aa r X i v : . [ m a t h . L O ] A ug Generalized Silver and Miller measurability
Giorgio LaguzziAugust 10, 2020
Abstract We present some results about the burgeoning research area concern-ing set theory of the “ κ -reals”. We focus on some notions of measurabilitycoming from generalizations of Silver and Miller trees. We present analo-gies and mostly differences from the classical setting. The study of the generalized version of the Baire space κ κ and Cantor space2 κ , for κ uncountable regular cardinal, is a burgeoning research area, whichintersects both the generalized descriptive set theory and the set theory of the“ κ -reals”, where we refer to the elements of κ κ and 2 κ as κ -reals. Basic Notation.
The dramatis personae of our work are the so-called tree-likeforcings. A tree T is a subset of either 2 <κ or κ <κ , closed under initial segments. Stem ( T ) denotes the longest node of T compatible with all the other nodes of T ; Succ ( t, T ) := { ξ < κ : t a ξ ∈ T } ; Split (T) is the set of splitting nodes of T ;we put ht ( T ) := sup { α : ∃ t ∈ T ( | t | = α ) } , while Term ( T ) denotes the terminal nodes of T . For α < κ , T ↾ α := { t ∈ T : | t | < α } . A branch through T is thelimit of an increasing cofinal sequence { t ξ : ξ < κ } of nodes in T , and [ T ] willdenote the set of all branches of T . Moreover we will assume κ <κ = κ and that κ is regular. Note that we will use the usual letters for denoting forcing notionslike Sacks, Silver, Miller and Cohen, omitting the symbol κ , as it is clear that,in this paper, we will always deal with some generalized version.Our attention will be particularly focused on the following types of trees: • T ⊆ <κ is called Sacks iff ∀ t ∈ T ∃ t ′ ∈ T ( t ⊆ t ′ ∧ t ′ ∈ Split ( T )) (we write T ∈ S ); • T ⊆ <κ is club Sacks iff it is Sacks and for every x ∈ [ S ] we have { α < κ : x ↾ α ∈ Split ( T ) } is closed unbounded (we write T ∈ S club );analogously we define S stat by requiring { α < κ : x ↾ α ∈ Split ( T ) } to bestationary; • T ⊆ <κ is Silver iff it is Sacks and moreover for every s, t ∈ T such that | s | = | t | one has s a i ⇔ t a i , for i ∈ { , } (we write T ∈ V ); Acknowledgement.
The author is particularly indebted to Philipp Schlicht, since someof the ideas in this paper were inspirired by his seminar given in Freiburg during January2013. Further, the author is grateful to Luca Motto Ros, for a detailed explanation of thedifferences and analogies between the Miller measurability and Hurewicz dichotomy, whichcan be found in Remark 29 of the present paper. Finally, the author thanks the referee forhis/her useful suggestions. T ⊆ <κ is club Silver iff it is Silver and Lev ( T ) := { α < κ : ∃ t ∈ T ( t ∈ Split ( T )) } is closed unbounded (we write T ∈ V club ); analogously for V stat ; • T ⊆ κ <κ is called Miller iff ∀ t ∈ T ∃ t ′ ∈ T ( t ⊆ t ′ ∧ t ′ ∈ Split ( T ) ∧| Succ ( t, T ) | = κ ) (we write T ∈ M ); • T ⊆ κ <κ is called club Miller ( T ∈ M club ) iff it is Miller and the followinghold: – for every x ∈ [ T ] one has { α < κ : x ↾ α ∈ Split ( T ) } is closedunbounded, – for every t ∈ Split ( T ) one has { α < κ : t a α ∈ T } is closed un-bounded. • T ⊆ κ <κ is called full Miller ( T ∈ M full ) iff it is Miller and for every t ∈ Split ( T ) for every α < κ , one has t a α ∈ T .The associated forcing notions are ordered by inclusion. We remark that astronger version of M club has been introduced by Friedman and Zdomskyy in[2], where they proved that such a version, combined with club Sacks, preserves κ + via < κ -support iteration.For our tree-forcings P , one can introduce a corresponding notion of regularityas follows. Definition 1.
A set X of κ -reals is said to be:- P - measurable iff ∀ T ∈ P ∃ T ′ ∈ P , T ′ ⊆ T ([ T ′ ] ⊆ X ∨ [ T ′ ] ∩ X = ∅ ) . - weakly - P - measurable iff ∃ T ∈ P ([ T ] ⊆ X ∨ [ T ] ∩ X = ∅ ) . When P is one of our tree forcings, we may also use the notation “Sacks mea-surable”, “Miller measurable” and so on.In [3], the authors show that the following generalization from the classicalsetting holds true: if Θ is a family of sets of κ -reals closed under intersectionwith closed sets and continuous pre-images, then ∀ X ∈ Θ( X is weakly- P -measurable) ⇔ ∀ X ∈ Θ( X is P -measurable) . Hence, when one investigates the validity of P -measurability for all sets in Θ, itis actually sufficient to investigate the weak- P -measurability on Θ.There are essentially two main reasons for which the investigation of regularityproperties in κ κ is interesting and more involved than the classical setting:1. the club filter is a Σ set without the Baire property, as was proved byHalko and Shelah in [4];2. there is not an analogue of the factoring lemma for the Levy collapse Coll ( κ, < λ ), for κ > ω and λ inaccessible. More precisely, there are x ∈ κ κ such that Coll ( κ, < λ ) /x is not equivalent to Coll ( κ, < λ ).e will call such reals bad , while on the opposite side, the good reals will bethose having quotients equivalent to the Levy collapse.Actually 2 is true even for the κ -Cohen forcing C , as it is well-know that onecan pick a Cohen κ -real x and then a forcing P shooting a club through thecomplement of x , and this two step iteration is equivalent to C . Hence, both C and Coll ( κ, < λ ) are not strongly homogeneous, unlike their counterparts in thestandard setting.Nevertheless, even if one cannot hope for a full factoring lemma, in [10] PhilippSchlicht has shown that one can recover a partial version. Indeed, he has proventhat when forcing with Coll ( κ, < λ ), one can obtain perfectly many good reals,in a sense, in order to use the usual Solovay’s argument and obtain that all On κ -definable subsets of 2 κ have the perfect set property. Inspired by his method,we prove some variants that will allow us to get the following two results:( ⋆ ) for κ inaccessible, a κ + -iteration of C with < κ -support forces all On κ -definable sets to be V stat -measurable;( ⋆⋆ ) for λ inaccessible, Coll ( κ, < λ ) forces all On κ -definable sets to be M full -measurable;Furthermore, we will also prove that ( ⋆ ) is no longer true when one replaces V stat with V club . We conclude this introductory section with a schema of thepaper: in section 2 we show some interesting construction involving Sacks treesand Miller trees, marking some difference from the standard setting; in section3 we present some results concerning adding perfect trees of Cohen branches; insection 4 we build the model to get all On κ -definable subsets of 2 κ to be V stat -measurable; in section 5, we prove that Coll ( κ, < λ ) forces all On κ -definablesubsets of κ κ to be M full -measurable; a concluding section is finally devoted todiscussing some further potential developments. This section may be read independently from the rest of the paper. It is devotedto analyzing some basic differences from the standard setting. Throughout thissection, we assume that κ is a regular successor. Let Γ := { λ α : α < κ } besuch that λ = 0 and { λ α : 1 ≤ α < κ } enumerates the limit ordinals < κ suchthat 2 λ α = κ . For t ∈ λ α and α < κ , let π ( t, α + 1) := { t ′ ∈ λ α +1 : t ′ ⊇ t } .Furthermore, fix a well-ordering W ( t, α + 1) = { t α +1 ξ : ξ < κ } of π ( t, α + 1).In the standard case when κ = ω , we know that 2 ω and ω ω are not homeo-morphic, even if they are connected via a Borel isomorphism. The followingsimple remark shows that the situation is different, when κ > ω is a successor.The following result was proved in [5]. We give a sketch of the proof, since theconstruction is needed later. Remark 2. κ and κ κ are homeomorphic. Moreover, there are many suchhomeomorphisms. Indeed, let f : κ <κ → <κ be defined recursively as follows:- f ( ∅ ) = ∅ - f ( h ξ i ) is the ξ th element of the well-ordering W ( ∅ , t ∈ κ α and ξ < κ , f ( t a ξ ) is the ξ th element in the well-ordering W ( f ( t ) , α + 1). given { t α : α < γ } increasing sequence, γ limit ordinal, put f (lim α<γ t α ) =lim α<γ f ( t α ).Notice that the range of f is not 2 <κ , but it is a strict subset of it, namely S { t ∈ λ α : λ α ∈ Γ } . This f provides a bijection h : κ κ → κ in the naturalway, that is h ( x ) = lim α<κ f ( x ↾ α ), for every x ∈ κ κ . It easily follows from thedefinition that h is a homeomorphism.The homeomorphism obviously depends on the well-orderings of W ( t, α ), andso it is far from being uniquely determined. We now want to use these home-omorphisms between κ κ and 2 κ to exhibit some particular situations which donot occur in the standard setting. Fact 3.
For every club Miller tree T ⊆ κ <κ with the property that for every t ∈ T , |{ ξ : t a ξ / ∈ T }| = κ , there exists a homeomorphism h such that h ′′ [ T ] does not contain the branches of a club Sacks tree.Proof. Let T ⊆ κ <κ be a club Miller tree. Instead of h , we actually define thefunction f : κ <κ → <κ , from which we will naturally obtain the desired h . Forevery club splitting node t ∈ T we define f satisfying the following requirement:let C t ⊆ κ denote the club set of successors of t , then ξ ∈ C t ⇒ f ( t a ξ ) ⊇ f ( t ) a α and every t ∈ f ′′ T with length λ α , we get that f ( t ) cannot be a splitting node. Hence, f ′′ T cannot contain aclub Sacks subtree. Lemma 4.
There exist a homeomorphism h : κ κ → κ and Y ⊆ κ κ such that Y is weakly club Miller measurable but h ′′ Y is not weakly club Sacks measurable.Proof. Consider f , h and T as in Fact 3. Note that, for every club Sacks tree S , [ S ] \ h ′′ [ T ] has cardinality 2 κ . This follows easily from Fact 3, since thereis α ∈ κ and t ∈ λ α such that t a ∈ S ∧ f − ( t a / ∈ T (actually there arecofinally many such α ’s).Let { S ξ : ξ < κ } be an enumeration of all club Sacks trees. Now we construct { Y ξ : ξ < κ } and { Z ξ : ξ < κ } recursively as follows:- Step − Y = [ T ] and Z = ∅ ;- Step ξ successor or ξ = 0: pick y ξ ∈ h − [ S ξ ] \ S ι ≤ ξ Y ι and put Y ξ +1 = Y ξ ∪ { y ξ } . Then pick z ξ ∈ h − [ S ξ ] \ S ι ≤ ξ Z ι such that z ξ / ∈ Y ξ +1 , and put Z ξ +1 = Z ξ ∪ { z ξ } . Note that the choice of y ξ can be done, since by Fact3 any club Sacks set contains 2 κ many branches which are not in h ′′ [ T ].- Step ξ limit: put Y ξ = S ι<ξ Y ι and Z ξ = S ι<ξ Z ι .Finally put Y = S ξ< κ Y ξ and Z = S ξ< κ Z ξ . Then for all club Sacks tree S both h ′′ Y ∩ [ S ] = ∅ and [ S ] * h ′′ Y (the latter because Z ∩ Y = ∅ .)On the opposite side, we have the following. Lemma 5.
Assume f : κ <κ → <κ is a map as above and satisfying thefollowing further property: for every α < κ and every t ∈ λ α , ( † ) f − { t ′ ∈ λ α +1 : t a ⊂ t ′ } , f − { t ′ ∈ λ α +1 : t a ⊂ t ′ } are stationary . Then for every club Miller tree T we have that f ′′ T contains a club Sacks tree.roof. Indeed, we are going to prove the following stronger conclusion: let S ∗ , club be the version of club Sacks forcing obtained by replacing 2 <κ with κ <κ , i.e., S ∗ , club consists of 2-branching trees in κ <κ with club splitting. Then we are goingto prove that for every T ∈ M club there exists T ⊂ T in S ∗ , club and S ∈ S club such that f ′′ T = S . We recursively construct T as follows. Step 0.
Let t ∅ = Stem ( T ) and put α ∅ = | t ∅ | . Then pick ξ , ξ ∈ Succ ( t ∅ ) suchthat f ( t ∅ a ξ ) ⊃ f ( t ∅ ) a f ( t ∅ a ξ ) ⊃ f ( t ∅ ) a
1; note that this can be doneby ( † ), since Succ ( t ∅ , T ) is club. Further, for i ∈ { , } , let t h i i be the leastsplitting node in T t a ∅ i := { s ∈ T : s ⊆ t ∅ a i ∨ s ⊇ t ∅ a i } . Successor step.
Assume the construction done for all σ ∈ β . For every σ ∈ β ,we use the same idea as step 0, and we pick ξ , ξ ∈ Succ ( t σ ) such that, for i ∈ { , } , f ( t σ a ξ i ) ⊃ f ( t σ ) a i . Analogously, t σ a i is the least splitting node in T t a σ i . Limit step.
For σ ∈ δ such that, for all β < δ , t σ ↾ β is already constructed, put t σ := S β<δ t σ ↾ β .Finally let T be the downward closure of S σ ∈ <κ t σ and S := f ′′ T . By con-struction, T and S have the required properties. Remark 6.
In a sense, the situation occurring in Lemma 4 is very unpleasant,as we would like to generally view Miller trees as particular kind of Sacks trees,and moreover that this fact is preserved under homeomorphism. Hence, Lemma4 and 5 may be understood as a way of separating good homeomorphisms frombad ones.
We present some results about adding certain types of generic perfect trees. Insection 4 and 5, it will be crucial to use specific kinds of perfect trees such thateach of their branches is Cohen over the ground model. We refer to such generictrees by saying “perfect trees of Cohen branches”.
Lemma 7.
Let κ be inaccessible. Let VT := { p : ∃ T ∈ V ∃ α ∈ κ ( p = T ↾ α ) } ,ordered by end-extension, i.e., p ′ ≤ p iff p ⊆ p ′ ∧∀ t ∈ p ′ \ p ∃ s ∈ Term ( p )( s ⊆ t ) .Let T G := S { p : p ∈ G } , with G being VT -generic filter over the ground model N . Then N [ G ] | = T G ∈ V ∧ ∀ x ∈ [ T G ]( x is Cohen over N ) ∧ Lev ( T G ) is stationary and co-stationary , where Lev ( T G ) denotes the set of splitting levels of T G . Moreover, VT is aforcing of size κ and is < κ -closed. So it is actually equivalent to κ -Cohenforcing.Proof. Fix p ∈ VT and D ⊆ C open dense and let { t α : α < δ < κ } , enumerateall terminal nodes of p (w.l.o.g. assume δ is a limit ordinal). We use the followingnotation: for every s, t ∈ <κ , put t ⊕ s := { t ′ ∈ <κ : ∀ α < | t | ( t ′ ( α ) = t ( α )) ∧ ∀ α ≥ | t | ( t ′ ( α ) = s ( α )) } . Then consider the following recursive construction:- pick s ⊇ t such that s ∈ D ;- for α + 1, pick s α +1 ⊇ t α +1 ⊕ s α such that s α +1 ∈ D . for α limit, put s ′ α = S ξ<α s ξ and pick s α ⊇ t α ⊕ s ′ α such that s α ∈ D .- once the procedure has been done for every α < δ , we put s δ := S α<δ t ⊕ s α and then t ′ α := t α ⊕ s δ .Note that to make sure that t α ⊕ s δ ∈ <κ we need to use the assumptionthat κ is inaccessible. Finally, let p ′ be the downward closure of S α<δ t ′ α . Byconstruction, p ′ ∈ VT , p ′ ≤ p and for every terminal node t ∈ p ′ , we get t ∈ D .Hence p ′ (cid:13) ∀ x ∈ [ T G ]( H x ∩ D = ∅ ), where H x := { s ∈ C : s ⊂ x } .We now want to further extend p ′ in order to catch the second property as well,i.e., Lev ( T G ) is both stationary and co-stationary. So fix ˙ C name for a clubof κ . Build sequences { q n : n ∈ ω } and { ξ n : n ∈ ω } such that: q = p ′ , and q n +1 ≤ q n such that q n +1 (cid:13) ξ n ∈ ˙ C and ξ n > ht ( q n ) and ht ( q n +1 ) > ξ n . Finallyput ξ ω = lim n<ω ξ n , q ω := S n ∈ ω q n , and then p ∗ := q ω ∪ [ { t a i : t ∈ Term ( q ω ) ∧ i ∈ { , }} . Hence p ∗ (cid:13) ∀ n ( ξ n ∈ ˙ C ), and then p ∗ (cid:13) ξ ω ∈ ˙ C . But ξ ω = ht ( q ω ), since the ξ n ’s and the | ht ( q n ) | ’s are mutually cofinal, and hence p ∗ (cid:13) ξ ω ∈ Lev ( T G ) ∩ ˙ C .This shows that Lev ( T G ) is stationary. For proving that it is co-stationaryas well, we can further extend p ∗ , by using the same procedure, in order tofind { q ′ n : n ∈ ω } and { ξ ′ n : n ∈ ω } as above and then p ∗∗ ≤ q ′ ω such that p ∗∗ := q ′ ω ∪ S { t a t ∈ Term ( q ′ ω ) } . Hence p ∗∗ (cid:13) ξ ω ∈ Lev ( T G ) ∩ ˙ C ∧ ξ ′ ω / ∈ Lev ( T G ) ∩ ˙ C, which completes the proof.About generic Miller trees of Cohen branches the situation is very different,since the above argument does not seem to work. The next method shows asimple different way to add a tree T ∈ M full of Cohen branches, which we will usein section 5. On the opposite side, Lemma 10 marks a necessary condition foradding a tree T ∈ M club of Cohen branches, generalizing some results obtainedby Spinas and Brendle in the classical setting (see [11] and [1]).We use the following notation: given a tree T ⊆ κ <κ ,- Split α ( T ) is recursively defined as: Split ( T ) = { Stem ( T ) } ; t ∈ Split α ( T ) iff t ∈ Split ( T ) and for every β < α there exists t β ⊂ t such that t β ∈ Split β ( T ).- T [ α ] := { s ∈ T : ∃ t ∈ Split α ( T ) ∃ i < κ ( t a i ∈ T ∧ s ⊆ t a i ) } . Lemma 8.
Define the forcing MT := { p : ∃ T ∈ M full ∃ α < κ ( p ⊒ T [ α ]) } ,ordered by end-extension. Then MT adds a full Miller tree of Cohen branches.Proof. Let D ⊆ C be open dense and p ∈ MT . Pick φ : Term ( p ) → κ <κ suchthat φ ( t ) ∈ D and φ ( t ) ⊇ t , and then define p ′ ≤ p as the downward closure of S { φ ( t ) a ξ : t ∈ Term ( p ) ∧ ξ ∈ κ } . Then p ′ (cid:13) ∀ x ∈ [ T G ]( H x ∩ D = ∅ ), where H x := { s ∈ C : s ⊂ x } . Remark 9.
Note that in the proof of both Lemma 7 and Lemma 8, we haveproved that one can add a certain type of generic tree whose branches are Cohenin the extension N [ G ], where G is VT - and MT -generic over N , respectively. Inthe application that we will see in the next sections, we actually need somethingtronger, i.e., that all branches of the generic tree have to be Cohen in any extension M ⊇ N [ G ] via a < κ -closed forcing. But, this is actually implicit inour proof. Indeed, in Lemma 8 we have proven that, for every D ⊆ C opendense in N , N [ G ] | = ϕ : ≡ ∃ F ⊆ κ <κ ∀ x ∈ κ κ ( x ∈ [ T G ] ⇒ ∃ t ∈ F ( t ⊂ x ∧ t ∈ D )) , and analogously for Lemma 7 with 2 κ in place of κ κ . Note that this formula ϕ is Σ ( κ κ ). Hence, it is upward absolute between N [ G ] and any extension M via <κ -closed forcing (this to ensure ( κ <κ ) M = ( κ <κ ) N [ G ] and then Σ -absoluteness).Hence, we get M | = ϕ , which means M | = ∀ x ∈ [ T G ]( H x ∩ D = ∅ ). Since D ∈ N was arbitrarily chosen, we have obtained: for every D ⊆ C ∩ N , for every x ∈ [ T G ] M , one has H x ∩ D = ∅ . Hence, M | = ∀ x ∈ [ T G ]( x is Cohen over N ). Lemma 10.
Let M be a ZFC-model extending the ground model N . If for all x ∈ κ κ ∩ M there exists y ∈ κ κ ∩ N such that ∀ α < κ ∃ β ≥ α ( x ( β ) < y ( β )) ,then in M there is no club Miller tree of Cohen branches. In other words, If oneadds a club Miller tree of Cohen branches, then one necessarily adds dominating κ -reals over the ground model.Proof. Let T ∈ M club and t ∈ Split ( T ). Define h t ( α ) := min {| t ′ | : ∃ ξ ≥ α ( t ′ ∈ Split ( T ) ∧ t ′ ⊇ t a ξ ) } + 1 . Further, given z ∈ κ ↑ κ ∩ N , define B ( z ) := { x ∈ κ κ : ∀ µ ∀ α ≤ µ ( z ( x ( α )) ≥ µ ) } . Claim 11. B ( z ) is closed nowhere dense.Proof of Claim. To see that B ( z ) is nowhere dense, fix s ∈ κ <κ . Then let s ′ = s a β , where 0 β is the sequence of 0s of length β , and β is sufficiently largethat | s ′ | > sup { z ( s ( α )) : α < | s |} . Hence [ s ′ ] ∩ B ( z ) = ∅ .Let T ∈ M club ∩ M be a tree of Cohen branches over N . Pick h ∈ κ κ suchthat ∀ t ∈ Split ( T ) ∃ α < κ ∀ ξ ≥ α ( h t ( ξ ) < h ( ξ )). To show that h is dominatingover N , we argue by contradiction; pick z ∈ κ ↑ κ ∩ N which is not eventuallydominated by h , and with the further property that z (0) > | Stem ( T ) | . Let usconstruct { t ξ : ξ < κ } recursively as follows: • t = Stem ( T ) and for λ limit ordinal let t λ = S ξ<λ t ξ . • Assume t ξ already defined. By the choice of z , there exists β ∈ κ suchthat h ( β ) < z ( β ). We distinguish two cases: – if t ξ a β ∈ T , then simply put t ξ +1 be the least splitting node extend-ing t ξ a β ; – if t ξ a β / ∈ T , then let γ ξ := min { γ : γ > β ∧ t ξ a γ ∈ T } . By construc-tion, h ( γ ξ ) = h ( β ) and so h ( γ ξ ) < z ( β ) ≤ z ( γ ξ ), since z is increasing.Then let t ξ +1 be the least splitting node of T extending t ξ a γ ξ .Note that when ξ is limit, by recursive construction, t ξ ∈ Split ( T ), as t ξ is a limit of splitting nodes in T . Hence the construction works evenfor ξ successor of a limit ordinal. Finally let x = S ξ<κ t ξ . It is left toshow that x ∈ [ T ] ∩ B ( z ), which will give us x ∈ [ T ] not Cohen over N ,since B ( z ) ∈ N is nowhere dense. Clearly x ∈ [ T ], since the constructionexplicitely gives us cofinally many α < κ such that x ↾ α ∈ T . To showthat x ∈ B ( z ), we argue as follows: for every α < κ , pick the least ξ < κ such that α < | t ξ | . By induction over ξ < κ : ξ = 0: for every α < | Stem ( T ) | , we have z ( x ( α )) > | Stem ( T ) | ; – ξ limit ordinal: trivial; – ξ + 1: if α < | t ξ | use inductive hypothesis. If | t ξ | ≤ α < | t ξ +1 | , then x ( | t ξ | ) = t ξ +1 ( | t ξ | ) = γ ξ , and so by choice of γ ξ , it follows that forevery α < | t ξ +1 | , z ( x ( α )) ≥ z ( γ ξ ) > | t ξ +1 | , since z is increasing. Corollary 12. C does not add a generic T ∈ M club of Cohen branches. Silver forcing may be introduced by using partial functions f : κ →
2, orderedby extension; simply identify such an f with the tree T f := { x ∈ κ : ∀ α ∈ dom( f )( f ( α ) = x ( α )) } . We will use T and f T interchangeably, depending onthe situation. Throughout this section, T f will denote the tree associated witha given f , and vice versa, f T will denote the partial function associated with agiven T . Note that dom( f ) = κ \ Lev ( T G ).In this section we want to investigate the family of V club -measurable and V stat -measurable sets. Lemma 13.
There exists a Σ set which is not V club -measurable (i.e., the clubfilter Cub ). Lemma 14.
Assume κ be inaccessible. Let C κ + be a κ + -iteration of κ -Cohenforcing with < κ support, and let G be the C κ + -generic filter over N . Then N [ G ] | = “ all On κ -definable sets in κ are V stat -measurable. ” Notation : we will abuse notation by saying that “ x ∈ κ is in Cub ”, instead ofthe more correct “ { α < κ : x ( α ) = 1 } is in Cub ”.We start with the proof of the easier of the two lemmata.
Proof of Lemma 13.
We will show that for every T ∈ V club , ∃ x ∈ κ ( x ∈ Cub ∩ [ T ]) ∧ ∃ y ∈ κ ( y ∈ NS ∩ [ T ]) , where NS is the ideal of non-stationary subsets of κ . Define x ∈ κ as follows: x ( α ) := ( f T ( α ) if α ∈ dom( f T ) , . Then obviously x ⊇ Lev ( T ) and so x ∈ Cub ∩ [ T ]. Analogously, we can define y ( α ) := ( f T ( α ) if α ∈ dom( f T ) , . Hence, y ∈ NS ∩ [ T ].The rest of this section is devoted to prove Lemma 14. We use a variant ofSchlicht’s method to only work with branches having good quotient. We needthe following key lemma. Hereafter, VT α denotes the < κ -support α -iterationof VT , introduced in section 3. emma 15. Let α < κ + . Let ˙ T be the canonical VT -name for the genericSilver tree added by VT , and ˙ x be a VT α -name for a Cohen branch through ˙ T . Let G be the VT α -generic filter over N and z = ˙ x G . Then VT α / ˙ x = z isequivalent to VT α . Note that, unlike Schlicht’s work, here the name for a branch comes from a“larger” forcing than the one adding the generic tree. So we need a slightgeneralization of his argument.
Notation : from now on, ˙ x , ˙ T , G will be as in the statement of Lemma 15, while x p will denote the initial segment of ˙ x decided by p := h ˙ p ( ξ ) : ξ < α i ∈ VT α .We prove some preliminary results. Claim 16. VT ∗ α := { p ∈ VT α : | x p | ≥ ht ( p (0)) } is dense in VT α .Proof. Given p ∈ VT α we have to find p ′ ≤ p in VT ∗ α . Start with p := p and then, for every n ∈ ω , pick p n +1 ≤ p n such that | x p n +1 | > ht ( p n (0)). Let p ω := S n ∈ ω p n and w := S n ∈ ω x p n . Then w ⊆ x p ω and | w | = ht ( p ω (0)). Hence p ′ := p ω has the required property.In the following two claims, we need to work with conditions forcing ˙ x ∈ ˙ T .Note that, for every p ∈ VT ∗ α we can always find p ≤ p such that p (cid:13) ˙ x ∈ ˙ T .Hence, from now on, we will always consider conditions p sufficiently strong toforce ˙ x ∈ ˙ T . Claim 17.
For every p ∈ VT ∗ α we have | x p | = ht ( p (0)) .Proof. Note that p (cid:13) ˙ x ∈ ˙ T ∧ p (0) ⊏ ˙ T , where ⊏ means “initial segment”;hence, there exists t ∈ Term ( p (0)) such that p (cid:13) t ⊂ ˙ x . By contradiction,assume x p = t a s , for some t ∈ Term ( p (0)) and non-empty s ∈ <κ . Let S bethe downward closure of S { t ⊕ t ′ : t ∈ Term ( p (0)) } , for some t ′ ⊥ t a s with t ′ ⊃ t . Let p ′ ∈ VT α be defined as p ′ ( ι ) := ( S if ι = 0 , ˙ p ( ι ) if ι > . Then pick p ∗ ≤ p ′ such that p ∗ ∈ VT ∗ α . Since p ∗ (cid:13) S ⊏ ˙ T and | x p ∗ | ≥ ht ( S ), itfollows that t ′ ⊆ x p ∗ and so x p ∗ ⊥ x p , contradicting p ∗ ≤ p . Claim 18. ∀ p ∈ VT ∗ α ∀ s ∈ <κ ( x p ⊆ s ⇒ ∃ p ∗ ∈ VT ∗ α ( s ⊆ x p ∗ )) . Proof.
The argument is very similar to the one above. Note that for every p ∈ VT ∗ α , there exists t ∈ Term ( p (0)) such that t = x p . Pick s ∈ <κ suchthat x p ⊆ s . Let S be the downward closure of S { t ⊕ s : t ∈ Term ( p (0)) } .Define p ′ ∈ VT α as follows : p ′ ( ι ) := ( S if ι = 0 , ˙ p ( ι ) if ι > . Then pick p ∗ ∈ VT ∗ α such that p ∗ ≤ p ′ . Since p ∗ (cid:13) S ⊏ ˙ T and | x p ∗ | ≥ ht ( S ), itfollows that s ⊆ x p ′ ⊆ x p ∗ . Corollary 19.
Let D ⊆ VT ∗ α be open dense. Then W q := { x p ∈ <κ : p ∈ D ∧ p ≤ q } is dense in C below x q .roof of Lemma 15. The proof is completely analogous to the one of Schlicht for C . We give it for completeness and because we actually deal with VT α -namesfor branches in ˙ T instead of VT -names only.We will prove the lemma for VT ∗ α , but since it is forcing equivalent to VT α , thesame will hold true for the latter as well (and then even for C α ). It is well-known that VT ∗ α / ˙ x = z = VT ∗ α \ S β<γ A β , where the elements of this union arerecursively defined in N [ z ] as follows: A := { p ∈ VT ∗ α : ∃ ξ < κ ( p (cid:13) ˙ x ( ξ ) = z ( ξ )) } .A β +1 := { p ∈ VT ∗ α : ∃ D ⊆ A β open dense below p , D ∈ N } .A λ := [ β<λ A β , for λ limit ordinal , and finally γ is chosen so that A γ = A γ +1 .Note that γ = 0; by contradiction, pick p ∈ A \ A . Since p ∈ A , it followsthat there exists D ⊆ A such that D ∈ N and D is dense below p . Then theset W p := { x p ′ ∈ <κ : p ′ ∈ D ∧ p ′ ≤ p } is dense in C below x p , by Corollary 19,and so there exists p ′ ∈ D such that x p ′ ⊂ z , as z is Cohen over N (and x p ⊂ z ,by p / ∈ A ). Also since D ⊆ A , it follows p ′ ∈ A . But, by definition, p ′ ∈ A ⇔ p ′ (cid:13) ˙ x ( ξ ) = z ( ξ ) , for some ξ < κ ⇔ x p ′ z, providing us with a contradiction. Hence we get VT ∗ α / ˙ x = z = { p ∈ VT ∗ α : ∀ ξ < κ ( p (cid:13) ˙ x ( ξ ) = z ( ξ )) } = { p ∈ VT ∗ α : x p ⊂ z } , which is a < κ -closed subset of a forcing equivalent to C , and so it is in turnequivalent to C .We now have all tools needed for proving the main lemma of this section. Proof of Lemma 14.
Let X ⊆ κ be a set defined by some formula ϕ withordinal parameters and v ∈ On κ , which we may assume to be absorbed into theground model, by the κ + -cc. Also, for any x ∈ [ T G ] N [ G ] , there is α < κ + anda C α -name ˙ x for such x . Note that, by Remark 9, x is Cohen over N , and byLemma 15, ˙ x has good quotient in C α , and hence in C κ + as well. Indeed, C κ + can be viewed as Q ˙ x ∗ ˙ R ∗ ˙ R , where Q ˙ x is the forcing generated by ˙ x (and soit is equivalent to C as x is Cohen over N ), while (cid:13) Q ˙ x ˙ R ∼ = C α (that means, N [ x ] | = ˙ R x ∼ = C α ), since x has good quotient, and finally ˙ R is just a “tail” of C κ + , and so it is equivalent to C κ + itself. So let us put ˙ R = ˙ R ∗ ˙ R , so to have N [ x ] | = ˙ R x ∼ = C κ + .Let x be Cohen over N with good quotient. Then N [ x ] | = “ (cid:13) ˙ R x ϕ ( x )” or N [ x ] | = “ (cid:13) ˙ R x ϕ ( x )” . Assume the former, and put θ ( x ) := “ (cid:13) ˙ R x ϕ ( x )” Then there exists s ∈ C suchthat s (cid:13) θ ( ˙ x ). Pick T stationary-Silver tree of good Cohen branches over N such that Stem ( T ) = s . Hence, for every z ∈ [ T ], we have N [ z ] | = θ ( z ), and so N [ z ] | = “ (cid:13) ˙ R z ϕ ( z ) ” . Since any z has good quotient, it follows that ˙ R z is C κ + . That means that thereexists H filter ˙ R z -generic (i.e., C κ + -generic) over N [ z ] such that N [ z ][ H ] = N [ G ].Hence N [ G ] | = ϕ ( z ), that gives us N [ G ] | = [ T ] ⊆ X .or the case N [ x ] | = “ (cid:13) ˙ R x ϕ ( x )”, simply note that “ (cid:13) ˙ R x ϕ ( x )” is equivalentto “ (cid:13) ˙ R x ¬ ϕ ( x )”, by weak homogeneity. Hence, a specular argument providesus with T ∈ V stat such that N [ G ] | = [ T ] ∩ X = ∅ . Remark 20.
Note that
Lev ( T ) is both stationary and co-stationary. As aconsequence, [ T ] is completely disjoint both from Cub and from NS , and sothere is no contradiction with Lemma 13. A word about the Silver game.
In the classical setting one can uniformlyintroduce an unfolding game associated with any notion of regularity comingfrom a certain tree forcing (see [8]). Here, we focus on the unfolding gameconnected to the Silver measurability. To this aim we need to introduce theideal I V of Silver small sets.
Definition 21. X ⊆ κ is said to be V -null iff for all T ∈ V there exists T ′ ≤ T , T ′ ∈ V such that [ T ′ ] ∩ X = ∅ . Further, we define I V as the κ + -ideal κ + -generated by the V -null sets.For emulating the classical unfolding game, we need to satisfy, for every X ⊆ κ ,(*) if II has a winning strategy in G ( X ) then X ∈ I V ;(**) if I has a winning strategy in G ( X ) then there exists T ∈ V such that[ T ] ⊆ X .Nevertheless, in the context of 2 κ the situation seems to be less clear. In ourgeneralized setting, the output of the game has to be a κ -real, and so we considergames of length κ . The basic idea is the same as the standard case, i.e., playerI and II play conditions such that each is stronger than the previous one. But what should the rule be at limit steps? First of all, note that at limits it isnatural to pick the intersection of all previous moves, and hence we want theforcing to be < κ -closed. This forces us to work with V club . We essentially havetwo choices, depending on who chooses first at limit steps. Definition 22.
We use the following notation: T ′ (cid:22) T iff T ′ ≤ T and | Stem ( T ′ ) | > | Stem ( T ) | . Given X ⊆ κ , we define two games G ( X ) and G ( X ) of length κ as follows:for n < ω , player I chooses T n (cid:22) T n − , and player II chooses T n (cid:22) T n . Fromthe first limit ordinal, G ( X ) and G ( X ) are defined differently:- in G ( X ) player I chooses first, i.e., player I first chooses T ω (cid:22) T ξ<ω T ξ ;then player II chooses T ω (cid:22) T ω . Then the game continues by followingthis order of choice (so in particular, at any limit λ , I chooses first).- in G ( X ) the situation is reversed: player II first chooses T ω (cid:22) T ξ<ω T ξ ;then player I chooses T ω (cid:22) T ω . Then the game continues by following thisorder of choice (so in particular, at any limit λ , II chooses first).The output of the game will then be x such that { x } := T ξ [ T ξ ], and we willsay that I wins iff x ∈ X , otherwise II wins.Unfortunately, both fail to have the desired properties (*) and (**) mentionedabove. In fact, the reason for that is strictly connected to the bad behaviour of V club -measurability. emma 23. Player II has a winning strategy in G ( Cub ) , while player I hasa winning strategy in G ( Cub ) .Proof. We recursively construct the winning strategy of II in G ( Cub ) as fol-lows: we only take care of limit steps λ : if h T ξ , T ξ : ξ < λ i is the partial play,then II chooses T λ (cid:22) T ξ<λ T ξ so that for α λ := | Stem (cid:0) T ξ<λ T ξ (cid:1) | , one has | Stem ( T λ ) | > α λ ∧ Stem ( T λ )( α λ ) = 0 . (1)Note that one can make such a choice since Stem ( T ξ<κ T ξ ) is a splitting node.Let us call σ such a strategy for player II. For every T ∗ := h T ξ : ξ < κ i playof I, one has that the output produced by σ ( T ∗ ) is not in Cub , since the set of { α λ : λ < κ limit ordinal } is closed unbounded.To check the second assertion, we can analogously build the winning strategy τ for player I in G ( Cub ). Player I chooses first at limit steps λ , and so, in (1), wecan freely choose Stem ( T λ )( α λ ) = 1. In such a way, for every T ∗ := h T ξ : ξ < κ i play of II, one has the output produced by τ ( T ∗ ) is in Cub .An interesting issue might be to switch the point of view in the following sense.Define X ⊆ κ to be G i -measurable iff G i ( X ) is determined. By Lemma 23, theclub filter Cub is measurable in both cases.
Question . Can we force all On κ -definable sets to be G i -measurable? Or,in other words, can one find a model where G i ’s are determined for allOn κ -definable sets? In this section, we prove that
Coll ( κ, < λ ) forces that all On κ -definable subsetsof κ κ are M full -measurable. We assume 2 κ = κ + . Consider the forcing MT introduced in section 3, for adding a full-Miller tree of Cohen reals. Claim 24. MT is forcing-equivalent to Coll ( κ, κ ) .Proof. MT is clearly < κ -closed and has size 2 κ . Moreover, MT collapses 2 κ to κ ; in fact, for every A := { a ξ : ξ < κ } ⊆ κ of size κ , A ∈ N , the set D A := { σ ∈ MT : ∃ t ∈ Split ( σ ) ∀ ξ < κ ( t a ξ a a ξ ∈ σ ) } is open dense. Hence the function H : Split ( T G ) → κ ∩ N defined by H ( t ) := { α : ∃ ξ < κ ( t a ξ a α ) } is surjective, and so 2 κ ∩ N collapses to κ . MT is then < κ -closed, of size 2 κ , collapsing 2 κ to κ , and hence equivalent to Coll ( κ, κ ). Claim 25.
Let Q = Coll ( κ, < λ ) , and let ˙ T , ˙ x be MT ∗ Q -names for the full-Miller generic tree added by G (0) and a branch of [ ˙ T ] , respectively. There exists MT ∗ P ⊆ MT ∗ Q dense subposet such that for every ( σ, ˙ p ) ∈ MT ∗ P thereexists t ∈ Term ( σ ) such that x ( σ, ˙ p ) = t , where x ( σ, ˙ p ) is the initial segment of ˙ x decided by ( σ, ˙ p ) .Proof. First of all, we want to prove an analogue of Claim 16. More precisely, wewant to prove that the set of conditions ( σ, ˙ p ) for which there exists t ∈ Term ( σ )such that t ⊆ x ( σ, ˙ p ) is dense in MT ∗ Q . To this aim, we start from a condition( σ , ˙ p ) and we inductively build ( σ n +1 , ˙ p n +1 ) ≤ ( σ n , ˙ p n ) such that there exists n ∈ Term ( σ n ) such that x ( σ n +1 , ˙ p n +1 ) ⊇ t n . Then put σ = S n ∈ ω σ n , pick ˙ p such that σ (cid:13) ˙ p ≤ ˙ p n for all n ∈ ω , and put w = S n ∈ ω t n . By construction, w ∈ Term ( σ ) and x ( σ, ˙ p ) ⊇ w , as ( σ, ˙ p ) ≤ ( σ n , ˙ p n ), for all n ∈ ω .The second part is an analogue of the proof of Claim 17, i.e., we want to showthat if x ( σ, ˙ p ) ⊇ t , for some t ∈ Term ( σ ), none of the extensions of t can beruled out, and so t = x ( σ, ˙ p ) . By contradiction, assume x ( σ,p ) = t a s , for some t ∈ Term ( σ ) and non-empty s ∈ <κ . Let σ ′ be the downward closure of σ ∪ S { t a ξ : ξ ∈ κ } , for some t ⊥ t a s with t ⊃ t . Then pick ( σ ′′ , ˙ q ) ≤ ( σ ′ , ˙ p )such that there exists t ∈ Term ( σ ′′ ) such that t ⊆ x ( σ ′′ , ˙ q ) . Hence, one has x ( σ ′′ , ˙ q ) ⊇ t ⊇ t ⊥ x ( σ, ˙ p ) , contradicting ( σ ′′ , ˙ q ) ≤ ( σ, ˙ p ).With a similar construction, we can get an analogue of Claim 18 and Corollary19 as well. Claim 26.
Let G be MT ∗ P -generic over N . Let ˙ T be the canonical name forthe generic Miller tree added by G (0) , ˙ x an MT ∗ P -name for a branch in ˙ T ,and z = ˙ x G . Then MT ∗ P / ˙ x = z is forcing-equivalent to MT ∗ P (and henceto Coll ( κ, < λ ) ).Proof. Use the notation ( MT ∗ P ) z := MT ∗ P / ˙ x = z . Claim 25, together withthe analogues of Claim 18 and Corollary 19, gives the same argument as in theproof of Lemma 15, and so we can obtain( MT ∗ P ) z = { ( σ, ˙ p ) ∈ MT ∗ P : x ( σ, ˙ p ) ⊂ z } . We work in N [ z ]. Note that( MT ∗ P ) z = { ( σ, ˙ p ) ∈ MT ∗ P : ∃ t ∈ Term ( σ )( t ⊂ z ∧ x ( σ, ˙ p ) = t ) } . ⊆ : clearly, if ∀ t ∈ Term ( σ )( t z ), then ( σ, ˙ p ) / ∈ ( MT ∗ P ) z , as ( σ, ˙ p ) (cid:13) σ ⊏ ˙ T . ⊇ : if there exists t ∈ Term ( σ ) such that t ⊂ z , then x ( σ, ˙ p ) = t ⊂ z .First, we prove that P := { σ ∈ MT : ∃ t ∈ Term ( σ ) ∃ ˙ p ∈ P ( t ⊂ z ∧ x ( σ, ˙ p ) = t ) } (2)is < κ -closed and collapses 2 κ to κ , and so it is equivalent to MT . Let { σ α : α <δ } , for δ < κ , be a decreasing sequence of conditions in P , and for every α < δ ,let t α ∈ Term ( σ α ) be such that t α ⊂ z and ˙ p α ∈ P such that x ( σ α , ˙ p α ) = t α .Then put σ δ = S α<δ σ α , t δ = S α<δ t α and pick ˙ p δ ∈ P such that σ δ (cid:13) ∀ α <δ ( ˙ p δ ≤ ˙ p α ). Hence, t δ ∈ Term ( σ δ ), t δ ⊂ z and t δ = x ( σ δ , ˙ p δ ) , which means σ δ ∈ P . Hence, the poset is < κ -closed. The proof that it also collapses 2 κ to κ is the same as the one given for Claim 24, since the sets D A ’s are dense in P aswell; simply, for every σ ∈ P , pick t ∈ Term ( σ ) such that t ⊥ x ( σ, ˙ p ) , for some˙ p ∈ P , and then let σ ′ ≤ σ be the downward closure of σ ∪ S { t a ξ a a ξ : ξ ∈ κ } ,where A := { a ξ : ξ ∈ κ } .Secondly, define ˙ P := { ˙ p ∈ P : ∃ σ ∈ MT (( σ, ˙ p ) ∈ ( MT ∗ P ) z ) } . (3)Let H be an arbitrary MT -generic filter over N [ z ]. Work in N [ z ][ H ]. Then Coll ( κ, < λ ) is equivalent to P . Indeed, first note that, the argument used inthe second part of the proof of Claim 25 actually gives the following: if ˙ p, ˙ q ∈ P are such that σ (cid:13) ˙ q ≤ ˙ p and ( σ, ˙ p ) ∈ ( MT ∗ P ) z , then x ( σ, ˙ p ) = x ( σ, ˙ q ) , and so( σ, ˙ q ) ∈ ( MT ∗ P ) z as well. (if we drop the assumption σ (cid:13) ˙ q ≤ ˙ p , the onlything that we can say in general is that ∃ t ∈ Term ( σ ) such that x ( σ, ˙ p ) = t nd ∃ t ∈ Term ( σ ) such that x ( σ, ˙ q ) = t , but t and t might be different).Furthermore, let { p ξ : ξ < δ } , for δ ≤ λ , be the set of minimal conditions in P (i.e., there is no q ≥ p ξ and q = p ξ such that q ∈ P ); we can build a partialfunction e : Coll ( κ, < λ ) → Coll ( κ, < λ ), satisfying: • for every ξ < δ , for all α ∈ λ and β , µ ∈ κ , there are α ′ ∈ λ and β ′ , µ ′ ∈ κ such that, e ( p ξ ∪ { (( α , β ) , µ ) } ) = { (( α ′ , β ′ ) , µ ′ ) } ; • for all α ′ ∈ λ and β ′ , µ ′ ∈ κ , there are ξ < δ , α ∈ λ and β , µ ∈ κ suchthat e ( p ξ ∪ { (( α , β ) , µ ) } ) = { (( α ′ , β ′ ) , µ ′ ) } ; • let q := p ξ ∪ { (( α , β ) , µ ) and q := p ξ ∪ { (( α , β ) , µ ). Then q ≤ q ⇒ e ( q ) ≤ e ( q ) and q ⊥ q ⇒ e ( q ) ⊥ e ( q ); • let P := P \ { p ξ : ξ < δ } ; then e | P : P → Coll ( κ, < λ ) is a denseembedding.This e can be constructed by a pretty standard argument, simply by following abijection δ × λ × κ × κ ↔ λ × κ × κ , and by using the homogeneity of Coll ( κ, < λ ).Hence, (2) and (3) give: ( MT ∗ P ) z ∼ = P ∗ ˙ P ∼ = Coll ( κ, < λ ). Lemma 27.
Let λ be inaccessible greater than κ , and let G be Coll ( κ, < λ ) -generic over N . Then N [ G ] | = “ all On κ -definable subsets of κ κ are M full -measurable ”.Proof. The argument is in strict analogy to the one of Lemma 14, and we justgive a sketch. Let X ⊆ κ κ be defined via some formula ϕ whose parameters canbe absorbed into the ground model N , by λ -cc. Let T be the generic tree in M full added by the first step, i.e., T is associated with G := G ∩ Coll ( κ, κ + ).By Claim 26, we know that each branch x ∈ [ T ] ∩ N [ G ] has good quotient, andso Coll ( κ, < λ ) can be viewed as ˙ Q x ∗ ˙ R , where ˙ Q x is the poset generated by x and N [ x ] | = ˙ R x ∼ = Coll ( κ, < λ ).Let x be Cohen over N with good quotient and assume N [ x ] | = “ (cid:13) ˙ R x ϕ ( x )”.Work into N [ x ]; pick s ∈ κ <κ such that s (cid:13) “ (cid:13) ˙ R x ϕ ( x )” (here we are using C ∼ =( κ <κ , ⊂ )). Pick T full-Miller tree of good Cohen branches with Stem ( T ) = s .Then proceed as in Lemma 14: for every z ∈ [ T ], N [ z ] | = “ (cid:13) ˙ R z ϕ ( z )”, whichimplies there exists a Coll ( κ, < λ )-generic filter H over N [ z ] with N [ z ][ H ] = N [ G ], and so N [ G ] | = ϕ ( z ), as ˙ R z ∼ = Coll ( κ, < λ ). The case N [ x ] | = “ (cid:13) ˙ R x ϕ ( x )”is analogous. Remark 28.
Our result cannot be improved by replacing T ∈ M full with treeshaving branches z satisfying { α < κ : y ↾ α is splitting } being club. Indeed,a similar argument to the one presented in Lemma 13 shows that Cub is not M club -measurable (see also [3, Thm 2.12] for a more general approach). Howeverit remains open whether one can get M statfull -measurability for all On κ -definablesubsets of κ κ . Remark 29.
In [9], the authors investigate two properties related to the Millermeasurability: the Hurewicz dichotomy and a strengthening called the Millertree Hurewicz dichotomy. These notions are related to the Miller measurability,but they are not in general equivalent. The authors of [9] prove that
Coll ( κ, < λ )orces all On κ -definable sets to have the Hurewicz dichotomy. Furthermore, theyprove that if κ is not weakly compact, then the Miller tree Hurewicz dichotomyfails for closed sets, whereas that cannot be true for the Miller measurabilitybecause of Lemma 27. On the contrary, for κ weakly compact, they prove thatthe two dichotomies are equivalent and they both imply the Miller measura-bility pointwise, but it is not clear which is the relation with the full-Millermeasurability. In section 4 we have proved that C κ + forces all On κ -definable sets to be stationary-Silver measurable, for κ inaccessible. The latter assumption was essential in ourproof to show that C adds a stationary Silver tree of Cohen branches. Therefore,the following question arises naturally.Q.1 Can one force all On κ -definable sets to be stationary-Silver measurable,for κ successor?Even if not strictly necessary for a positive answer to Q.1, another issue strictlyrelated is the following.Q.2 Does C add a stationary Silver tree of Cohen branches even for κ successor?About Q.2, my intuition inclines to a negative answer.Another interesting issue is the role of the inaccessible λ concerning full-Millermeasurability and Miller measurability.Q.3 Can one force all On κ -definable sets to be Miller measurable without usinginaccessible cardinals?Q.4 What about the same question for full-Miller measurability instead?The key point here is that we do not have an analogous study in the classicalsetting; indeed, in the standard case, projective Baire property implies projec-tive Miller measurability (and even projective full-Miller measurability) and soShelah’s amalgamation and sweetness provide us with a model for those notionsof regularity without any need of an inaccessible. But , in our generalized con-text, the Baire property fails for Σ , and hence we really need a direct methodto get Miller measurability. A possible solution might be to consider an amoebaforcing adding a Miller tree of Cohen branches in a gentler way than Coll ( κ, κ ),in order to get: 1) κ + will not be collapsed, and 2) one could obtain sufficientlymany good Cohen branches.The issue of separating different regularities classwise has been developed in theclassical setting: in particular a method for separating Silver and Miller on allsets has been presented in [6]. A similar questions arises here.Q.5 Can one force all sets to be Silver measurable but there exists a non-Millermeasurable set?Finally, a last important research branch regards the ∆ -level. In fact, becauseof the ∆ -well ordering of ( κ κ ) L , one obtains ∆ non-regular sets in L . Asa consequence, some arguments used in the standard setting for ∆ sets holdtrue for ∆ in the generalized context. The investigation of this topic has beeninitiated by Friedman, Khomskii and Kulikov in [3]. We also believe that thistopic be strictly connected to the study of cardinal characteristics associatedith the ideals generated by tree-forcings, and hence a careful study of theamoeba forcings is necessary. In the standard setting, amoeba forcings havebeen studied in [11] and [7], where the authors have presented some applicationsto regularity properties and cardinal characteristics. In the generalized settingsuch a topic has not been suitably developed yet, and we aim at extending suchan investigation. References [1] J¨org Brendle,
Mutual generics and perfect free subsets , Acta MathematicaHungarica (1999), Vol. 82, pp. 143-161.[2] Sy D. Friedman, Lyubomir Zdomskyy,
Measurable cardinals and the cofinal-ity of the symmetric group , Fundamenta Matematicae, Vol. 207 (2010), pp101-122.[3] Sy D. Friedman, Yurii Khomskii, Vadim Kulikov,
Regularity properties onthe generalized reals , preprint (2014).[4] Aapo Halko, Saharon Shelah,
On strong measure zero subsets of κ , Funda-menta Matematicae 170(3) (2001), pp 219-229.[5] H. Hung, S. Negrepontis, Spaces homeomorphic to (2 α ) α , Bull. Amer. Math.Soc. 79 (1973), pp 143-146.[6] Giorgio Laguzzi, On the separation of regularity properties of the reals , sub-mitted.[7] Giorgio Laguzzi,
Some considerations on amoeba forcing notions , Archivefor Mathematical Logic (2014), DOI: 10.1007/s00153-014-0375-x.[8] Benedikt L¨owe,
Uniform unfolding and analytic measurability , Archive forMathematical Logic 97 (1998), pp 505-520.[9] Philipp L¨ucke, Luca Motto Ros, Philipp Schlicht,
The Hurewicz dichotomyfor generalized Baire space , preprint (2014).[10] Philipp Schlicht,
Perfect subsets in the generalized Baire space , preprint(2013).[11] Otmar Spinas,