is a limit ordinal, we define τ ξ to be the topology generated by S ζ<ξ τ ζ ,which equals the topology generated by S ζ<ξ B ζ .For κ ≤ ξ < κ + , although the definition of d ξ ( A ) depends on our choice ofbijection b µ,ξ : µ → ξ used to compute f µξ , Lemma 6.1 states that our definition of τ ξ is independent of this choice.In Section 6, following [Bag19], we generalize the notions of ξ -stationarity, ξ -s-stationarity and ξ -reflecting cardinals and use these generalizations to characterizethe nonisolated points in ( µ, τ ξ ) for all ξ < µ + . Then we show that, using ourdefinition of Π ξ -indescribability (Definition 3.4), if µ has the property that { α < µ : α is f µξ ( α )-indescribable } is stationary in µ (for example, this will occur if µ is Π ξ +1 -indescribable) then thereis an α < µ which is nonisolated in the space ( µ, τ ξ +1 ).2. Canonical reflection functions
In this section we establish the basic properties of the canonical reflection func-tions at a regular cardinal. Although some of these results are folklore, we includeproofs for the reader’s convenience and because these arguments seem to be absentfrom the literature.We begin this section with an easy result which will be needed later, and whichshows that, for regular κ , the ξ th canonical reflection function F κξ (see Definition1.1) represents a useful object in any generic ultrapower obtained from a normalideal on κ . Proposition 2.1.
Suppose κ is a regular cardinal and I is a normal ideal on κ . Let G be generic for P ( κ ) /I , let U be the V -normal V -ultrafilter obtained from G andlet j : V → V κ /U be the corresponding generic ultrapower embedding. Then, forall ξ < κ + , the ξ th canonical reflection function F κξ represents j ” ξ in the genericultrapower, that is, j ( F κξ )( κ ) = j ” ξ .Proof. Let j : V → V κ /U be the generic ultrapower obtained from a generic filter G ⊆ P ( κ ) /I over V . Then the critical point of j is κ and κ equals the equivalenceclass of the identity function id : κ → κ . Thus, for all X ∈ P ( κ ) V we have X ∈ U if and only if κ ∈ j ( X ). Since crit( j ) = κ , it is easy to see that for ξ ≤ κ we have j ( F κξ )( κ ) = j ” ξ . Now suppose κ < ξ < κ + and let b κ,ξ : κ → ξ be the bijectionsuch that F κξ ( α ) = b κ,ξ [ α ] for all α < κ . By elementarity, j ( b κ,ξ ) : j ( κ ) → j ( ξ )is a bijection in M and j ( b κ,ξ )( α ) = j ( b κ,ξ ( α )) for all α < κ . Thus, j ( F κξ )( κ ) = j ( b κ,ξ )[ κ ] = j ” ξ . (cid:3) Corollary 2.2.
Suppose U is a normal measure on κ and j : V → M is thecorresponding ultrapower embedding. For all ξ < κ + , the ξ th canonical reflectionfunction F κξ represents j ” ξ in the ultrapower, that is, j ( F κξ )( κ ) = j ” ξ . Next we show that at least some of the canonical reflection functions at a regular κ are, in fact, canonical; in Remark 2.4, we show that this partial canonicity resultis the best possible. BRENT CODY
Lemma 2.3.
Suppose κ is regular. (1) For all ξ < κ + the set { α < κ : F κζ ( α ) ( F κξ ( α ) } contains a club for all ζ < ξ . (2) If ξ < κ + is a limit ordinal then the set { α < κ : F κξ ( α ) = [ ζ ∈ F κξ ( α ) F κζ ( α ) } is club in κ . (3) If κ ≤ ξ < κ + and ξ is a limit ordinal the function F κξ is canonical inthe sense that whenever F : κ → P κ κ + is a function such that for all ζ < ξ the set { α < κ : F κζ ( α ) ( F ( α ) } contains a club, then the set { α < κ : F κξ ( α ) ⊆ F ( α ) } contains a club.Proof. Let ~b = h b κ,ξ : ξ < κ + i be the sequence of bijections used to define ~f and ~F . For (1), clearly the result holds for ξ ≤ κ . Suppose κ < ξ < κ + and ζ < ξ .If ζ < κ then the set { α < κ : ζ ( F κξ ( α ) } = { α < κ : ζ ( b κ,ξ [ α ] } is a club.Suppose κ < ζ < ξ < κ + . We will show that the set C = { α < κ : F κζ ( α ) ( F κξ ( α ) } is unbounded and that there is a δ < κ such that C \ δ is closed in κ . To seethat C is unbounded, fix α < κ . Given that α n has been defined, choose α n +1 such that α n < α n +1 < κ , F κζ ( α n ) ⊆ F κξ ( α n +1 ) and F κξ ( α n +1 ) ∩ ( ξ \ ζ ) = ∅ . Let α ω = S n<ω α n and notice that F κζ ( α ω ) ( F κξ ( α ω ). Thus, C is unbounded. Nowlet δ < κ be such that F κξ ( δ ) ∩ ( ξ \ ζ ) = ∅ . Then it follows that C \ δ is closed in κ , because if h α η : η < γ i is a strictly increasing sequence of elements of C \ δ , and α = S η<γ α η , then α ∈ C because F κξ ( α ) ∩ ( ξ \ ζ ) = ∅ .Notice that it is easy to see that (2) holds when ξ ≤ κ is a limit ordinal. Nowsuppose κ < ξ < κ + and ξ is a limit ordinal. First we will prove that C = { α < κ : F κξ ( α ) ⊇ [ ζ ∈ F κξ ( α ) F κζ ( α ) } is a club subset of κ . It is easy to see that C is closed below κ because for all α < β < κ we have F κξ ( α ) = b κ,ξ [ α ] ⊆ b κ,ξ [ β ] = F κξ ( β ). So, it remains to prove that C is unbounded in κ . Fix α < κ . Assuming α n has been defined, choose α n +1 such that α n < α n +1 < κ and F κξ ( α n +1 ) ⊇ [ ζ ∈ b κ,ξ [ α n ] F κζ ( α n ) . (1)Let α ω = S n<ω α n . We will show that α ω ∈ C . Suppose η ∈ S ζ ∈ b κ,ξ [ α ω ] F κζ ( α ω ),then η ∈ S ζ ∈ b κ,ξ [ α m ] F κζ ( α ω ) for some m < ω . Thus, for some fixed ζ ∈ b κ,ξ [ α m ]we have η ∈ F κζ ( α ω ) = S n<ω F κζ ( α n ). Let k < ω be such that η ∈ F κζ ( α k ). Let M = max( m, k ) and notice that η ∈ F κζ ( α M ) and ζ ∈ b κ,ξ [ α M ]. By (1), we have η ∈ F κξ ( α M +1 ) = b κ,ξ [ α M +1 ] ⊆ b κ,ξ [ α ω ] = F κξ ( α ω ).Next, let us show that D = { α < κ : F κξ ( α ) ⊆ [ ζ ∈ b κ,ξ [ α ] F κζ ( α ) } is a club subset of κ . It is easy to see that D is closed below κ , so it only remainsto show that D is unbounded in κ . Fix α < κ . Assuming α n has been defined, wedefine α n +1 as follows. IGHER INDESCRIBABILITY AND DERIVED TOPOLOGIES 9 • First, suppose the set F κξ ( α n ) = b κ,ξ [ α n ] is bounded below the limit ordinal ξ , then there is some σ n < ξ such that σ n > κ and b κ,ξ [ α n ] ⊆ σ n . Choose β n < κ such that b σ n [ β n ] ⊇ b κ,ξ [ α n ], and then choose α n +1 < κ such that b κ,ξ [ α n +1 ] ⊇ b σ n [ β n ] ∪ { σ n } . • Next, suppose F κξ ( α n ) is unbounded in ξ . Notice that, cf( ξ ) < κ . Wefix a sequence ~ζ = h ζ η : η < cf( ξ ) i cofinal in ξ such that ζ η ∈ ξ \ κ forall η < cf( ξ ). For each η < cf( ξ ), since b ζ η : κ → ζ η is a bijection and b κ,ξ [ α n ] ∩ ζ η ∈ [ ζ η ] <κ we may choose β η < κ such that α n < β η and b ζ η [ β η ] ⊇ b κ,ξ [ α n ] ∩ ζ η . Then, for each η < cf( ξ ), choose δ η < κ such that β η < δ η and b κ,ξ [ δ η ] ⊇ b ζ η [ β η ] ∪ { ζ η } . Now, let α n +1 = S η< cf( ξ ) δ η .This defines the sequence h α n : n < ω i , which is strictly increasing. Let us showthat α ω = S n<ω α n ∈ D . Suppose γ ∈ F κξ ( α ω ) = b κ,ξ [ α ω ]. Then γ ∈ b κ,ξ [ α n ]for some n < ω . First, suppose F κξ ( α n ) = b κ,ξ [ α n ] is bounded below ξ . Then γ ∈ b κ,ξ [ α n ] ⊆ b σ n [ β n ] ⊆ F κσ n ( α ω ) where σ n ∈ b κ,ξ [ α n +1 ] ⊆ b κ,ξ [ α ω ]. Thus, γ ∈ F κσ n ( α ω ) ⊆ [ ζ ∈ b κ,ξ [ α ω ] F κζ ( α ω ) . Now, suppose that F κξ ( α n ) = b κ,ξ [ α n ] is unbounded in ξ . Choose an η < cf( ξ ) suchthat γ < ζ η , and notice that by our choice of β η we have γ ∈ b κ,ξ [ α n ] ∩ ζ η ⊆ b ζ η [ β η ].By our choice of δ η we have ζ η ∈ b κ,ξ [ δ η ] ⊆ b κ,ξ [ α ω ]. Thus γ ∈ F κζ η ( α ω ) = b ζ η [ α ω ]where ζ η ∈ b κ,ξ [ α ω ], and hence γ ∈ S ζ ∈ b κ,ξ [ α ω ] F κξ ( α ω ).For (3), suppose ξ = κ and for all ζ < κ the set C ζ = { α < κ : F κζ ( α ) ( F ( α ) } = { α < κ : ζ ( F ( α ) } contains a club. Then C = △ ζ<κ C ζ contains a club and if α ∈ C then for all ζ < α we have F κζ ( α ) = ζ ( F ( α ). Thus, for every limit ordinal α ∈ C we have α ⊆ F ( α ). This implies { α < κ : α ⊆ F ( α ) } = { α < κ : F κ ( α ) ⊆ F ( α ) } contains a club.Now suppose κ < ξ < κ + and ξ is a limit. Assume that for each ζ < ξ the set C ζ = { α < κ : F κζ ( α ) ( F ( α ) } contains a club. It follows that the set C = △ ζ<ξ C ζ = { α < κ : α ∈ \ β ∈ b κ,ξ [ α ] C β } contains a club. Furthermore, since ξ is a limit, the set D = { α < κ : F κξ ( α ) = [ ζ ∈ b κ,ξ [ α ] F κζ ( α ) } contains a club. Notice that if α ∈ C then for all ζ ∈ b κ,ξ [ α ] we have F κζ ( α ) ( F ( α ).Thus C ∩ D ⊆ { α < κ : F κξ ( α ) ⊆ F ( α ) } . (cid:3) Remark 2.4.
Let us point out that Lemma 2.3(3) does not hold if ξ < κ or if ξ < κ + and ξ is a successor ordinal. For example, if ξ = κ + 1, the assumption that { α < κ : F κκ ( α ) ( F ( α ) } = { α < κ : α ( F ( α ) } contains a club, does not implythat { α < κ : F κκ +1 ( α ) ⊆ F ( α ) } contains a club. To see this, note that C = { α <κ : F κκ +1 ( α ) = α ∪ { κ }} contains a club (by Lemma 2.8) and let F : κ → P κ κ + bethe function defined by F ( α ) = α ∪ { κ + 1 } . Then { α < κ : α ( F ( α ) } contains a Technically we should notate this sequence as ~ζ n = h ζ nη : η < cf( ξ ) i to indicate that it dependson n , but for notational simplicity, and since it will not lead to confusion, we omit this additionalsuperscript. club but C ∩ { α < κ : F κκ +1 ( α ) ⊆ F ( α ) } = ∅ , and hence { α < κ : F κκ +1 ( α ) ⊆ F ( α ) } is nonstationary in κ .The following lemma shows that the canonical reflection functions at a regularcardinal satisfy a natural kind of coherence property. Lemma 2.5.
Suppose κ is a regular cardinal and ξ < κ + is a limit ordinal. Thenthe set C = { α < κ : ( ∀ ζ ∈ F κξ ( α )) F κξ ( α ) ∩ ζ = F κζ ( α ) } is a club in κ .Proof. First, let us check that C is closed below κ . Suppose h α η : η < γ i is astrictly increasing sequence from C and let α = S η<γ α η . To show that α ∈ C ,fix ζ ∈ F κξ ( α ) = S η<γ F κξ ( α η ) and let η < γ be such that ζ ∈ F κξ ( α η ). Then forall η ∈ γ \ η we see that ζ ∈ F κξ ( α η ), and since α η ∈ C we have F κξ ( α η ) ∩ ζ = F κζ ( α η ). Since F κξ ( α ) = S η<γ F κξ ( α η ) and F κζ ( α ) = S η<γ F κζ ( α η ), this implies that F κξ ( α ) ∩ ζ = F κζ ( α ) and hence α ∈ C .Now let us show that C is unbounded in κ . For each ζ < ξ , let D ζ = { α < κ : F κζ ( α ) ⊆ F κξ ( α ) } and recall that D ζ is club in κ . Let △ ζ<ξ D ζ = { α < κ : α ∈ \ ζ ∈ b κ,ξ [ α ] D ζ } and D = △ ζ<ξ D ζ ! ∩ { α < κ : F κξ ( α ) = [ ζ ∈ F κξ ( α ) F κζ ( α ) } and notice that D is club by Lemma 2.3(2). Fix α < κ . Given α n , choose α n +1 ∈ D such that α n < α n +1 and for every ζ ∈ F κξ ( α n ) we have F κη ( α n ) ∩ ζ ⊆ F κζ ( α n +1 ) forall η ∈ F κξ ( α n ). We will show that α ω = S n<ω α n is in C . We must show that forevery ζ ∈ F κξ ( α ω ) we have F κξ ( α ω ) ∩ ζ = F κζ ( α ω ). Fix ζ ∈ F κξ ( α ω ) = S n<ω F κξ ( α n ).Since α ω ∈ D , it follows that α ω ∈ D ζ and thus, F κξ ( α ω ) ∩ ζ ⊇ F ζ ( α ω ). Nowsuppose β ∈ F κξ ( α ω ) ∩ ζ . Then β < ζ and β ∈ F κξ ( α m ) for some m < ω . Since ζ ∈ F κξ ( α ω ), we can fix a k < ω such that ζ ∈ F κξ ( α k ). Now let M = max( m, k ).Since α m ∈ D we have β ∈ F κξ ( α M ) ∩ ζ = [ η ∈ F κξ ( α M ) F κη ( α M ) ∩ ζ and hence β ∈ F η ( α M ) ∩ ζ for some η ∈ F κξ ( α M ). But we chose α M +1 suchthat F η ( α M ) ∩ ζ ⊆ F ζ ( α M +1 ). Therefore, β ∈ F ζ ( α M +1 ) ⊆ F ζ ( α ω ). Hence F κξ ( α ω ) ∩ ζ = F ζ ( α ω ), and since ζ ∈ F κξ ( α ω ) was arbitrary, we see that α ω ∈ C . (cid:3) Next we will show that for all limit ordinals ξ < κ + , for club many α < κ , thevalue of f κξ ( α ) is determined by the values of f κζ ( α ) for ζ ∈ F κξ ( α ). Lemma 2.6.
Suppose κ is regular and ξ < κ + is a limit ordinal. Then the set D = { α < κ : f κξ ( α ) = [ ζ ∈ F κξ ( α ) f κζ ( α ) } IGHER INDESCRIBABILITY AND DERIVED TOPOLOGIES 11 contains a club subset of κ .Proof. Let C = { α < κ : F κξ ( α ) = [ ζ ∈ F κξ ( α ) F κζ ( α ) } be the club subset of κ from Lemma 2.3(3) and let C = { α < κ : ( ∀ ζ ∈ F κξ ( α )) F κξ ( α ) ∩ ζ = F κζ ( α ) } be the club subset of κ from Lemma 2.5. Let us show that C ∩ C ⊆ D . Suppose α ∈ C ∩ C . Then we have F κξ ( α ) = [ ζ ∈ F κξ ( α ) F κζ ( α ) = [ ζ ∈ F κξ ( α ) ( F κξ ( α ) ∩ ζ )and thus f κξ ( α ) = ot( F κξ ( α )) = [ ζ ∈ F κξ ( α ) ot( F κξ ( α ) ∩ ζ ) = [ ζ ∈ F κξ ( α ) ot( F κζ ( α )) = [ ζ ∈ F κξ ( α ) f κζ ( α ) . (cid:3) The next two Lemmas confirm our intuition that for regular κ and ξ < κ + forclub-many α < κ the value f κξ ( α ) behaves like α ’s version of ξ . Lemma 2.7.
Suppose κ is regular and ξ < κ + is a limit ordinal. Then the set D = { α < κ : f κξ ( α ) is a limit ordinal } is a club subset of κ .Proof. It is easy to see that D is closed below κ . Let us show that D is unbounded.If cf( ξ ) < κ then we fix a strictly increasing sequence h ζ η : η < cf( ξ ) i cofinalin ξ and let α < κ be such that { ζ η : η < cf( ξ ) } ⊆ F κξ ( α ). It follows that f κξ ( α ) = ot( F κξ ( α )) is a limit ordinal for all α > α , so D is unbounded. On theother hand, suppose cf( ξ ) = κ and fix α < κ . Given that α n has been defined, let α n +1 < κ be such that α n < α n +1 and ∃ β ∈ F κξ ( α n +1 ) such that β > sup F κξ ( α n ).Then, if we let α ω = S n<ω α n , it follows that α ω ∈ D . (cid:3) Now let us consider the canonical reflection functions F κξ when ξ is a successorordinal. Lemma 2.8.
Suppose κ is regular. For all ζ < κ + the following sets are closedunbounded in κ . D = { α < κ : F κζ +1 ( α ) ∩ ζ = F κζ ( α ) } D = { α < κ : F κζ +1 ( α ) = F κζ ( α ) ∪ { ζ }} D = { α < κ : f κζ +1 ( α ) = f κζ ( α ) + 1 } Proof.
The fact that D is a club subset of κ follows directly from the fact that D is a club subset of κ , so we just need to consider D and D . It is easy to see thatthe result holds for ζ < κ . Suppose κ ≤ ζ < κ + . It is easy to see that D and D are closed below κ , so we only need to show that D and D are unbounded in κ .Let us show that D is unbounded in κ . Let C = { α < κ : F κζ ( α ) ⊆ F κζ +1 ( α ) } be the club from Lemma 2.3(1). Fix α < κ . Given that α n has been defined,choose α n +1 < κ such that α n < α n +1 , α n +1 ∈ C and F κζ +1 ( α n ) ∩ ζ ⊆ F κζ ( α n +1 ).Let α ω = S n<ω α n . Since α ω ∈ C we have F κζ +1 ( α ω ) ∩ ζ ⊇ F κζ ( α ω ). Furthermore, F κζ +1 ( ω ) ∩ ζ = ( S n<ω F κζ +1 ( α n )) ∩ ζ ⊆ F κζ ( α ω ). Hence α ω ∈ D and α ω > α .It only remains to show that D is unbounded in κ , but this is easy. Since b κ,ζ +1 : κ → ζ + 1 is a bijection, we may let δ < κ be such that ζ ∈ F κζ +1 ( δ ) = b κ,ζ +1 [ δ ].Then it follows that D \ δ ⊆ D . (cid:3) Next we prove a highly useful proposition which draws a connection between thecanonical reflection functions at a regular cardinal κ and the canonical reflectionfunctions at regular α < κ . This result will be applied in the proof of Proposition5.7. Proposition 2.9.
Suppose κ is regular and ξ < κ + . For each α < κ let π ξ,α : F κξ ( α ) → f κξ ( α ) be the transitive collapse of F κξ ( α ) . Then there is a club C κξ ⊆ κ such that for allregular uncountable α ∈ C κξ the set D αξ = { β < α : π ξ,α [ F κξ ( β )] = F αf κξ ( α ) ( β ) } is in the club filter on α .Proof. We proceed by induction. Suppose ξ < κ . Then for all α < κ , F κξ ( α ) = f κξ ( α ) = ξ and thus π ξ,α : ξ → ξ is the identity function. Thus, if α ∈ κ \ ξ thenfor any β < α we have π ξ,α [ F κξ ( β )] = ξ = F αξ ( β ) = F αf κξ ( α ) ( β ).Now suppose ξ = κ . Then for all α < κ we have F κκ ( α ) = f κκ ( α ) = α and thus π κ,α : α → α is the identity function. Thus, if α < κ is regular and uncountable,then for any β < α we have π κ,α [ F κκ ( β )] = π κ,α [ β ] = β = F αα ( β ) = F αf κκ ( α ) ( β ) . Suppose ξ = ζ + 1 is a successor with κ < ζ + 1 < κ + and the result holds for ζ .Recall that the set D = { α < κ : F κζ +1 ( α ) = F κζ ( α ) ∪ { ζ }} is in the club filter on κ . Let’s show that C κζ +1 = D ′ ∩ C κζ is the desired club,where C κζ is the club subset of κ obtained from the inductive hypothesis. Notice foreach α ∈ C κζ +1 we have f κζ +1 ( α ) = ot( F κζ +1 ( α )) = f κζ ( α ) + 1 and that the transitivecollapse π ζ +1 ,α : F κζ ( α ) ∪ { ζ } → f κζ ( α ) + 1 satisfies π ζ +1 ,α ↾ F κζ ( α ) = π ζ,α and π ζ +1 ,α ( ζ ) = f κζ ( α ). Suppose α ∈ C κζ +1 is a fixed regular and uncountable cardinal.By Lemma 2.8, E α = { β < α : F αf κζ ( α )+1 ( β ) = F αf κζ ( α ) ( β ) ∪ { f κζ ( α ) }} is in the club filter on α , and since α ∈ D ′ , the set E α ∩ D ∩ D αζ is in the club filteron α where D αζ = { β < α : π ζ,α [ F κζ ( β )] = F αf κζ ( α ) ( β ) } IGHER INDESCRIBABILITY AND DERIVED TOPOLOGIES 13 is obtained by the inductive hyothesis. We will show that E α ∩ D ∩ D αζ ⊆ D αζ +1 . If β ∈ E α ∩ D ∩ D αζ then we have π ζ +1 ,α [ F κζ +1 ( β )] = π ζ +1 ,α [ F κζ ( β ) ∪ { ζ } ]= π ζ +1 ,α [ F κζ ( β )] ∪ { f κζ ( α ) } = π ζ,α [ F κζ ( β )] ∪ { f κζ ( α ) } = F αf κζ ( α ) ( β ) ∪ { f κζ ( α ) } = F αf κζ ( α )+1 ( β )= F αf κζ +1 ( α ) ( β )Suppose ξ is a limit ordinal with κ ≤ ξ < κ + . It follows from Lemma 2.3,Lemma 2.5, Lemma 2.6 and Lemma 2.7 that there is a club D ⊆ κ such that forevery α ∈ D the following properties hold.(1) F κξ ( α ) = S ζ ∈ F κξ ( α ) F κζ ( α )(2) ( ∀ ζ ∈ F κξ ( α )) F κξ ( α ) ∩ ζ = F κζ ( α )(3) f κξ ( α ) = S ζ ∈ F κξ ( α ) f κζ ( α ) is a limit ordinal.(4) f κζ ( α ) < f κξ ( α ) for all ζ ∈ F κξ ( α ).First, notice that by (2), it follows that for all ζ ∈ F κξ ( α ) we have π ξ,α ↾ F κζ ( α ) = π ζ,α . By the inductive hypothesis, for each ζ < ξ there is a club C κζ in κ such thatfor all regular uncountable α ∈ C κζ the set D αζ = { β < α : π ζ,α [ F κζ ( β )] = F αf κζ ( α ) ( β ) } is in the club filter on α . Let us show that C κξ = D ′ ∩ △ ζ<ξ C κζ ! = D ′ ∩ { α < κ : α ∈ \ ζ ∈ F ξ ( α ) C κζ } is the desired club subset of κ where D ′ = { α < κ : D ∩ α is unbounded in α } .Suppose α ∈ C κξ is uncountable and regular.Now, for each ζ ∈ F κξ ( α ) we have α ∈ C κζ , and therefore, the set D αζ is in theclub filter on α . It follows by normality that the set E = △ ζ ∈ F ξ ( α ) D αζ = { β < α : β ∈ \ ζ ∈ F κξ ( β ) D αζ } is in the club filter on α . Since f κξ ( α ) < α + is a limit ordinal, it follows from Lemma2.5 that the set E = { β < α : ( ∀ ζ ∈ F αf κξ ( α ) ( β )) F αf κξ ( α ) ( β ) ∩ ζ = F αζ ( α ) } is in the club filter on α . Furthermore, by Lemma 2.3(1), for each ζ < f κξ ( α ), theset E ,ζ = { β < α : F αζ ( β ) ⊆ F αf κξ ( α ) ( β ) } is in the club filter on α , and thus E = △ ζ Suppose κ is regular and ξ < κ + . Then there is a club C κξ ⊆ κ such that for all regular uncountable α ∈ C κξ the set D αξ = { β < α : f κξ ( β ) = f αf κξ ( α ) ( β ) } is in the club filter on α . Restricting Π ξ formulas and consistency of higher Π ξ -indescribability We begin this section with a precise definition of Π ξ and Σ ξ formulas over V κ ,where κ is a regular cardinal and ξ is an ordinal. The following definition is similarto [Bag19, Definition 4.1], the only difference being that we allow for first andsecond order parameters from V κ . Definition 3.1. Suppose κ is a regular cardinal. We define the notion of Π ξ formula over V κ for all ordinals ξ as follows. IGHER INDESCRIBABILITY AND DERIVED TOPOLOGIES 15 (1) A formula ϕ is Π , or equivalently Σ , over V κ if it is a formula in thelanugage of set theory with free-variables of two types, namely first andsecond-order, with any number of first-order quantifiers, no second-orderquantifiers, and parameters from V κ ∪ P ( V κ ).(2) A formula ϕ is Π ξ +1 over V κ if it is of the form ∀ X · · · ∀ X m ψ ( X , . . . , X m )where ψ ( X , . . . , X m ) is Σ ξ over V κ . Similarly, ϕ is Σ ξ +1 over V κ if it isof the form ∃ X · · · ∃ X m ψ ( X , . . . , X m ) where ψ ( X , . . . , X m ) is Π ξ over V κ . Note that in both of the preceeding cases, ψ ( X , . . . , X m ) may haveadditional undisplayed first and second-order free variables and parameters.(3) When ξ is a limit ordinal, a formula ϕ , with finitely many second-order freevariables and finitely many second-order parameters, is Π ξ over V κ if it isof the form ^ ζ<ξ ϕ ζ where ϕ ζ is Π ζ over V κ for all ζ < ξ . Similarly, ϕ is Σ ξ if it is of the form _ ζ<ξ ϕ ζ where ϕ ζ is Σ ζ over V κ for all ζ < ξ . Definition 3.2. Suppose κ is a regular cardinal, ξ < κ + and let h F κζ : ζ < κ + i bea sequence of canonical reflection functions at κ . Given a formula ϕ ( X , . . . , X m , A , . . . , A n ) , which is either Π ξ or Σ ξ with all second-order variables X , . . . , X n and second-order parameters A , . . . , A n ⊆ V κ displayed, and an ordinal α < κ , we define the restriction of ϕ from κ to α , written as ϕ | κα , by using the fixed sequence of canonical reflection functions as follows. If ξ < κ we define ϕ | κα to be the formula obtained from ϕ by replacing each second-orderparameter A i with A i ∩ V α , that is, ϕ | κα = ϕ [ A / ( A ∩ V α ) , . . . , A n / ( A n ∩ V α )] . If ξ ≥ κ , then we define ϕ | κα in two cases as follows.(1) Suppose ξ = ζ + 1 is a successor ordinal, and ϕ = QX · · · QX k ψ ( X , . . . , X m , A , . . . , A n ) , where Q denotes either an existential or universal quantifier and ψ is eitherΠ ζ or Σ ζ repectively. Then we define ϕ | κα = QX · · · QX k ( ψ | κα ) . (2) If ξ is a limit ordinal and ϕ is a Π ξ formula, that is ϕ is of the form V ζ<ξ ϕ ζ where each ϕ ζ is Π ζ , then we define ϕ | κα = ^ ζ ∈ F ξ ( α ) ( ϕ ζ | κα ) . Similarly, if ϕ is Σ ξ , that is of the form W ζ<ξ ϕ ζ where each ϕ ζ is Σ ζ , thenwe define ϕ | κα = _ ζ ∈ F ξ ( α ) ( ϕ ζ | κα ) . Let us show that, modulo the nonstationary ideal on κ , the definition of ϕ | κα isindependent of which sequence of canonical reflection functions is used. Lemma 3.3. Suppose κ is a regular cardinal and suppose that ~F = h F κξ : ξ < α i and ~F = h ¯ F κξ : ξ < κ + i are two sequences of canonical reflection functions at κ .If a formula ϕ is Π ξ or Σ ξ over V κ where ξ < κ + and α < κ , let ϕ | κα denote therestriction of ϕ to α computed using ~F and let ϕ ¯ | κα denote the restriction of ϕ to α computed using ~F . For all ξ < κ + , if ϕ is Π ξ or Σ ξ over V κ then the set { α < κ : ϕ | κα = ϕ ¯ | κα } is in the club filter on κ .Proof. We proceed by induction on ξ < κ + . The result clearly holds for ξ < κ sincein that case, ϕ | κα = ϕ = ¯ ϕ | κα for all α < κ .Suppose ξ = ζ +1 < κ + is a successor ordinal, and the result holds for ζ . Suppose ϕ = ∃ X · · · X m ψ ( X , . . . , X m )where ψ ( X , . . . , X m ) is Σ ζ over V κ . Then, by our inductive hypothesis, there is aclub C in κ such that for all α ∈ C we have ψ | κα = ψ ¯ | κα and thus ϕ | κα = ϕ ¯ | κα .Suppose ξ < κ + is a limit ordinal and the result holds for ζ < ξ . Suppose ϕ = V ζ<ξ ϕ ζ where ϕ ζ is Π ζ over V κ . For each ζ < ξ , by our inductive hypothesis,there is a club C ζ in κ such that for all α ∈ C ζ we have ϕ ζ | κα = ϕ ζ ¯ | κα . Now let D = △ ζ<ξ C ζ = { α < κ : α ∈ \ ζ ∈ F κξ ( α ) C ζ } . By normality, it follows that D in the club filter on κ . It is easy to check that theset D = { α < κ : F κξ ( α ) = ¯ F κξ ( α ) } is in the club filter on κ . Thus C = D ∩ D is in the club filter on κ and if α ∈ C then ϕ | κα = ^ ζ ∈ F κξ ( α ) ( ϕ ζ | κα ) = ^ ζ ∈ ¯ F κξ ( α ) ( ϕ ζ ¯ | κα ) = ϕ ¯ | κα . (cid:3) Definition 3.4. Suppose κ is a cardinal and ξ < κ + . A set S ⊆ κ is Π ξ -indescribable if for every Π ξ sentence ϕ over V κ , if V κ | = ϕ then there is some α ∈ S such that V α | = ϕ | κα . In the remainder of this section we will show that the definition of Π ξ -indescribabilitygiven in Definition 3.4 is nontrivial. First, notice that the ranks of the first-orderparameters of the Π ξ sentence ϕ might be unbounded in κ , and so, one might beconcerned that ϕ | κα contains parameters that are not elements of V α ∪ P ( V α ) andhence V α | = ϕ | κα would be meaningless. However, as shown below in Lemma 3.6,the set of α < κ such that the parameters of ϕ | κα are elements of V α ∪ P ( V α ) is in theclub filter on κ . Second, notice that our definition of Π ξ -indescribability uses the IGHER INDESCRIBABILITY AND DERIVED TOPOLOGIES 17 restriction of formulas from Definition 3.2, and hence, it may seem to depend onwhich sequence of canonical reflection functions is used to compute ϕ | κα . However,Lemma 3.7 below states that our definition of Π ξ -indescribability is completely in-dependent of which canonical reflection functions are used to compute ϕ | κα . Third,in Proposition 3.8 below, we show our notion of indescribability is consistent byshowing that every measurable cardinal κ is Π ξ -indescribable for all ξ < κ + and, interms of consistency strength, the existence of a cardinal κ which is Π ξ -indescribablefor all ξ < κ + is strictly weaker than the existence of a measurable cardinal.We will need the following easy corollary of Lemma 2.5. Corollary 3.5. Suppose κ is a regular cardinal and ξ < κ + is a limit ordinal. Foreach α < κ , let π ξ,α : F κξ ( α ) → f κξ ( α ) denote the transitive collapse of F κξ ( α ) . Thenthe set C = { α < κ : ( ∀ ζ ∈ F κξ ( α )) f κζ ( α ) = π ξ,α ( ζ ) } is in the club filter on κ .Proof. Let D = { α < κ : ( ∀ ζ ∈ F κξ ( α )) F κξ ( α ) ∩ ζ = F κζ ( α ) } be the club subset of κ from Lemma 2.5. Let us show that D ⊆ C . Suppose α ∈ D and ζ ∈ F κξ ( α ). Since F κξ ( α ) ∩ ζ = F κζ ( α ) we see that f κζ ( α ) = ot( F κζ ( α )) =ot( F κξ ( α ) ∩ ζ ) and hence f κζ ( α ) = π ξ,α ( ζ ). (cid:3) Next we show that for regular κ and any Π ξ formula ϕ over V κ , where ξ < κ + ,there is a club of α < κ such that the parameters of ϕ | κα are in V α ∪ P ( V α ). Lemma 3.6. Suppose κ is a regular cardinal. For any ξ < κ + , if ϕ = ϕ ( X , . . . , X n ) is any Π ξ formula over V κ , then the set C ϕ = { α < κ : ϕ | κα is Π f κξ ( α ) over V α } contains a club. Similarly, if ϕ is any Σ ξ formula over V κ , then C ϕ = { α < κ : ϕ | κα is Σ f ξ ( α ) over V α } contains a club.Proof. Suppose ξ < κ . Suppose the first-order parameters for ϕ are h a γ : γ < | ξ |i and the second-order parameters of ϕ are h A γ : γ < | ξ |i . There is some fixed δ < κ such that all first-order parameters of ϕ are in V δ , and thus, if δ ≤ α < κ , then ϕ | κα = ϕ [ A γ /A γ ∩ V α : γ < | ξ | ]is Π ξ over V α .Suppose ξ is a limit ordinal with κ ≤ ξ < κ + and assume ϕ is a Π ξ formula over V κ . We note that an argument similar to that which follows works if ϕ is Σ ξ over V κ . We can write ϕ = V ζ<ξ ϕ ζ where each ϕ ζ is Π ζ over V κ . For each ζ < ξ , let C ϕ ζ be the club obtained from the induction hypothesis, let C = △ ζ<ξ C ϕ ζ = { α < κ : α ∈ \ ζ ∈ F κξ ( α ) C ϕ ζ } and note that C is club in κ . Let D = { α < κ : F κξ ( α ) ∩ κ = α } and note that D is club in κ . Let π ξ,α : F κξ ( α ) → f κξ ( α ) be the transitive collapseof F κξ ( α ). By Corollary 3.5, the set E = { α < κ : ( ∀ ζ ∈ F κξ ( α )) f κζ ( α ) = π ξ,α ( ζ ) } is in the club filter on κ . We will show that C ∩ D ∩ E ⊆ C ϕ . Suppose α ∈ C ∩ D ∩ E .Then we have ϕ | κα = ^ ζ ∈ F κξ ( α ) ( ϕ ζ | κα )= ^ ζ<α ϕ ζ ∧ ^ ζ ∈ F ξ ( α ) \ κ ( ϕ ζ | κα ) (since α ∈ D )= ^ ζ<α ϕ ζ ∧ ^ η ∈ f ξ ( α ) \ α ( ϕ π − ξ,α ( η ) | κα ) . It is easy to see that V ζ<α ϕ ζ is Π α over V α with parameters from α since α ∈ C ϕ ζ for all ζ ∈ F ξ ( α ) = b κ,ξ [ α ]. Now, suppose η ∈ f ξ ( α ) \ α . Since π − ξ,α ( η ) ∈ F κξ ( α ) and α ∈ C it follows that α ∈ C ϕ π − ξ,α ( η ) and hence ϕ π − ξ,α ( η ) | κα is Π f γαη ( α ) over V α withparameters from α . But since α ∈ E , we see that ϕ π − ξ,α ( η ) | κα is Π η over V α withparameters from α . This implies that ϕ | κα is Π f κξ ( α ) over V α with parameters from α . Now suppose ξ = ζ + 1 is a successor ordinal with κ < ξ < κ + . Then ϕ is of theform ϕ = ∀ Xψ ( X , . . . , X n , X )where ψ is Σ ζ . Let C ψ be the club obtained from the inductive hypothesis from ψ and let D = { α < κ : f κζ +1 ( α ) = f κζ ( α ) + 1 } be the club subset of κ from Lemma 2.8. For any α ∈ C ψ ∩ D we have ϕ | κα = ∀ X ( ψ | κα )where ψ | κα is Σ f κζ ( α ) over V α . Thus, ϕ is Σ f κζ ( α )+1 , but since α ∈ D we see that ϕ is Σ f κζ +1 ( α ) over V α . (cid:3) Now we show that the definition of Π ξ -indescribability given in Definition 3.4above is independent of which sequence of canonical reflection functions is used tocompute ϕ | κα . Lemma 3.7. Assume κ > ω is a regular cardinal. As in 3.3, let ~F = h F κξ : ξ < κ + i and ~F = h ¯ F κξ : ξ < κ + i be two sequence of canonical reflection functions at κ , andlet | κα and ¯ | κα be the notions of restriction defined using ~F and ~F respectively. Forall S ⊆ κ and all ξ < κ + , the following are equivalent. (1) For all Π ξ sentences ϕ over V κ , if V κ | = ϕ then there is an α ∈ S such that V α | = ϕ | κα . (2) For all Π ξ sentences ϕ over V κ , if V κ | = ϕ then there is an α ∈ S such that V α | = ϕ ¯ | κα .Proof. Suppose (1) holds. To show that (2) holds, fix a Π ξ sentence ϕ over V κ suchthat V κ | = ϕ . By Lemma 3.3, there is a club C in κ such that for all α ∈ C we IGHER INDESCRIBABILITY AND DERIVED TOPOLOGIES 19 have ϕ | κα = ϕ ¯ | κα . Let ψ ( C ) be the usual Π sentence over V κ expressing that C is aclosed unbounded subset of κ . Then V κ | = ϕ ∧ ψ ( C ). By (1), there is an α ∈ S suchthat V α | = ( ϕ ∧ ψ ( C )). Since ( ϕ ∧ ψ ( C )) | κα = ϕ | κα ∧ ψ ( C ∩ α ), we see that α ∈ S ∩ C .Hence V α | = ϕ ¯ | κα . A similar argument shows that (2) implies (1). (cid:3) Finally, we show that the notion of Π ξ -indescribability given in Definition 3.4above is strictly weaker than a measurable cardinal in consistency strength. Proposition 3.8. Suppose U is a normal measure on a measurable cardinal κ .Then κ is Π ξ -indescribable for all ξ < κ + and the set X = { α < κ : α is Π ξ -indescribable for all ξ < α + } is in U .Proof. Let j : V → M be the usual ultrapower embedding obtained from U where M is transitive and j has critical point κ . Notice that if follows directly fromLemma 3.6 that if ϕ is any Π ξ formula over V κ then, in M , j ( ϕ ) | j ( κ ) κ is Π ξ over V κ (because κ ∈ j ( C ϕ )). We will show that even more is true: let us argue byinduction that for any ξ < κ + , if ϕ is a Π ξ or Σ ξ formula over V κ then j ( ϕ ) | j ( κ ) κ = ϕ. (2)Suppose ϕ is a Π ξ formula over V κ , let the first-order parameters of ϕ be h a γ : γ < | ξ |i and let the second-order parameters of ϕ be h A γ : γ < | ξ |i . If ξ < κ then j ( ϕ ) = ϕ [ A γ /j ( A γ ) : γ < | ξ | ] and hence j ( ϕ ) | j ( κ ) κ = ϕ .Now suppose ξ is a limit ordinal with κ ≤ ξ < κ + and that the result holds for ζ < ξ . Let ϕ be a Π ξ sentence over V κ . Then ϕ = V ζ<ξ ϕ ζ where each ϕ ζ is Π ζ over V κ . By elementarity, j ( ϕ ) is a Π j ( ξ ) sentence over V j ( κ ) with parameter j ( A ) in M . In fact, j ( ϕ ) is of the form V ζ Suppose κ is a regular cardinal and α < κ is a cardinal. We define ϕ | κα for all formulas ϕ of L κ + ,κ + by induction on complexity of ϕ .(1) If ϕ is a term equation t = t or a relational formula of the form R ( t , . . . , t k )we define ϕ | κα to be ϕ .(2) If ϕ is of the form ¬ ψ where ψ | κα has already been defined, we let ϕ | κα bethe formula ¬ ( ψ | κα ).(3) If ϕ is of the form V ζ<ξ ϕ ζ where ξ < κ + and ϕ ζ | κα has been defined for all ζ < ξ , then we define ϕ | κα to be V ζ ∈ F κξ ( α ) ( ϕ ζ | κα ). (4) If ϕ is of the form ∃h x ζ : ζ < ξ i ψ where ξ < κ + and ψ | κα has already beendefined, we let ϕ | κα be the formula ∃h x ζ : ζ ∈ F κξ ( α ) i ( ψ | κα ). As for the notion of restriction of Π ξ formulas considered in Section 3 above, onecan show that the notion of restriction of L κ + ,κ + formulas is well-defined modulo thenonstationary ideal. That is, if ~F = h F κξ : ξ < κ + i and ~F = h F κξ : ξ < κ + i are twosequence of canonical reflection functions at κ , then, by induction on subformulas,one can show that for all L κ + ,κ + formulas ϕ , there is a club of α < κ such that ϕ | κα = ϕ ¯ | κα where ϕ | κα is the notion of restriction computed using ~F and ϕ | κα is Notice that when ξ < κ we have F κξ ( α ) = ξ , and thus ϕ | κα = V ζ<ξ ( ϕ ζ | κα ). Notice that it is possible that some of the bound variables of ϕ could become free variablesof ϕ | κα , but this will not matter in the cases we are interested in. IGHER INDESCRIBABILITY AND DERIVED TOPOLOGIES 21 the notion of restriction computed using ~F . Furthermore, also using induction onsubformulas, one can show that for all formulas ϕ in L κ + ,κ + , there is a club subsetof κ , C such that for all regular α ∈ C the formula ϕ | κα is in L α + ,α + . Finally, onecan show that the following definition of L κ + ,κ + -indescribability is not dependenton which sequence of canonical reflection functions is used to compute restrictionsof L κ + ,κ + formulas. Definition 4.2. Suppose κ is a regular cardinal. A set S ⊆ κ is L κ + ,κ + -indescribable if for all sentences ϕ of L κ + ,κ + , if V κ | = ϕ then there is some α < κ such that V α | = ϕ | κα .Let us show that the existence of a cardinal κ which is L κ + ,κ + -indescribable isstrictly weaker than the existence of a measurable cardinal. First we need a lemma. Lemma 4.3. Suppose j : V → M is the ultrapower by a normal measure U on κ .Then for all formulas ϕ ( x η : η < ν ) of L κ + ,κ + and all matching tuples ( c η : η < ν ) we have j ( ϕ [ c η : η < ν ]) | j ( κ ) κ = ϕ [ c η : η < ν ] . Proof. When ϕ is either a term equation or a relational formula ν is finite and thus ϕ is first order, so the result holds. If the result holds for ψ and ϕ is of the form ¬ ψ , then clearly it holds for ϕ too. Now suppose ϕ ( x η : η < ν ) is of the form ^ ζ<ξ ϕ ζ and ( c η : η < ν ) is a sequence of parameters from V κ . For each ζ < ξ , we let η ζ = ( η i : i < δ ζ ) be a sequence of ordinals less than ν such that the sequence x ζ = ( x η i : i < δ ζ ) consists of the free variables of ϕ ζ and we let c ζ = ( c η i : i < δ ζ )be the corresponding sequence of parameters. It follows that ϕ [ c η : η < ν ] = ^ ζ<ξ ( ϕ ζ [ c η i : i < δ ζ ]) . Define sequences ~ϕ = ( ϕ ζ : ζ < ξ ), ~η = ( η ζ : ζ < ξ ), ~x = ( x η : η < ν ), ~c = ( c η : η < ν ) and ~δ = ( δ ζ : ζ < ξ ). Let j ( ~ϕ ) = ( ¯ ϕ ζ : ζ < j ( ξ )), j ( ~η ) = (¯ η ζ : ζ < j ( ξ ), j ( ~x ) = (¯ x η : η < j ( ν )), j ( ~c ) = (¯ c η : η < j ( ν )) and j ( ~δ ) = (¯ δ ζ : ζ < j ( ξ )). It followsthat ¯ η ζ = (¯ η i : i < ¯ δ ζ )), and, by elementarity, for each ζ < j ( ξ ) the free variablesfor ¯ ϕ ζ are (¯ x ¯ η i : i < ¯ δ ζ )). Now we have j ( ϕ ) = V ζ Proposition 4.4. Suppose U is a normal measure on a measurable cardinal κ .Then κ is L κ + ,κ + -indescribable and the set { α < κ : α is L α + ,α + -indescribable } is in U . Higher Π ξ -indescribability ideals In this section, given a regular cardinal κ , we prove the existence of universalΠ ξ formulas for all ξ < κ + . We then use universal formulas to show that Π ξ -indescribability is, in a sense, expressible by a Π ξ +1 formula. This leads to severalhierarchy results and a characterization of Π ξ -indescribability in terms of the nat-ural filter base consisting of the Π ξ -club subsets of κ . IGHER INDESCRIBABILITY AND DERIVED TOPOLOGIES 23 Universal Π ξ formulas and normal ideals. For a regular cardinal κ > ω ,we define the notion of universal Π ξ formula, where ξ < κ + , as follows. Definition 5.1. Suppose κ is a regular cardinal, ξ is an ordinal with κ ≤ ξ <κ + and n < ω . We say that a Π ξ formula Ψ( X , . . . , X n , Y ξ ) over V κ , where X , . . . , X n , Y ξ are second-order variables, is a universal Π ξ formula at κ for for-mulas with n free second order variables if for all Π ξ formulas ϕ ( X , . . . , X n ), withall free variables displayed, there is a K ϕ ⊆ κ and there is a club C ⊆ κ such thatfor all A , . . . , A n ⊆ κ and all α ∈ C ∪ { κ } we have V α | = ϕ ( A , . . . , A n ) | κα if and only if V α | = Ψ( A , . . . , A n , K ϕ ) | κα . When n = 0, the intended meaning is that ϕ is a Π ξ sentence and Ψ ξ, ( Y ) hasone free-variable. The notion of universal Σ ξ formula at κ for formulas with n freesecond-order variables is defined similarly.We will use the following lemma to prove that universal Π ξ formulas exist atregular κ where κ ≤ ξ < κ + . Lemma 5.2. Suppose κ is regular and ≤ ζ < κ + . Suppose ψ ζ ( W , . . . , W n , X, Z ) is any Π ζ formula over V κ , ψ ′ ζ ( W , . . . , W n , X, Z ) is any Σ ζ formula over V κ and ϕ ( Y, Z ) is any Π formula over V κ where all free second-order variables are dis-played. Then the formula ∀ Y ∀ W · · · ∀ W n ( ϕ ( Y, Z ) ∨ ψ ζ ( W , . . . , W n , X, Y )) is equivalent to a Π ζ formula over V κ and ∃ Y ∃ W · · · ∃ W n ( ϕ ( Y, Z ) ∧ ψ ′ ζ ( W , . . . , W n , X, Y )) is equivalent to a Σ ζ formula over V κ .Proof. We provide a proof the case in which the formulas are Π ζ , the other case,where the formulas are Σ ζ is similar. We proceed by induction on ζ . If ζ =1 then ψ ( W , . . . , W n , X, Y ) is of the form ∀ W ψ ( W , . . . , W n , W, X, Y ) where ψ ( W , . . . , W n , W, X, Y ) is Π over V κ and we see that ∀ Y ∀ W · · · W n ( ϕ ( Y, Z ) ∨ ψ ( W , . . . , W n , X, Y ))is equivalent over V κ to the Π formula ∀ Y ∀ W · · · ∀ W n ∀ W ( ϕ ( Y, Z ) ∨ ψ ( W , . . . , W n , W, X, Y )) . If ζ = η + 1 < κ + is a successor ordinal, then ψ η +1 ( W , . . . , W n , X, Z ) is of theform ∀ W ψ η ( W , . . . , W n , W, X, Z ) where ψ η is Σ η over V κ and thus the formula ∀ Y ∀ W · · · ∀ W n ( ϕ ( Y, Z ) ∨ ψ η +1 ( W , . . . , W n , X, Y ))is equivalent over V κ to ∀ Y ∀ W · · · ∀ W n ∀ W ( ϕ ( Y, Z ) ∨ ψ η ( W , . . . , W n W, X, Y )) , which is Π η +1 over V κ by the inductive hypothesis.If ζ < κ + is a limit ordinal, then ψ ζ ( W , . . . , W n , X, Z ) = ^ η<ζ ψ η ( W , . . . , W n , X, Z ) where ψ η is Π η over V κ for all η < ζ . In this case, the formula ∀ Y ∀ W · · · ∀ W n ( ϕ ( Y, Z ) ∨ ψ ζ ( W , . . . , W n , X, Y ))is equivalent over V κ to ^ η<ζ ∀ Y ∀ W · · · ∀ W n ( ϕ ( Y, Z ) ∨ ψ η ( W , . . . , W n , X, Y )) , which is Π ζ over V κ by the inductive hypothesis. (cid:3) The following proposition generalizes results of L´evy [L´ev71] and Bagaria [Bag19];Levy proved the case in which ξ < ω and Bagaria proved the case in which ξ < κ . Theorem 5.3. Suppose κ > ω is a regular cardinal and ξ is an ordinal with ξ <κ + . For each n < ω there is a universal Π ξ formula Ψ κξ,n ( X , . . . , X n , Y ξ ) and auniversal Σ ξ formula ¯Ψ κξ,n ( X , . . . , X n , Y ξ ) at κ for formulas with n free second-order variables.Proof. The case in which ξ < κ follows directly from the proof of [Bag19, Propo-sition 4.4]. Suppose ξ = ζ + 1 is a successor ordinal with κ < ζ + 1 < κ + andthe result holds for all η ≤ ζ . Let us show that there is a universal Π ζ +1 formulaat κ for formulas with n free second-order variables; a similar argument works forΣ ζ +1 formulas, which we leave to the reader. Let ¯Ψ ζ,n +1 ( X , . . . , X n , X n +1 , Y ξ )be a universal Σ ζ formula at κ for formulas with n + 1 free second-order variablesobtained from the induction hypothesis. We will show thatΨ ζ +1 ,n ( X , . . . , X n , Y ξ ) = ∀ W ¯Ψ ζ,n +1 ( X , . . . , X n , W, Y ξ )is the desired formula. Suppose ϕ ( X , . . . , X n ) = ∀ W ϕ ζ ( X , . . . , X n , W ) is anyΠ ζ +1 formula with n free second-order variables, where ϕ ζ is Σ ζ with n + 1 freesecond-order variables. Let C ϕ ζ and K ϕ ζ be as obtained from the inductive hy-pothesis. Fix A , . . . , A n . Then for all α ∈ C ϕ ζ ∪ { κ } we have V α | = ϕ ( A , . . . , A n ) | κα ⇐⇒ ( ∀ W ⊆ V α ) V α | = ϕ ζ ( A , . . . , A n , W ) | κα ⇐⇒ ( ∀ W ⊆ V α ) V α | = ¯Ψ ζ,n +1 ( A , . . . , A n , W, K ϕ ζ ) | κα ⇐⇒ V α | = ∀ W ¯Ψ ζ,n +1 ( A , . . . , A n , W, K ϕ ζ ) | κα ⇐⇒ V α | = Ψ ζ +1 ,n ( A , . . . , A n , K ϕ ζ ) | κα , which establishes the successor case of the induction.Suppose ξ is a limit ordinal and the result holds for all ζ < ξ . We will show thatthere is a universal Π ξ formula at κ for formulas with 1 free second-order variable;the proof for n free second-order variables is essentially the same but one mustreplace the single variable X with a tuple X , . . . , X n in the appropriate places.We let Γ : κ × κ → κ be the usual definable pairing function and for A ⊆ κ and η < κ we let ( A ) η = { β < κ : Γ( η, β ) ∈ A } be the “ η th slice” of A . Using the inductive hypothesis, for each ζ < ξ we let Ψ κζ, bea universal Π ζ formula at κ for formulas with n free variables and we let Θ ζ, ( X, Y ξ ) Note that if we had here a block of quantifiers ∀ W · · · ∀ W k , they could be collapsed to asingle one by modifying ϕ ζ without changing the fact that ϕ ζ is Σ ζ . IGHER INDESCRIBABILITY AND DERIVED TOPOLOGIES 25 be the following formula which is equivalent to Ψ κζ, ( X, ( Y ξ ) b − κ,ξ ( ζ ) ), that is,Θ ζ, ( X, Y ξ ) = ∀ Y ( ∀ β ( β ∈ Y ↔ Γ( b − κ,ξ ( ζ ) , β ) ∈ Y ξ ) → Ψ ζ, ( X, Y )) . By Lemma 5.2, Θ ζ, ( X, Y ξ ) is Π ζ over V κ . We letΨ ξ, ( X, Y ξ ) = ^ ζ<ξ Θ ζ, ( X, Y ξ ) . and note that Ψ ξ, ( X, Y ξ ) is Π ξ . Also notice that the first-order parameters ofΨ ξ, ( X, Y ξ ) are in the set { b − κ,ξ ( ζ ) : ζ < ξ } = κ .Suppose ϕ ( X ) = V ζ<ξ ϕ ζ ( X ) is any Π ξ formula over V κ with one free second-order variable, where each ϕ ζ is Π ζ over V κ . Let K ϕ = { Γ( b − κ,ξ ( ζ ) , β ) : ζ < ξ ∧ β ∈ K ϕ ζ } code the sequence h K ϕ ζ : ζ < ξ i where K ϕ ζ = k ϕ ζ is obtained from [Bag19,Proposition 4.4] for ζ < κ and from the inductive hypothesis for κ ≤ ζ < κ + . Fix A ⊆ V κ . Notice that it follows easily from the definitions of K ϕ and Ψ ξ, ( X, Y ξ )that V κ | = ϕ ( A ) if and only if V κ | = Ψ ξ, ( A, K ϕ ).Let h C ϕ ζ : ζ < ξ i be the clubs obtained from the induction hypothesis. Bynormality, the set C = △ ζ<ξ C ϕ ζ = { α < κ : α ∈ \ ζ ∈ F κξ ( α ) C ϕ ζ } is in the club filter on κ . Choose α ∈ C . We have ϕ ( A ) | κα = ^ ζ ∈ F κξ ( α ) ( ϕ ζ ( A ) | κα )and Ψ ξ, ( A, K ϕ ) | κα = ^ ζ ∈ F ξ ( α ) (Θ ζ, ( A, K ϕ ) | κα ) . Since α ∈ C , it follows that for every ζ ∈ F κξ ( α ) we have V α | = ϕ ζ ( A ) | κα ⇐⇒ V α | = Ψ ζ, ( A, K ϕ ζ ) | κα ⇐⇒ V α | = Ψ ζ, ( A, ( K ϕ ) b − κ,ξ ( ζ ) ) | κα ⇐⇒ V α | = Θ ζ, ( A, K ϕ ) | κα . Therefore, V α | = ϕ ( A ) | κα ⇐⇒ V α | = Ψ ξ, ( A, K ϕ ) | κα . (cid:3) Remark 5.4. Notice that the proof of Theorem 5.3 is essentially a construction viatransfinite recursion of a universal Π ξ formula at κ for formulas with n free second-order variables. Furthermore, when ξ is a limit, let us emphasize that the definitionof Ψ κξ,n ( X , . . . , X n , Y ξ ) depends not only on the chosen bijection b κ,ξ : κ → ξ , buton the entire history of bijections b κ,ζ : κ → ζ chosen at previous limit steps ζ < ξ in the construction.Generalizing work of Bagaria [Bag19], as our first application of the existence ofuniversal formulas, we show that there are natural normal ideals on κ associatedto Π ξ -indescribability for all ξ < κ + . Theorem 5.5. If a cardinal κ is Π ξ -indescribable where ξ < κ + , then the collection Π ξ ( κ ) = { X ⊆ κ : X is not Π ξ -indescribable } is a nontrivial normal ideal on κ .Proof. Suppose κ is Π ξ -indescribable where ξ < κ + . It is easy to see thatΠ ξ ( κ ) = { X ⊆ κ : X is not Π ξ -indescribable } is a nontrivial ideal on κ , so we just need to prove it is normal. Suppose S ∈ Π ξ ( κ ) + and fix a regressive function f : S → κ . For the sake of contradiction, assume thatfor all η < κ the set { α ∈ S : f ( α ) = η } is not in Π ξ ( κ ) + . Then, for each η < κ there is some Π ξ formula ϕ η ( X ) over V κ and some A η ⊆ V κ such that V κ | = ϕ η ( A η )but V α | = ¬ ϕ η ( A η ) | κα for all α ∈ S such that f ( α ) = η . Let Ψ κξ, ( X, Y ξ ) be theuniversal Π ξ formula for formulas with one free second-order variables, let K ϕ η ⊆ κ and let C ϕ η be as in Theorem 5.3. Then for all η < κ we have V κ | = Ψ κξ, ( A η , K ϕ η )and yetfor all α ∈ X ∩ C ϕ η such that f ( α ) = η we have V α | = ¬ Ψ κξ, ( A η , K ϕ η ) | κα . (3)Let A = { Γ( η, β ) : η < κ ∧ β ∈ A η } ⊆ κ and K = { Γ( η, β ) : η < κ ∧ β ∈ K ϕ η } ⊆ κ code the sequences h A η : η < κ i and h K ϕ η : η < κ i respectively. Let C = △ η<κ C ϕ η = { ζ < κ : ζ ∈ \ η<ζ C ϕ η } and notice that C is closed unbounded in κ . We define ϕ ( A, K, C ) to be thefollowing formula over V κ with all second-order parameters displyaed( C is unbounded) ∧ ∀ η ∀ X ∀ Y ( X = ( A ) η ∧ Y = ( K ) η → Ψ ξ, ( X, Y )) . It follows from Lemma 5.2 that ϕ ( A, K, C ) is Π ξ over V κ . Clearly ϕ ( A, K, C ) isequivalent over V κ to ^ η<κ Ψ κξ, ( A η , K ϕ η )and V κ | = ϕ ( A, K, C ).Let us consider the case in which ξ = ζ + 1 is a successor ordinal. Then, by thedefinition of the universal formula Ψ κξ, in the proof of Theorem 5.3, ϕ ( A, K, C ) isof the form( C is unbounded) ∧ ∀ η ∀ X ∀ Y ( X = ( A ) η ∧ Y = ( K ) η → ∀ W ( ¯Ψ κζ, ( X, W, Y ))) , and by the proof of Lemma 5.2, ϕ ( A, K, C ) is equivalent over V κ to( C is unbounded) ∧ ∀ η ∀ X ∀ Y ∀ W ( X = ( A ) η ∧ Y = ( K ) η → ¯Ψ κζ, ( X, W, Y )) . Since V κ | = ϕ ( A, K, C ) and S is Π ξ -indescribable, there is an α ∈ S ∩ C such that V α | = ∀ η ∀ X ∀ Y ∀ W ( X = ( A ) η ∧ Y = ( K ) η → ( ¯Ψ κζ, ( X, W, Y ) | κα )) . Hence for every η < α we have V α | = ∀ W ¯Ψ κζ, ( A η , W, K ϕ η ) | κα and thus V α | = Ψ κζ, ( A η , K ϕ η ) | κα . IGHER INDESCRIBABILITY AND DERIVED TOPOLOGIES 27 Since f ( α ) < α , this contradicts the fact that V α | = ¬ Ψ κξ, ( A η , K ϕ η ) | κα . Let us consider the case in which ξ is a limit ordinal. Notice that, by thedefinition of Ψ κξ, in the proof of Theorem 5.3, and by the proof of Lemma 5.2, ϕ ( A, K, C ) is equivalent over V κ to the Π ξ sentence ^ ζ<ξ ( C is unbounded) ∧ ( ∀ η ∀ X ∀ Y ( X = ( A ) η ∧ Y = (( K ) η ) b − κ,ξ ( ζ ) → Ψ κζ, ( X, Y ))) . Since V κ | = ϕ ( A, K, C ) and since S is Π ξ -indescribable, there is an α ∈ S such that V α | = ϕ ( A, K, C ) | κα . But this implies that α ∈ C and V α | = ^ ζ ∈ F ξ ( α ) ∀ η ∀ X ∀ Y ( X = ( A ) η ∧ Y = (( K ) η ) b − κ,ξ ( ζ ) → (Ψ κζ, ( X, Y ) | κα )) . Thus V α | = ^ ζ ∈ F ξ ( α ) ∀ η Ψ κζ, ( A η , (( K η ) b − κ,ξ ( ζ ) )) | κα ))and hence, by definition of Ψ κξ, in the proof of Theorem 5.3, we have V α | = ∀ η (Ψ κξ, ( A η , K η )) | κα Now, since f ( α ) < α we see that V α | = Ψ κξ, ( A f ( α ) , K f ( α ) ) | κα and since α ∈ C ϕ f ( α ) this contradicts (3). (cid:3) A hierarchy result. In order to prove the hierarchy results below (Corollary5.9 and Corollary 5.12), we first need establish a connection between universalformulas at κ and universal formulas at regular α < κ . Lemma 5.6. Suppose κ > ω is regular. Fix any ξ < κ + and n < ω , let Ψ κξ,n ( X , . . . , X n , Y ξ ) and ¯Ψ κξ,n ( X , . . . , X n , Y ξ ) be, respectively, universal Π ξ and Σ ξ formulas at κ forformulas with n free second-order variables, which were defined by transfinite re-cursion in the proof of Theorem 5.3. There are clubs C ξ,n and D ξ,n in κ such thatthe following hold. (1) For all regular α ∈ C ξ,n the formula Ψ κξ,n ( X , . . . , X n , Y ξ ) | κα is a universal Π f κξ ( α ) formula at α for formulas with n free second-order variables. (2) For all regular α ∈ D ξ,n the formula ¯Ψ κξ,n ( X , . . . , X n , Y ξ ) | κα is a universal Σ f κξ ( α ) formula at α for formulas with n free second-order variables.Proof. We proceed by induction on ξ . It is easy to see that the result holds for ξ < κ . For example, if ξ < κ is a limit ordinal and n < ω we haveΨ κξ,n ( X , . . . , X n , Y ξ ) = ^ ζ<ξ Θ κζ,n ( X , . . . , X n , Y ξ ) where Θ κζ,n ( X , . . . , X n , Y ξ ) is equivalent to Ψ κζ,n ( X , . . . , X n , ( Y ξ ) ζ ) over V κ for each ζ < ξ . If ξ < α < κ we haveΨ κξ,n ( X , . . . , X n , K ) | κα = ^ ζ<ξ (Θ κζ,n ( X , . . . , X n , Y ξ ) | κα )= ^ ζ<ξ Θ αζ,n ( X , . . . , X n , Y ξ )= Ψ αξ,n ( X , . . . , X n , Y ξ )by the inductive hypothesis. Thus Ψ κξ,n ( X , . . . , X n , Y ξ ) | κα is a universal Π ξ formulaat κ for formulas with n free variables.Suppose κ ≤ ξ < κ + . Let us suppose ξ is a limit ordinal. The case in which ξ is a successor ordinal is easier and is left to the reader. We will now prove (1); theproof of (2) is similar. Recall thatΨ κξ, ( X, Y ξ ) = ^ ζ<ξ Θ κζ, ( X, Y ξ )where each Θ κζ, ( X, Y ξ ) is the following Π ζ formula over V κ equivalent to Ψ κζ, ( X, ( Y ξ ) b − κ,ξ ( ζ ) ):Θ κζ, ( X, Y ξ ) = ∀ Y ( ∀ β ( β ∈ Y ↔ Γ( b − κ,ξ ( ζ ) , β ) ∈ Y ξ ) → Ψ κζ, ( X, Y )) . For each ζ < ξ , let C ζ, be the club subset of κ obtained from the inductivehypothesis. By normality the set E = △ ζ<ξ C ζ, = { α < κ : α ∈ \ ζ ∈ F κξ ( α ) C ζ, } is in the club filter on κ . In what follows we will need to use the fact that E = { α < κ : ( ∀ ζ < f κξ ( α )) f κπ − ξ,α ( ζ ) ( α ) = ζ } is in the club filter on α ; to see this notice that E = { α < κ : ( ∀ η ∈ F κξ ( α )) f κη ( α ) = π ξ,α ( η ) } = { α < κ : ( ∀ η ∈ F κξ ( α )) π η,α [ F κη ( α )] = π ξ,α ( η ) }⊇ { α < κ : ( ∀ η ∈ F κξ ( α )) F κη ( α ) = F κξ ( α ) ∩ η } where the last set is in the club filter on κ by Lemma 2.5.Since E ∩ E is in the club filter, we may let C ξ, ⊆ E ∩ E be a club subsetof κ . Now suppose α ∈ C ξ, is regular. Then we haveΨ κξ, ( X, Y ξ ) | κα = ^ ζ ∈ F κξ ( α ) ∀ Y ( ∀ β ( β ∈ Y ↔ Γ( b − κ,ξ ( ζ ) , β ) ∈ Y ξ ) → (Ψ κζ, ( X, Y ) | κα ))(4)where Ψ κζ, ( X, Y ) | κα is a universal Π f κζ ( α ) formula at α for all ζ ∈ F κξ ( α ) since α ∈ E . Let π ξ,α : F κξ ( α ) → f κξ ( α ) be the transitive collapse of F κξ ( α ). Using π ξ,α See the proof of Theorem 5.3 for details regarding this notation. IGHER INDESCRIBABILITY AND DERIVED TOPOLOGIES 29 to reindex (4) we see thatΨ κξ, ( X, Y ξ ) | κα = ^ ζ Suppose κ is regular and ξ < κ + . For any formula ϕ which iseither Π ξ or Σ ξ over V κ , there is a club D ϕ ⊆ κ such that for all regular uncountable α ∈ D ϕ the set of β < α such that ( ϕ | κα ) | αβ = ϕ | κβ is in the club filter on α .Proof. Clearly the result holds if ξ < κ . Suppose κ ≤ ξ < κ + and ξ is a limitordinal. Let us assume ϕ is Σ ξ over V κ . The proof of the case in which ϕ is Π ξ is similar. We have ϕ = W ζ<ξ ϕ ζ where ϕ ζ is Σ ζ over V κ for each ζ < ξ . Foreach ζ < ξ let D ϕ ζ be the set in the club filter on κ obtained from the inductivehypothesis. By Proposition 2.9, we may fix a club C κξ ⊆ κ such that for all regularuncountable α ∈ C κξ there are club-many β < α such that π ξ,α [ F κξ ( β )] = F αf κξ ( α ) ( β ).Thus, the set D ϕ = C κξ ∩ △ ζ<ξ D ϕ ζ = C κξ ∩ { α < κ : α ∈ \ ζ ∈ F κξ ( α ) D ϕ ζ } is in the club filter on κ .Suppose α ∈ D ϕ is regular and uncountable. Since α ∈ C κξ we may fix a club E in α such that for all β ∈ E we have π ξ,α [ F κξ ( β )] = F αf κξ ( α ) ( β ). For each ζ ∈ F κξ ( α )we have α ∈ D ϕ ζ , and thus we may fix a club E ϕ ζ in α such that for all β ∈ E ϕ ζ we have ( ϕ ζ | κα ) | αβ = ϕ ζ | κβ . Notice that the set E ∗ = E ∩ △ ζ ∈ F κξ ( α ) E ϕ ζ = E ∩ { β < α : β ∈ \ ζ ∈ F κξ ( β ) E ϕ ζ } See Remark 5.4. is in the club filter on α . It will suffice to show that for every β ∈ E ∗ we have( ϕ | κα ) | αβ = ϕ | κβ . Suppose β ∈ E ∗ . Then we have( ϕ | κα ) | αβ = _ ζ<ξ ϕ ζ (cid:12)(cid:12)(cid:12) κα (cid:12)(cid:12)(cid:12) αβ = _ ζ ∈ F κξ ( α ) ϕ ζ | κα (cid:12)(cid:12)(cid:12) αβ (definition of | κα )= _ ζ Suppose κ > ω is regular and ξ < κ + . There is a Π ξ +1 formula Φ κξ ( Z ) over V κ and a club C ⊆ κ such that for all S ⊆ κ we have S is a Π ξ -indescribable subset of κ if and only if V κ | = Φ κξ ( S ) and for all regular α ∈ C we have S ∩ α is a Π f κξ ( α ) -indescribable subset of α if and only if V α | = Φ κξ ( S ) | κα . Proof. We let R ⊆ κ be a set, defined as follows, coding information about which α < κ and which a ⊆ α satisfy V α | = Ψ κξ, ( a ) | κα . For each regular α < κ let h a αβ : β < δ α i be a sequence of subsets of α such that for all a ⊆ α we have V α | = Ψ κξ, ( a ) | κα if and only if a = a αβ for some β < δ α . Let Γ : κ × κ × κ → κ bethe usual definable bijection. We let R = { Γ( α, β, γ ) : ( α is regular) ∧ β < α δ ∧ γ ∈ a αβ } . IGHER INDESCRIBABILITY AND DERIVED TOPOLOGIES 31 For α, β < κ we define R ( α,β ) = { γ : Γ( α, β, γ ) ∈ R } to be the ( α, β ) th slice of R so that when α is regular and β < δ α we have R ( α,β ) = a αβ . Now we letΦ κξ ( Z ) = ∀ X [Ψ κξ, ( X ) → ( ∃ Y ⊆ Z ∩ REG)( Y ∈ NS + κ ) ∧ ( ∀ η ∈ Y )( ∃ β )( X ∩ η = R ( η,β ) )] . Since the part of Φ κξ to the right of the → is Σ over V κ , and since Ψ κξ, ( X ) is Π ξ over V κ and appears to the left of the → in Φ κξ , it follows that Φ κξ is Π ξ +1 over V κ .First let us show that S ⊆ κ is Π ξ -indescribable in κ if and only if V κ | = Φ κξ ( S ).Suppose S is Π ξ -indescribable in κ . To see that V κ | = Φ κξ ( S ), fix K ⊆ κ suchthat V κ | = Ψ κξ, ( K ). Then D = { α < κ : V α | = Ψ κξ, ( K ) | κα } is in Π ξ ( κ ) ∗ andthus Y = S ∩ D ∩ REG is, in particular, stationary in κ . If η ∈ Y then we haveΨ κξ, ( K ) | κα and hence Ψ κξ, ( K ∩ α ) | κα , which implies that K ∩ η = R ( η,β ) for some β < δ α . Conversely, suppose V κ | = Φ κξ ( S ) and let us show that S is Π ξ -indescribablein κ . Fix a Π ξ sentence ϕ such that V κ | = ϕ . Then, by Theorem 5.3, V κ | = Ψ κξ, ( K ϕ )and thus there is a Y ⊆ S ∩ REG stationary in κ such that for all α ∈ Y we have V α | = Ψ κξ, ( K ϕ ) | κα . By Theorem 5.3 the set D ϕ = { α < κ : V α | = ϕ | κα ⇐⇒ V α | = Ψ ξ, ( K ϕ ) | κα } is in the club filter on κ . Thus we may choose a regular α ∈ Y ∩ D ϕ ∩ REG ⊆ S and observe that V α | = ϕ | κα . Thus S is Π ξ -indescribable in κ .For the second part of the statement of the corollary, let us begin by noticingthat, by Lemma 5.6, there is a club C in κ such that for all regular α ∈ C theformula Ψ κξ, ( X ) | κα is a universal Π f κξ ( α ) formula at α for Π f κξ ( α ) sentences over V α .Furthremore, by Proposition 5.7 the set C = { α < κ : there are club-many ζ < α such that (Ψ κξ, ( X ) | κα ) | αζ = Ψ κξ, ( X ) | κζ } is in the club filter on κ . Now, let C = C ∩ C , suppose S ⊆ κ and α ∈ C isregular. We must show that S is a Π f κξ ( α ) -indescribable subset of α if and only if V α | = Φ κξ ( S ) | κα . Notice that,Φ κξ ( S ) | κα = ∀ X [Ψ κξ, ( X ) | κα → ( ∃ Y ⊆ S ∩ α ∩ REG)( Y ∈ NS + α ) ∧ ( ∀ η ∈ Y )( ∃ β )( X ∩ η = R ( η,β ) ∩ α )] . Suppose S is Π f κξ ( α ) -indescribable in α . We want to show that V α | = Φ κξ ( S ) | κα .Suppose K ⊆ α is such that V α | = Ψ κξ, ( K ) | κα . Then D = { ζ < α : V α | = (Ψ κξ, ( K ) | κα ) | αζ } is in Π f κξ ( α ) ( α ) ∗ . We want to show that there is a Y ⊆ S ∩ α ∩ REG stationary in α such that for all ζ ∈ Y we have K ∩ ζ = R ( ζ,β ) ∩ α , or just that for all ζ ∈ Y wehave V ζ | = Ψ κξ, ( K ∩ ζ ) | κζ . Since α ∈ C , the set D = { ζ < α : (Ψ κξ, ( K ) | κα ) | αζ = Ψ κξ, ( K ) | κζ } is in the club filter on α . Thus Y = D ∩ D ∩ S ∩ α ∩ REG is the desired stationarysubset of α .For the converse, suppose V α | = Φ κξ ( S ) | κα and suppose V α | = ϕ where ϕ is Π f κξ ( α ) over V α . Since α ∈ C the formula Ψ κξ, ( X ) | κα is a universal Π f κξ ( α ) formula at κ for Π f κξ ( α ) sentences. Thus V α | = Ψ κξ, ( K αϕ ) | κα where K αϕ ⊆ α is the appropriate code for ϕ obtained from Theorem 5.3. Since V α | = Φ κξ ( S ) | κα we see that there is a Y ⊆ S ∩ α ∩ REG stationary in α such that for all ζ ∈ Y we have V ζ | = Ψ κξ, ( K ϕ ) | κζ .Now, by Theorem 5.3 applied to the universal Π f κξ ( α ) formula Ψ κξ, ( X ) | κα at α , theset D = { ζ < α : V ζ | = ϕ | αζ ⇐⇒ V ζ | = (Ψ κξ, ( K ϕ ) | κα ) | αζ } is in the club filter on α . Thus the set D ∩ Y ∩ REG is stationary in α and hencewe can choose a regular ζ ∈ D ∩ Y and observe that V ζ | = ϕ | αζ . (cid:3) We obtain our first hierarchy result as an easy corollary of Theorem 5.8. Corollary 5.9. Suppose S ⊆ κ is Π ξ -indescribable in κ where ξ < κ + and let ζ < ξ . Then the set C = { α < κ : S ∩ α is Π f κζ ( α ) -indescribable } is in the filter Π ξ ( κ ) ∗ .Proof. Since ζ < ξ , it follows that S is Π ζ -indescribable in κ , and thus V κ | = Φ κζ ( S ),where Φ κζ ( Z ) is the Π ζ +1 formula over V κ obtained from Theorem 5.8. By Theorem5.8, there is a club D in κ such that for every regular α ∈ D , S ∩ α is Π f κζ ( α ) -indescribable if and only if V α | = Φ κζ ( S ) | κα . Since the set D ∩ { α < κ : V α | = Φ κζ ( S ) | κα } is in the filter Π ζ ( κ ) ∗ , we see that { α < κ : S ∩ α is Π f κζ ( α ) -indescribable } ∈ Π ζ ( κ ) ∗ ⊆ Π ξ ( κ ) ∗ . (cid:3) Next, in order to show that when κ is Π ξ -indescribable, we have a proper con-tainment Π ζ ( κ ) ( Π ξ ( κ ) for all ζ < ξ , we need two lemmas. Recall that for anuncountable regular cardinal κ , if S ⊆ κ is stationary in κ and for each α ∈ S we have a set S α ⊆ α which is stationary in α , then it follows that S α ∈ S S α isstationary in κ . We generalize this to Π ξ -indescribability for all ξ < κ + as follows. Lemma 5.10. Suppose S is a Π ξ -indescribable subset of κ where ξ < κ + . Furthersuppose that S α is a Π f κξ ( α ) -indescribable subset of α for each α ∈ S . Then S α ∈ S S α is a Π ξ -indescribable subset of κ .Proof. If ξ < κ the result follows directly from [Cod, Lemma 3.1]. Suppose κ ≤ ξ < κ + and ϕ is some Π ξ sentence over V κ such that V κ | = ϕ . By Lemma 3.6, C ϕ = { α < κ : ϕ | κα is Π f κξ ( α ) over V κ } is in the club filter on κ . By Proposition 5.7, there is a club D ϕ ⊆ κ such that forall regular α ∈ D ϕ the set of β < α such that ( ϕ | κα ) | αβ = ϕ | κβ is in the club filter on α . Thus, S ∩ C ϕ ∩ D ϕ is Π ξ -indescribable in κ . Hence there is a regular uncountable α ∈ S ∩ C ϕ ∩ D ϕ such that V α | = ϕ | κα . Let E be a club subset of α such that forall β ∈ E we have ( ϕ | κα ) | αβ = ϕ | κβ . Since S α ∩ E is Π f κξ ( α ) -indescribable in α and ϕ | κα is Π f κξ ( α ) over V α , there is some β ∈ S α ∩ E such that V β | = ( ϕ | κα ) | αβ . Since( ϕ | κα ) | αβ = ϕ | κβ , it follows that S α ∈ S S α is Π ξ -indescribable in κ . (cid:3) IGHER INDESCRIBABILITY AND DERIVED TOPOLOGIES 33 Proposition 5.11. For all ordinals ξ , if S ⊆ κ is Π ξ -indescribable in κ where ξ < κ + , then the set T = { α < κ : S ∩ α is not Π f κξ ( α ) -indescribable in α } is Π ξ -indescribable in κ .Proof. We proceed by induction on ξ . For ξ < ω this is a well-known result, whichfollows directly from [Cod, Lemma 3.2]. Suppose ξ ∈ κ + \ ω and, for the sake ofcontradiction, suppose S is Π ξ -indescribable and T is not Π ξ -indescribable in κ .Then κ \ T is in the filter Π ξ ( κ ) ∗ and is thus Π ξ -indescribable in κ . By Corollary2.10, there is a club C ⊆ κ such that for all regular uncountable α ∈ C , the set D α = { β < α : f κξ ( β ) = f αf κξ ( α ) ( β ) } is in the club filter on α . Let D be the set of regular uncountable cardinals lessthan κ , and note that D ∈ Π ( κ ) ∗ ⊆ Π ξ ( κ ) ∗ . Notice that ( κ \ T ) ∩ C ∩ D is Π ξ -indescribable in κ . For each α ∈ ( κ \ T ) ∩ C ∩ D , it follows by induction that theset T α = { β < α : S ∩ β is not Π f αfκξ ( α ) ( β ) -indescribable } is Π f κξ ( α ) -indescribable in α . Thus, for each α ∈ ( κ \ T ) ∩ C ∩ D the set T α ∩ D α isΠ f κξ ( α ) -indescribable in α . Now it follows by Lemma 5.10 that the set [ α ∈ ( κ \ T ) ∩ C ∩ D ( T α ∩ D α ) ⊆ T is Π ξ -indescribable in κ , a contradiction. (cid:3) Now we show that for regular κ , whenever ζ < ξ < κ + and the ideals underconsideration are nontrivial, we have Π ζ ( κ ) ( Π ξ ( κ ). Corollary 5.12. Suppose κ is Π ξ -indescribable where ξ < κ + . Then for all ζ < ξ we have Π ζ ( κ ) ( Π ξ ( κ ) .Proof. The fact that Π ζ ( κ ) ⊆ Π ξ ( κ ) follows easily from the fact that the class ofΠ ξ formulas includes the Π ζ formulas. To see that the proper containment holds,consider the set C = { α < κ : α is Π f κζ ( α ) -indescribable } . By Corollary 5.9 and Proposition 5.11, we have κ \ C ∈ Π ξ ( κ ) \ Π ζ ( κ ). (cid:3) Higher Π ξ -clubs. Now we present a characterization of the Π ξ -indescribabilityof sets S ⊆ κ in terms of a natural base for the filter Π ξ ( κ ) ∗ . Definition 5.13. Suppose κ is a regular cardinal. We define the notion of Π ξ -clubsubset of κ for all ξ < κ + by induction.(1) A set C ⊆ κ is Π -club if it is closed and unbounded in κ .(2) We say that C is Π ζ +1 -club in κ if C is Π ζ -indescribable in κ and there is aclub C ∗ in κ such that for all α ∈ C ∗ , whenever C ∩ α is Π f κζ ( α ) -indescribablein α we must have α ∈ C . (3) If ξ is a limit, we say that C ⊆ κ is Π ξ -club in κ if C is Π ζ -indescribablefor all ζ < ξ and there is a club C ∗ in κ such that for all α ∈ C ∗ , whenever C ∩ α is Π ζ -indescribable for all ζ < f κξ ( α ), we must have α ∈ C . Remark 5.14. Let us note that in the previous definition, the use of the club C ∗ seems to be necessary in order to prove Theorem 5.15(2). The issue is that we seemto need C ∗ to show that χ ξ ( C ) | κα expresses, over V α , that C ∩ α is f κξ ( α )-club in α .Next we show that, when the Π ξ -indescribability ideal Π ξ ( κ ) is nontrivial, theΠ ξ -club subsets of κ form a filter base for the dual filter Π ξ ( κ ) ∗ , a set being Π ξ -club in κ is expressible by a Π ξ sentence and a set being Π ξ -indescribable in κ isexpressible by a Π ξ +1 sentence. We will need the following. Theorem 5.15. Suppose κ is a regular cardinal. For all ξ < κ + , if κ is Π ξ -indescribable then the following hold. (1) A set S ⊆ κ is Π ξ -indescribable if and only if S ∩ C = ∅ for all Π ξ -clubs C ⊆ κ . (2) There is a Π ξ formula χ κξ ( X ) over V κ such that for all C ⊆ κ we have C is Π ξ -club in κ if and only if V κ | = χ κξ ( C ) and there is a club D ξ in κ such that for all α ∈ D ξ and all C ⊆ κ we have C ∩ α is Π f κξ ( α ) -club in α if and only if V α | = χ κξ ( C ) | κα . Proof. Sun proved that the theorem holds for ξ = 1, and Hellsten generalized thisto the case in which ξ < ω . We provide a proof of the case in which κ ≤ ξ < κ + isa limit ordinal; the case in which ξ < κ + is a successor and the case in which ξ < κ is a limit are similar, but easier.Suppose κ ≤ ξ < κ + is a limit ordinal and that both (1) and (2) hold for allordinals ζ < ξ . For the forward direction of (1), suppose S ⊆ κ is Π ξ -indescribableand fix C ⊆ κ a Π ξ -club subset of κ . Then, in particular, for each ζ < ξ , C isΠ ζ -indescribable and, by Theorem 5.8, we see that V κ | = ^ ζ<ξ Φ κζ ( C ) . Let D = { α < κ : f κξ ( α ) = [ ζ ∈ F ξ ( α ) f κζ ( α ) is a limit ordinal } and notice that D contains a club subset of κ by Lemma 2.6 and Lemma 2.7. Foreach ζ < ξ , let C ζ be the club subset of κ obtained from Theorem 5.8 such that forall regular α ∈ C ζ we have C ∩ α is Π f κζ ( α ) -indescribable in α ⇐⇒ V α | = Φ κζ ( C ) | κα . Now let ¯ C = D ∩ ( △ ζ<ξ C ζ ) = C ∩ { α < κ : α ∈ \ ζ ∈ F ξ ( α ) C ζ } and notice that S ∩ ¯ C is Π ξ -indescribable in κ because ¯ C is in the club filter on κ .Thus we may choose α ∈ S ∩ ¯ C such that V α | = ^ ζ ∈ F κξ ( α ) Φ κζ ( C ) | κα IGHER INDESCRIBABILITY AND DERIVED TOPOLOGIES 35 and since α ∈ T ζ ∈ F κξ ( α ) C ζ , we see that C ∩ α is Π f κζ ( α ) -indescribable for all ζ ∈ F κξ ( α ). Since α ∈ D this implies that C ∩ α is Π ζ -indescribable for all ζ < f κξ ( α )and since C is Π ξ -club, it follows that α ∈ C and hence S ∩ C = ∅ .For the reverse direction of (1) when ξ is a limit, it suffices to show that if ϕ = V ζ<ξ ϕ ζ is any Π ξ sentence over V κ such that V κ | = ϕ , then the set C = { α < κ : V α | = ϕ | κα } contains a Π ξ -club in κ . Notice that ¬ ϕ is equivalent over V κ to the Σ ξ sentence W ζ<ξ ¬ ϕ ζ where ¬ ϕ ζ is Σ ζ over V κ for each ζ < ξ . It follows from Lemma 3.6 thatthe set C ¬ ϕ = { α < κ : ¬ ϕ | κα is Σ f κξ ( α ) over V α } as well as the sets C ¬ ϕ ζ = { α < κ : ¬ ϕ ζ | κα is Σ f κζ ( α ) over V κ } for ζ < ξ , are all in the club filter on κ . Furthermore, for each ζ < ξ , by Proposition5.7, we may let D ¬ ϕ be a club subset of κ such that for all regular α ∈ D ¬ ϕ thereare club-many β < α such that ( ¬ ϕ ζ | κα ) | αβ = ¬ ϕ ζ | κβ . We let D = C ¬ ϕ ∩ △ ζ<ξ ( C ¬ ϕ ζ ∩ D ¬ ϕ ζ )= C ¬ ϕ ∩ { α < κ : α ∈ \ ζ ∈ F κξ ( α ) ( C ¬ ϕ ζ ∩ D ¬ ϕ ζ ) } and note that D is in the club filter on κ . To prove the reverse direction of (1), wewill show that C ∩ D is Π ξ -club in κ .First, let us show that C ∩ D is Π ζ -indescribable in κ for all ζ < ξ . Suppose not.Then for some fixed ζ < ξ , the set C ∩ D is not Π ζ -indescribable in κ and hence κ \ ( C ∩ D ) = { α < κ : α / ∈ D or V α | = ¬ ϕ | κα } is in the filter Π ζ ( κ ) ∗ . Since κ is Π ξ -indescribable by assumption, and sinceΠ ζ ( κ ) ∗ ⊆ Π ξ ( κ ) ∗ ⊆ Π ξ ( κ ) + , we see that κ \ ( C ∩ D ) is Π ξ -indescribable in κ .But D is club in κ , and thus D ∩ ( κ \ ( C ∩ D )) = ( κ \ C ) ∩ D is Π ξ -indescribable.But now, from V κ | = ϕ , it follows that for some α ∈ κ \ C we have V α | = ϕ | κα , whichcontradicts the definition of C . Thus, C ∩ D is Π ζ -indescribable in κ for all ζ < ξ .Now let us show that C ∩ D is ξ -closed in κ . Suppose α < κ and C ∩ D ∩ α isΠ ζ -indescribable in α for all ζ < f κξ ( α ), but α / ∈ C ∩ D . Since D is club in κ , itfollows that α ∈ D and α / ∈ C . By the definition of C we have V α | = _ ζ ∈ F κξ ( α ) ¬ ϕ ζ | κα , where the formula ¬ ϕ | κα = W ζ ∈ F κξ ( α ) ¬ ϕ ζ | κα is Σ f κξ ( α ) over V α because α ∈ D ⊆ C ¬ ϕ .Fix some η ∈ F κξ ( α ) such that V α | = ¬ ϕ η | κα and notice that because α ∈ △ ζ<ξ C ¬ ϕ ζ ,it follows that ¬ ϕ η | κα is Σ f κη ( α ) over V α where f κη ( α ) < f κξ ( α ). Since α ∈ D and η ∈ F κξ ( α ) we have α ∈ D ¬ ϕ η , and thus there is a ¯ C ⊆ α in the club filter on α suchthat for all β ∈ ¯ C we have ( ¬ ϕ η | κα ) | αβ = ¬ ϕ η | κβ . Since C ∩ ¯ C ∩ α is Π ζ -indescribablein α for all ζ < f κξ ( α ), there must be a β ∈ C ∩ ¯ C ∩ α such that V β | = ( ¬ ϕ η | κα ) | αβ , but this contradicts the definition of C since ( ¬ ϕ η | κα ) | αβ = ¬ ϕ η | κβ . This establishesthat (1) holds for the limit ordinal ξ .Now, let us show that (2) holds for the limit ordinal ξ . The definition of “ X isΠ ξ -club” is equivalent over V κ to ^ η<ξ Φ κη ( X ) ∧ ( ∃ C ∗ ) ( C ∗ is club) ∧ ( ∀ β ∈ C ∗ ) ^ ζ Suppose µ > ω is a regular cardinal and h f µξ : ξ < µ + i and h ¯ f µξ : ξ < µ + i are two sequences of canonical functions at µ . Let h τ ξ : ξ < µ + i and h ¯ τ ξ : ξ < µ + i be the corresponding sequences of derived topologies defined using h f µξ : ξ < µ + i and h ¯ f µξ : ξ < µ + i respectively (see Section 1 for definitions). Thenfor all ξ < µ + we have τ ξ = ¯ τ ξ . This is where the club C ∗ in the definition of Π ξ -club seems to be necessary. We need C ∩ α being f κξ ( α )-closed in α (which uses f αζ ’s) to follow from the fact that for every β ∈ C ∗ , whenever C ∩ β is Π ζ -indescribable in β for all ζ < f κξ ( β ), then β must be in C . But, we seem to only getthis on a club of β ’s less than α . Proof. We proceed by induction on ξ < µ + . For ξ < µ , this is trivial because wecan assume without loss of generality that f µξ and ¯ f µξ are constant functions withvalue ξ , and thus the definitions of τ ξ and ¯ τ ξ are clearly the same. In this case, τ ξ = ¯ τ ξ is the ξ th derived topology defined in [Bag19].Now suppose µ ≤ ξ < µ + . If ξ is a limit, then τ ξ = ¯ τ ξ because τ ξ is the topologygenerated by using S ζ<ξ τ ζ as a subbase and ¯ τ ξ is the topology generated by using S ζ<ξ ¯ τ ζ as a subbase. Thus, by the inductive hypothesis, τ ξ = ¯ τ ξ .It remains to show that τ ξ +1 = ¯ τ ξ +1 . Let C ξ +1 be the base for τ ξ +1 obtainedfrom the subbase B ξ +1 = B ξ ∪ { d ξ ( A ) : A ⊆ µ } and let ¯ C ξ +1 be the base for ¯ τ ξ +1 obtained from the subbase ¯ B ξ +1 = ¯ B ξ ∪ { ¯ d ξ ( A ) : A ⊆ µ } .Suppose U ∈ τ ξ +1 and β ∈ U . Then there is some B ∈ C ξ +1 with β ∈ B ⊆ U .Furthermore, B is of the form B = I ∩ d ξ ( A ) ∩ d ξ ( A ) ∩ · · · ∩ d ξ n − ( A n − )where I ∈ B , n < ω , ξ ≤ ξ ≤ · · · ≤ ξ n − < ξ and A i ⊆ µ for all i < n . If ξ n − < µ , then it follows that B ∈ τ µ = ¯ τ µ and hence U ∈ ¯ τ µ ⊆ ¯ τ ξ . On the otherhand, if ξ n − ≥ µ , let m be the least natural number less than n such that ξ m ≥ µ .Then for i < m we have d ξ ( A i ) = ¯ d ξ i ( A i ), and for m ≤ i < n we have d ξ i ( A i ) = { α < µ : α is a limit point of A in the topology τ f µξi ( α ) } and ¯ d ξ i ( A i ) = { α < µ : α is a limit point of A in the topology τ ¯ f µξi ( α ) } For each i with m ≤ i < n , let C i ⊆ µ be a club such that for all α ∈ C i we have f µξ i ( α ) = ¯ f µξ i ( α ). Then it follows that for m ≤ i < n we have d ξ i ( A i ) ∩ C i = ¯ d ξ i ( A i ) ∩ C i . Since the clubs C i , for m ≤ i < n are all in B , it follows that the set¯ B = I ∩ ¯ d ξ ( A ) ∩· · ·∩ ¯ d ξ m − ( A m − ) ∩ ( ¯ d ξ m ( A m ) ∩ C m ) ∩· · ·∩ ( ¯ d ξ n − ( A n − ) ∩ C n − ) ∪{ β } is in the basis ¯ C ξ +1 for ¯ τ ξ +1 . Since β ∈ ¯ B ⊆ B , it follows that U ∈ ¯ τ ξ +1 . Thisshows that τ ξ +1 ⊆ ¯ τ ξ +1 and a similar argument shows ¯ τ ξ +1 ⊆ τ ξ +1 . (cid:3) Next we introduce a generalization of Bagaria’s ξ -stationarity. The followingdefinition is a slight generalization of [Bag19, Definition 2.6], which allows for morenontrivial cases; the new ingredient here is the use of canonical functions. Definition 6.2. A set A ⊆ µ is 0 -stationary in α if and only if A is unboundedin α . For 0 < ξ < α + , we say that A is ξ -stationary in α if and only if for every ζ < ξ , for every subset S of µ that is ζ -stationary in α , there is a β ∈ A ∩ α suchthat S is f αζ ( β )-stationary in β . We say that an ordinal α is ξ -reflecting if as asubset of µ it is ξ -stationary in α . Remark 6.3. Bagaria defined a set A ⊆ µ to be ξ -stationary in α < µ if and onlyif for every ζ < ξ , for every S ⊆ µ that is ζ -stationary in α there is a β ∈ A ∩ α suchthat S is ζ -stationary in β . Since f αζ equals the constant function ζ when ζ < α ,it follows that Bagaria’s notion of A being ξ -stationary in α is equivalent to ourswhen ξ < α . Bagaria comments in the paragraphs following [Bag19, Definition Note that α might not be regular, but nonetheless, f αζ still makes sense because we canconsider a club C in α with order-type cf( α ) and use it to turn a sequence of canonical functionsat cf( α ) into such a sequence at α . IGHER INDESCRIBABILITY AND DERIVED TOPOLOGIES 39 α can be ( α + 1)-reflecting, because if α is the least such ordinal there is a β < α such that α ∩ β = β is α -stationaryand thus ( β + 1)-stationary in β . Let us show that such an argument does not work to rule out the existence of ordinals α which are α + 1-reflecting under ourdefinition. Suppose α is ( α + 1)-reflecting, as in Definition 6.2. Then there is some β < α that is f αα ( β )-reflecting, but f αα ( β ) = β and thus the conclusion is that β is β -reflecting, and Bagaria shows that some ordinals (namely some large cardinals) β can be β -reflecting.The next definition is agrees with that given in [Bag19] as long as ξ < µ . Definition 6.4. Suppose µ is a regular cardinal and α < µ . We say that A ⊆ µ is0 -simultaneously stationary in a regular α (0-s-stationary for short), if and only if A is unbounded in α . For 0 < ξ < α + , we say that A is ξ -simultaneously stationaryin α ( ξ -s-stationary in α , for short) if and only if for every ζ < ξ , every pair ofsubsets S and T of µ that are ζ -s-stationary in α simultaneously f αζ ( β ) -reflect tosome β ∈ A , i.e., S and T are both f αζ ( β )-s-stationary in β . We say that α is ξ -s-reflecting if it is ξ -s-stationary in α .We will require the following lemma, which generalizes [Bag19, Proposition 2.7and Proposition 2.9]. The proof is similar to the arguments given in [Bag19] and isthus left to the reader. Lemma 6.5. Suppose µ > ω is a regular cardinal and α < κ is a cardinal withuncountable cofinality. For all ξ ∈ α + \ { } , whenever A ⊆ µ is ξ -s-stationary in α and C is a club subset of α , then A ∩ C is also ξ -s-stationary in α . In order to characterize the nonisolated points of the spaces ( µ, τ ξ ), for ξ < µ + ,in terms of η -s-reflecting cardinals, we will need the following proposition, whichgeneralizes [Bag19, Proposition 2.10]. Proposition 6.6. Suppose µ is a regular cardinal. (1) For all ξ < µ + , there is a club C ξ ⊆ µ such that for all A ⊆ µ we have d ξ ( A ) ∩ C ξ = { α < µ : A is f µξ ( α ) -s-stationary in α } ∩ C ξ . (2) For all ξ < µ + there is a club D ξ ⊆ µ such that for all α ∈ D ξ andall A ⊆ µ we have that A is ( f µξ ( α ) + 1) -s-stationary in α if and only if A ∩ d ζ ( S ) ∩ d ζ ( T ) ∩ α = ∅ (equivalently, if and only if A ∩ d ζ ( S ) ∩ d ζ ( T ) is f µζ ( α ) -s-stationary in α ) for every ζ ∈ F µξ ( α ) ∪ { ξ } and every pair S , T of subsets of α that are f µζ ( α ) -s-stationary in α . (3) For all ξ < µ + there is a club E ξ ⊆ µ such that for all α ∈ E ξ and all A ⊆ µ ,if A is f µξ ( α ) -s-stationary in α and A i is f µζ i ( α ) -s-stationary in α for some ζ i < ξ , all i < n , then A ∩ d ζ ( A ) ∩ · · · ∩ d ζ n − ( A n − ) is f µξ ( α ) -s-stationaryin α .Proof. If ξ < µ then f µξ : µ → µ is the function with constant value ξ , and thus theresult follows directly from [Bag19, Proposition 2.10], taking C ξ = D ξ = E ξ = µ .Let us show that if ξ is a limit ordinal with µ ≤ ξ < µ + and (1) holds forall ζ < ξ , then (1) holds for ξ . For each ζ < ξ , let C ζ be the club subset of µ obtained from the inductive hypothesis. By normality of the club filter on µ and by Lemma 2.6, there is a club C ξ ⊆ µ such that for all α ∈ C ξ we have α ∈ T ζ ∈ F µξ ( α ) C ζ and f µξ ( α ) = S ζ ∈ F µξ ( α ) f µζ ( α ) < α + is a limit ordinal. Now fix A ⊆ µ and suppose α ∈ d ξ ( A ) ∩ C ξ . Then α is a limit point of A in the τ f µξ ( α ) topology on µ . Thus, for all ζ ∈ F µξ ( α ), α is a limit point of A in the τ f µζ ( α ) topology, that is, α ∈ T ζ ∈ F µξ ( α ) d ζ ( A ). Since α ∈ T ζ ∈ F µξ ( α ) C ζ , it follows by ourinductive hypothesis that A is f µζ ( α )-s-stationary in α for all ζ ∈ F µξ ( α ), and since f µξ ( α ) = S ζ ∈ F µξ ( α ) f µζ ( α ), we see that A is f µξ ( α )-s-stationary in α . Conversely,suppose A is f µξ ( α )-s-stationary in α and α ∈ C ξ . Since f µξ ( α ) = S ζ ∈ F µξ ( α ) f µζ ( α )is a limit ordinal, we see that A is f µζ ( α )-s-stationary in α for all ζ ∈ F µξ ( α ). Since α ∈ T ζ ∈ F µξ ( α ) C ζ , it follows by our inductive hypothesis that α ∈ T ζ ∈ F µξ ( α ) d ζ ( A ).Since the set { f µζ ( α ) : ζ ∈ F µξ ( α ) } is cofinal in f µξ ( α ), the topology τ f µξ ( α ) equalsthe topology generated by S ζ ∈ F µξ ( α ) τ f µζ ( α ) . Thus, it follows that α is a limit pointof A in the τ f µξ ( α ) topology, i.e. α ∈ d ξ ( A ).Similarly, one can prove that if ξ is a limit ordinal with µ ≤ ξ < µ + and (3)holds for all ζ < ξ , then (3) holds for ξ by taking E ξ to be a club subset of µ suchthat if α ∈ E ξ then α ∈ T ζ ∈ F µξ ( α ) E ζ and f µξ ( α ) = S ζ ∈ F µξ ( α ) f µζ ( α ).Next, let us show that if ξ is a limit ordinal with µ ≤ ξ < µ + and both (1) and(3) hold for all ζ ≤ ξ , then (2) holds for ξ . By normality of the club filter on µ , wemay let D ξ be a club subset of µ contained in C ′ ξ ∩ E ξ ∩ { α < µ : α ∈ \ ζ ∈ F µξ ( α ) C ′ ζ ∩ E ζ } and by Lemma 2.5, Lemma 2.6 and Corollary 2.10 we can assume that for all α ∈ D ξ we have(i) ( ∀ ζ ∈ F µξ ( α )) F µζ ( α ) = F µξ ( α ) ∩ ζ ,(ii) f µξ ( α ) = S ζ ∈ F µξ ( α ) f µζ ( α ) is a limit ordinal and(iii) if α is regular and uncountable then for all ζ ∈ F µξ ( α ) ∪ { ξ } there is a clubsubset of α , call it C ζα , contained in the set { β < α : f µζ ( β ) = f αf µζ ( α ) ( β ) } . To prove the forward direction of (2), suppose α ∈ D ξ and A ⊆ µ is ( f µξ ( α ) + 1)-s-stationary in α . Suppose S, T ⊆ α are both f µζ ( α )-s-stationary in α for somefixed ζ ∈ F µξ ( α ) ∪ { ξ } . Since A ∩ C ζ ∩ C ζα is ( f µξ ( α ) + 1)-s-stationary in α , there issome β ∈ A ∩ C ζ ∩ C ζα such that S and T are both f αf µζ ( α ) ( β ) = f µζ ( β )-s-stationaryin β . Since β ∈ C ζ it follows by (1) that β ∈ A ∩ d ζ ( S ) ∩ d ζ ( T ). To see that A ∩ d ζ ( S ) ∩ d ζ ( T ) is, in fact, f µζ ( α )-s-stationary in α , suppose X and Y are η -s-stationary in α for some η < f µζ ( α ). Since (3) holds for ζ and α ∈ E ζ , it followsthat S ∩ d η ( X ) and T ∩ d η ( Y ) are f µζ ( α )-s-stationary in α . Now we have ∅ = A ∩ d ζ ( S ∩ d η ( X )) ∩ d ζ ( T ∩ d η ( Y )) ⊆ A ∩ d ζ ( S ) ∩ d ζ ( T ) ∩ d η ( X ) ∩ d η ( Y ) , and hence A ∩ d ζ ( S ) ∩ d ζ ( T ) is f µζ ( α )-s-stationary in α .For the converse of (2), suppose that α ∈ D ξ and for all ζ ∈ F µξ ( α ) ∪ { ξ } , if S, T ⊆ α are both ζ -s-stationary in α then A ∩ d ζ ( S ) ∩ d ζ ( T ) = ∅ . By (i) and (ii),in order show that A is ( f µξ ( α ) + 1)-s-stationary in α , it suffices to show that for all ζ ∈ F µξ ( α ) ∪ { ξ } , if S, T ⊆ α are both f µζ ( α )-s-stationary in α , then there is a β ∈ A such that S and T are both f αf µζ ( α ) ( β )-s-stationary in β . Fix ζ ∈ F µξ ( α ) ∪ { ξ } andsuppose S, T ⊆ α are f µζ ( α )-s-stationary in α . Since α ∈ D ξ and ζ ∈ F µξ ( α ) ∪ { ξ } IGHER INDESCRIBABILITY AND DERIVED TOPOLOGIES 41 we have a club subset of α : C ζα ⊆ { β < α : f µζ ( β ) = f αf µζ ( α ) ( β ) } . By Lemma 6.5, it follows that S ∩ C ζ ∩ C ζα and T are both f µζ ( α )-s-stationary in α . Thus, by assumption, there is a β ∈ A ∩ d ζ ( S ∩ C ζ ∩ C ζα ) ∩ d ζ ( T ). Now since β ∈ C ζ ∩ C ζα ∩ d ζ ( S ) ∩ d ζ ( T ), it follows by (1) that S and T are both f µζ ( β ) = f αf µζ ( α ) ( β )-s-stationary in β . Hence A is ( f µξ ( α ) + 1)-s-stationary in α .Now let us suppose that (1) and (2) hold for ξ where µ ≤ ξ < µ + and provethat (1) holds for ξ + 1. Let us suppose ξ is a limit ordinal; the case in which ξ isa successor is left to the reader. Let C ξ +1 be a club subset of µ such that for all α ∈ C ξ we have(i) ( ∀ ζ ∈ F µξ ( α )) F µζ ( α ) = F µξ ( α ) ∩ ζ ,(ii) α ∈ C ξ ∩ T ζ ∈ F µξ ( α ) C ζ ,(iii) f µξ ( α ) = S ζ ∈ F µξ ( α ) f µζ ( α ) is a limit ordinal,(iv) f µξ +1 ( α ) = f µξ ( α ) + 1Fix A ⊆ µ and suppose α ∈ d ξ +1 ( A ). Then α ∈ d ξ ( A ) ∩ C ξ and hence A is f µξ ( α )-s-stationary in α . To show that A is f µξ +1 ( α ) = f µξ ( α ) + 1-s-stationary in α ,suppose S, T ⊆ α are both f µζ ( α )-s-stationary in α for some fixed ζ ∈ F µξ ( α ) ∪ { ξ } .Since α ∈ C ζ , it follows by (1) that α ∈ d ζ ( S ) ∩ d ζ ( T ). Since α ∈ d ξ +1 ( A ) and d ζ ( S ) ∩ d ζ ( T ) ∩ C ζ is an open neighborhood of α in the τ ξ +1 topology, we seethat A ∩ d ζ ( S ) ∩ d ζ ( T ) ∩ C ζ \ { α } 6 = ∅ . Let β ∈ A ∩ d ζ ( S ) ∩ d ζ ( T ) ∩ C ζ \ { α } and notice that, by hypothesis, S and T are f µζ ( β )-s-stationary in β . Thus A is f µξ +1 ( α )-s-stationary in α . (cid:3) Now we are ready to characterize the nonisolated points of the spaces ( µ ξ , τ ξ )in terms of η -s-reflecting cardinals. The following is a generalization of [Bag19,Theorem 2.11]. Theorem 6.7. Suppose µ is a regular cardinal. For all ξ < µ + there is a club C ⊆ µ such that for all α ∈ C , α is not isolated in the τ ξ topology on µ if and onlyif α is f µξ ( α ) -s-reflecting.Proof. For ξ < µ , the function f µξ : µ → µ is constantly equal to ξ and hence theresult follows directly from [Bag19, Theorem 2.11], taking C = µ .Suppose µ ≤ ξ < µ + . For each ζ < ξ let C ζ be the club obtained from Proposi-tion 6.6(1), so that d ζ ( A ) ∩ C ζ = { α < µ : A is f µζ ( α )-s-stationary in α } ∩ C ζ . Let E ξ be the club subset of µ obtained in Proposition 6.6(3). By normality of theclub filter on µ , the set E ξ ∩ △ ζ<ξ C ζ ! = { α ∈ E ξ : α ∈ \ ζ ∈ F µξ ( α ) C ζ } contains a club subset of µ . It follows from Lemma 2.5, Corollary 2.10 and repeatedapplication of Lemma 2.8, that we may let C ⊆ E ξ ∩ (cid:16) △ ζ<ξ C ζ (cid:17) be a club subsetof µ such that for all α ∈ C we have(i) ( ∀ ζ ∈ F µξ ( α )) F µζ ( α ) = F µξ ( α ) ∩ ζ and(ii) if α is regular and uncountable then for all ζ ∈ F µξ ( α ) ∪ { ξ } there is a clubsubset of α , call it C αζ , contained in the set { β < α : f µζ ( β ) = f αf µζ ( α ) ( β ) } .Now suppose α ∈ C and let us show that α is not isolated in the τ ξ topology on µ if and only if α is f µξ ( α )-s-reflecting.Suppose α is not f µξ ( α )-s-reflecting. Then for some η < f µξ ( α ) there are sets S, T ⊆ α which are η -s-stationary in α but for all β < α the sets S ∩ β and T ∩ β arenot f αη ( β )-s-stationary in β . Since f µξ ( α ) equals the transitive collapse of F µξ ( α ), itfollows by (i), that η = f µζ ( α ) for some ζ ∈ F µξ ( α ). Thus S, T ⊆ α are sets whichare f µζ ( α )-s-stationary in α , but for all β < α the sets S ∩ β and T ∩ β are not f αf µζ ( α ) ( β ) = f µζ ( β )-s-stationary. Since α ∈ C ⊆ △ ζ<ξ C ζ , it follows by Proposition6.6 that C ∩ d ζ ( S ) ∩ d ζ ( T ) = { α } and since ζ < ξ the set C ∩ d ζ ( S ) ∩ d ζ ( T ) is an open neighborhood of α in the τ ξ topology.Conversely, suppose α is f µξ ( α )-s-reflecting and let α ∈ U where U is a basicopen set in τ ξ . Then U is of the form U = I ∩ d ζ ( A ) ∩ · · · ∩ d ζ ( A n − ) , where ζ < ξ and I is an open interval. Since I ∩ α is a club subset of α , it followsby Lemma 6.5 and Proposition 6.6(3) that U is f µξ ( α )-s-stationary in α . Thus U \ { α } 6 = ∅ . (cid:3) In order to show that Π ξ -indescribability can be used to obtain the nondiscrete-ness of the topologies τ ξ on µ for ξ < µ + , we need the following expressibilityresult. Lemma 6.8. Suppose κ is a regular cardinal. For all ξ < κ + there is a formula Π ξ formula ϕ ξ ( X ) over V κ and a club C ξ subset of κ such that for all A ⊆ κ we have A is ξ -s-stationary in κ if and only if V κ | = ϕ ξ ( A ) and for all α ∈ C ξ we have A is f κξ ( α ) -s-stationary in α if and only if V α | = ϕ ξ ( A ) | κα Proof. We follow the proof of [Bag19, Proposition 4.3] and proceed by inductionon ξ . We let ϕ ( X ) be the natural Π formula asserting that X is 0-s-stationary(i.e. unbounded) in κ .Suppose ξ < κ + is a limit ordinal and the result holds for ζ < ξ . We let ϕ ξ ( X ) = ^ ζ<ξ ϕ ζ ( X ) . Notice that Lemma 2.5 implies that ( ∀ ζ ∈ F µξ ( α )) F µζ ( α ) = F µξ ( α ) ∩ ζ holds for club-many α < µ when ξ is a limit. Using Lemma 2.5 and a repeated application of Lemma 2.8 one can showthat this same equation holds for club-many α < µ for all ordinals ξ < µ + . IGHER INDESCRIBABILITY AND DERIVED TOPOLOGIES 43 Clearly ϕ ξ ( X ) is Π ξ over V κ . By Lemma 2.6, the set D = { α < κ : f κξ ( α ) = [ ζ ∈ F κξ ( α ) f κζ ( α ) } is in the club filter on κ , and by normality so is the set C ξ = D ∩ ( △ ζ<ξ C ζ ) = D ∩ { α < κ : α ∈ \ ζ ∈ F κξ ( α ) C ζ } . Suppose A ⊆ κ . Using our inductive assumption about the ϕ ζ ’s, it is easy to verifythat A is ξ -s-stationary in κ if and only if V κ | = ϕ ξ ( A ). Now suppose α ∈ C ξ . Wehave ϕ ( A ) | κα = ^ ζ ∈ F κξ ( α ) ( ϕ ζ ( A ) | κα ) . Suppose V α | = ϕ ( A ) | κα . Then, for all ζ ∈ F κξ ( α ) we have that A ∩ α is f κζ ( α )-s-stationary in α , and since α ∈ D , it follows that A ∩ α is f κξ ( α )-s-stationary in α .Similarly, if we assume A ∩ α is f κξ ( α )-s-stationary in α , it is easy to verify that V α | = ϕ ( A ) | κα .Suppose ξ = ζ + 1 < κ + is a successor. We let ϕ ζ +1 ( X ) be the formula ^ η<ζ ϕ η ( X ) ∧ ∀ S ∀ T ( ϕ ζ ( S ) ∧ ϕ ζ ( T ) → ( ∃ β ∈ A )( S and T are f κζ ( β )-s-stationary in β )) . Note that, by an argument similar to that for Theorem 5.15(2), we can code infor-mation about which subsets of β are f κζ ( β )-s-stationary in β into a subset of κ andverify that ϕ ζ +1 ( X ) is Π ζ +1 over V κ . As we proceed, let us consider the case inwhich ζ is a limit ordinal; the case in which ζ is a successor is easier. By Lemma2.6 the set D = { α < κ : f κζ ( α ) = [ η ∈ F κζ ( α ) f κη ( α ) } is in the club filter on κ . Furthermore, by Corollary 2.10, there is a club C κζ in κ such that for all regular α ∈ C κζ the set D αζ = { β < α : f κζ ( β ) = f αf κζ ( α ) ( β ) } is in the club filter on β . By normality there is a club subset of κ , call it C ξ +1 contained in D ∩ C κζ ∩ C ζ ∩ ( △ η<ζ C η ) , where C ζ is the club subset of κ obtained from the inductive hypothesis. We leaveit to the reader to verify that C ξ +1 is the desired club subset of κ . (cid:3) The next proposition, which is a generalization of [Bag19, Proposition 4.3], willallow us to obtain the nondiscreteness of the topologies τ ξ from an indescribabilityhypothesis. Proposition 6.9. If a cardinal κ is Π ξ -indescribable for some ξ < κ + , then it is ( ξ + 1) -s-reflecting. Proof. Suppose κ is Π ξ -indescribable and suppose that S and T are ζ -s-stationaryin κ where ζ ≤ ξ . Then we have V κ | = ϕ ζ ( S ) ∧ ϕ ζ ( T )where ϕ ζ ( X ) is the Π ζ formula obtained in Lemma 6.8. Let C ζ be the club subsetof κ from the statement of Lemma 6.8. Since κ is Π ξ -indescribable, there is an α ∈ C ζ such that V α | = ϕ ζ ( S ) | κα ∧ ϕ ζ ( T ) | κα , which implies that S and T are both f κζ ( α )-s-stationary in α . Hence κ is ( ξ + 1)-s-stationary. (cid:3) Finally, we show that from an indescribability hypothesis, one can prove thatthe τ ξ +1 topology is not discrete. Corollary 6.10. Suppose µ is a regular cardinal and ξ < µ + . If the set S = { α < µ : α is f µξ ( α ) -indescribable } is stationary in µ (for example, this will occur if µ is Π ξ +1 -indescribable), thenthere is an α < µ which is nonisolated in the space ( µ, τ ξ +1 ) .Proof. Fix ξ < µ + . Let C be the club subset of µ obtained from Theorem 6.7; thatis, C ⊆ µ is club such that for all α ∈ C we have α is f µξ ( α )-s-reflecting if and onlyif α is not isolated in the τ ξ topology. Let D = { α < µ : f µξ +1 ( α ) = f µξ ( α ) + 1 } be the club subset of µ obtained from Lemma 2.8. Now, if α ∈ S ∩ C then α is f µξ +1 ( α )-s-reflecting and is hence not isolated in the τ ξ +1 topology. (cid:3) References [Bag19] Joan Bagaria. Derived topologies on ordinals and stationary reflection. Trans. Amer.Math. Soc. , 371(3):1981–2002, 2019.[Bar75] K. Jon Barwise. Mostowski’s collapsing function and the closed unbounded filter. Fund.Math. , 82:95–103, 1974/75.[BMM20] Joan Bagaria, Menachem Magidor, and Salvador Mancilla. The consistency strength ofhyperstationarity. J. Math. Log. , 20(1):2050004, 35, 2020.[BMS15] Joan Bagaria, Menachem Magidor, and Hiroshi Sakai. Reflection and indescribabilityin the constructible universe. Israel J. Math. , 208(1):1–11, 2015.[CGLH] Brent Cody, Victoria Gitman, and Chris Lambie-Hanson. Forcing a (cid:3) ( κ )-like principleto hold at a weakly compact cardinal. ( Accepted at Annals of Pure and Applied Logic ,available at https://arxiv.org/abs/1902.04146 ).[Cod] Brent Cody. A refinement of the Ramsey hierarchy via indescribability. ( Accepted atJournal of Symbolic Logic , available at https://arxiv.org/abs/1907.13540 ).[Cod19] Brent Cody. Adding a Nonreflecting Weakly Compact Set. Notre Dame J. Form. Log. ,60(3):503–521, 2019.[Cod20] Brent Cody. Characterizations of the weakly compact ideal on P κ λ . Ann. Pure Appl.Logic , 171(6):102791, 2020.[CS20] Brent Cody and Hiroshi Sakai. The weakly compact reflection principle need not implya high order of weak compactness. Arch. Math. Logic , 59(1-2):179–196, 2020.[Dic75] M. A. Dickmann. Large infinitary languages . North-Holland Publishing Co.,Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. Modeltheory, Studies in Logic and the Foundations of Mathematics, Vol. 83.[For10] Matthew Foreman. Ideals and generic elementary embeddings. In Handbook of set the-ory. Vols. 1, 2, 3 , pages 885–1147. Springer, Dordrecht, 2010.[GH75] Fred Galvin and Andr´as Hajnal. Inequalities for cardinal powers. Ann. of Math. (2) ,101:491–498, 1975. IGHER INDESCRIBABILITY AND DERIVED TOPOLOGIES 45 [Hel03] Alex Hellsten. Diamonds on large cardinals. Ann. Acad. Sci. Fenn. Math. Diss. ,(134):48, 2003. Dissertation, University of Helsinki, Helsinki, 2003.[Jec10] Thomas Jech. Stationary sets. In Handbook of set theory. Vols. 1, 2, 3 , pages 93–128.Springer, Dordrecht, 2010.[Kan03] Akihiro Kanamori. The higher infinite . Springer Monographs in Mathematics. Springer-Verlag, Berlin, second edition, 2003. Large cardinals in set theory from their beginnings.[Kue77] David W. Kueker. Countable approximations and L¨owenheim-Skolem theorems. Ann.Math. Logic , 11(1):57–103, 1977.[L´ev71] Azriel L´evy. The sizes of the indescribable cardinals. In Axiomatic Set Theory (Proc.Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967) ,pages 205–218. Amer. Math. Soc., Providence, R.I., 1971.[Sun93] Wen Zhi Sun. Stationary cardinals. Arch. Math. Logic , 32(6):429–442, 1993.[SW11] I. Sharpe and P. D. Welch. Greatly Erd¨os cardinals with some generalizations to theChang and Ramsey properties. Ann. Pure Appl. Logic , 162(11):863–902, 2011.(Brent Cody) Virginia Commonwealth University, Department of Mathematics andApplied Mathematics, 1015 Floyd Avenue, PO Box 842014, Richmond, Virginia 23284,United States Email address , B. Cody: [email protected] URL ::