Characterizing the spectra of cardinalities of branches of Kurepa trees
aa r X i v : . [ m a t h . L O ] D ec CHARACTERIZING THE SPECTRA OF CARDINALITIES OFBRANCHES OF KUREPA TREES
MÁRK POÓR ∗ AND SAHARON SHELAH † Abstract.
We give a complete characterization of the sets of cardinals thatin a suitable forcing extension can be the Kurepa spectrum, that is, the set ofcardinalities of branches of Kurepa trees. This answers a question of the firstnamed author. Introduction
A tree is a Kurepa tree if it is of height ω , each of its levels is countable, and ithas more than ω -many cofinal (that is of order type ω ) branches. In this paperwe study the possible values of the branch spectrum of Kurepa trees, i.e. the setSp ω = { λ : there exists a Kurepa tree T s.t. |B ( T ) | = λ } ⊆ [ ω , ω ](where B ( T ) stands for the set of cofinal branches of T ).The spectrum is related to the model theoretical spectrum of maximal modelsof L ω ,ω -sentences [SS17]. Also canonical topological and combinatorial structuresare associated with branches of Kurepa trees possessing a remarkably wide rangeof nonreflecting properties [Kos05]. For higher Kurepa trees (of weakly compactheight) the consistency strength of certain types of the branch spectrum was studiedin [HM19].It was first shown by Silver that the Kurepa Hypothesis (i.e. the existence ofa Kurepa tree) is independent [Sil67], or see [Kun83, Ch VIII, 3.]. Moreover thenon-existence of Kurepa trees is equiconsistent with the existence of an inaccessiblecardinal [Kun83, Ch VII, Ex. B8.].Questions about the possible values of the spectrum were addressed by Jin andShelah in [JS92]. They proved (assuming an inaccessible cardinal) that consistentlythere are only Kurepa trees with ω -many cofinal branches while 2 ω = ω .Building on ideas of Jin and Shelah, the first named author provided a sufficientcondition for a set to be equal to Sp ω in a forcing extension in [Poo]. Formally, itwas shown that if GCH holds, and 0 , / ∈ S is a set of ordinals such that S satisfieseitherCase A:(i) 2 ∈ S , Mathematics Subject Classification.
Primary 03E35; Secondary 03E05, 03E45.
Key words and phrases.
Kurepa tree, Constructible universe, Cardinal spectra. ∗ The first author was supported by the National Research, Development and Innovation Office– NKFIH, grants no. 124749, 129211. † The second author was partially supported by the European Research Council (ERC) grant338821, and by the Israel Science Foundation grant 1838/19. Paper 1189 on Shelah’s list.
Supported by the ÚNKP-19-3 New National Excellence Program of the Min-istry of Human Capacities. . (ii) { sup C : C ∈ [ S ] ≤ ω } ⊆ S ,(iii) ( ∀ α ∈ S ) : ( ω ≤ cf( α ) < ω ) → ( α + 1 ∈ S ),or Case B:(i) ∃ an inaccessible κ ,(ii) { sup C : C ∈ [ S ] <κ } ⊆ S ,(iii) ( ∀ α ∈ S ) : ( ω ≤ cf( α ) < κ ) → ( α + 1 ∈ S ),then in a forcing extension we have { α : ℵ α ∈ Sp ω } = S (cardinals are onlycollapsed in Case B, from ( ω , κ )). It can be easily seen that if cf( µ ) = ω and(Sp ω ∩ µ ) is cofinal in µ , then there exists a Kurepa tree with µ -many branches, asthe union of countably many Kurepa trees is a Kurepa tree, and it is not difficultto see that the same holds if cf( µ ) = ω , therefore Case A / ( ii ) and Case B / ((ii))are in fact necessary. However, it remained a question whether the last clauses canbe dropped.In this paper as the main result we prove that assuming CH + (2 ω = ω )conditions (i), (ii) (in both cases) are in fact sufficient by forcing a model of { α : ℵ α ∈ Sp ω } = S . Also, we can arbitrarily prescribe 2 ω to be any cardinal λ ≥ sup(Sp ω ) if in Case A the equality λ <ω = λ holds, or in Case B λ <κ = λ holdstoo.Moreover, when we do not want Kurepa trees with ω -many cofinal branches,we prove that the inaccessible is necessary by verifying that if ω is a successorin L , then there exists a Kurepa tree with only ω -many cofinal branches in V .It was known that these assumptions imply that there exists a Kurepa tree evenin L [ A ] for some A ⊆ ω [Kun83, Ch VII, Ex. B8.] (possibly having more than ω -many cofinal branches in V ). Our proof not only utilizes countable elementarysubmodels of initial segments of L [ A ], but the nodes of the tree are such elementarysubmodels, and each cofinal branch uniquely corresponds to an initial segment of L [ A ]. 2. Preliminaries, notations
Under ordinals we always mean Neumann ordinals. For a fixed cardinal χ wewill use the notation H ( χ ) for the collection of sets of hereditary size less than χ ,i.e. H ( χ ) = { x : | trcl( x ) | < χ } , where trcl( x ) stands for the transitive closure of x . In terms of forcing we will usethe notations of [Kun13], e.g. p ≤ q means that p is the stronger. If it is clearfrom the context and won’t make any confusion we will identify the set x in theground model with its canonical name ˇ x . For a set A the symbol P ( A ) denotesthe powerset of A , and [ A ] λ stands for { X ∈ P ( A ) : | X | = λ } . For a function s = {h β, s ( β ) i : β ∈ dom( s ) } we will also use the following notation and refer to s as h s β : β ∈ dom( s ) i . Under a sequence we mean a function defined on a set of ordinals. For sequences s, t the relation s = t ↾ dom( s ) (or equivalently s ⊆ t ) will be also denoted by s ⊳ t . Definition 2.1.
A tree h T, ≺ T i is a partially ordered set (poset) in which for each x ∈ T the set T ≺ x = { y ∈ T : y ≺ T x } HARACTERIZING THE SPECTRA OF CARDINALITIES OF BRANCHES OF KUREPA TREES3 is well ordered by ≺ T . Definition 2.2.
The height of x in the tree T is the order type of T ≺ x ht( x ) = otp( T ≺ x ) . Definition 2.3.
For each ordinal α the restriction of T to α is T <α = { t ∈ T : ht( t ) < α } . Definition 2.4.
The height of the tree T (in symbols ht( T )), is the least β suchthat ∄ t ∈ T : ht( t ) = β. We will need the following lemma [Kun83, Ch II. Thm. 1.6.] which we will referto as the ∆-system Lemma.
Lemma 2.5.
Let κ be an infinite cardinal, let θ > κ be regular, and satisfy ∀ α < θ ( | α <κ | < θ ). Assume that |A| ≥ θ , and ∀ x ∈ A ( | x | < κ ). Then there is a D ⊆ A ,such that |D| = θ , and D forms a ∆ -system, i.e. there is a kernel set y such that ∀ x = x ′ ∈ D : x ∩ x ′ = y. The forcing
Now we can state our main theorem.
Theorem 3.1.
Let S • be a set of infinite cardinals such that ω, ω / ∈ S • . Assume CH , and that eitherCase 1:(i) ω ∈ S • ,(ii) ω = ω ,(iii) { sup C : C ∈ [ S • ] <ω } ⊆ S • ,or Case 2:(i) there exists an inaccessible κ such that S • ∩ ( ω , κ ) = ∅ ,(ii) { sup C : C ∈ [ S • ] <κ } ⊆ S • .Then there exists a forcing extension V P such that V P | = S • = Sp ω , where P only collapses cardinals in ( ω , κ ) in Case2 . The key will be Lemma 3 .
26. After Lemma 3 .
29 we will put together the piecesin a short argument. Before these we need some preparation.
Definition 3.2.
In Case 1 (i.e. ω ∈ S • ) define the cardinal κ to be ω . Corollary 3.3.
No cardinal µ / ∈ ( ω , κ ) is collapsed. Theorem 3.4.
Suppose that all conditions from Theorem . hold, and κ is definedin Definition . . Assume further that λ is a cardinal which is an upper bound of S • such that λ <κ = λ (thus cf( λ ) ≥ κ ). Then there exists a forcing extension V P with V P | = ( S • = { µ : there exists a Kurepa tree T s.t. |B ( T ) | = µ } ) ∧ (2 ω = λ ) . Definition 3.5.
Let S + • = S • ∪ { κ, λ } . MÁRK POÓR AND SAHARON SHELAH
Definition 3.6.
For a cardinal θ ∈ S • let Q θ be the following notion of forcing.The triplet p = h T p , u p , η p i is an element of Q θ iff(a) T p is a countable tree of height δ for some δ < ω on the underlying set ω · δ ,where the β ’th level is [ ω · β, ω · ( β + 1)), i.e. T p, ≤ β \ T p,<β = [ ω · β, ω · ( β + 1))for each β < δ ,(b) for each t ∈ T p and β < δ there exists t ′ ∈ T p \ T p,<β s.t. t ≺ T p t ′ ,(c) u p ∈ [ θ ] ≤ ω ,(d) η p = h η p,α : α ∈ u p i , where η p,α ⊆ T p is a branch in T p,<γ for some γ ∈ { β + 1 : β < δ = ht( T p ) } (we do it for a technical reason, we also couldhave stored only the maximal element instead of a chain with a maximalelement).Then Q θ is a poset with the obvious order, i.e. q ≤ p , if T q is an end-extension of T p , formally T q,< ht( T p ) = T p , and for each α ∈ u p the inclusion η p,α ⊆ η q,α holds.Let T ∼ θ , η ∼ θ be the names for the generic tree and sequence, i.e. denoting thegeneric filter by G θ Q θ (cid:13) T ∼ θ = ∪{ T p : p ∈ G θ } and1 Q θ (cid:13) η ∼ θ = (cid:28) η ∼ θ,α = ∪{ η p,α : p ∈ G θ } : α ∈ θ (cid:29) . Definition 3.7.
For a cardinal θ ∈ S • let Q ∗ θ ⊆ Q θ be the following subposet. p ∈ Q ∗ θ , iff ht( T p ) is a successor, and ( ∀ α ∈ u p ) : η p,α is a branch through T p . Definition 3.8. If λ / ∈ S • then let Q λ be the countable supported product of h <ω , ⊳ i -s of length λ , i.e. Q λ = { p = h η α : α ∈ u p i : ( ∀ α ∈ u p ) η α ∈ <ω , for some u p ∈ [ λ ] ≤ ω } . Definition 3.9. If κ / ∈ S • (and then κ > ω V is inaccessible), then let Q κ be thecountable supported product of h <ω γ, ⊳ i ’s ( γ < κ ), a forcing which collapses eachcardinal in ( ω , κ ): Q κ = { p = h η α : α ∈ u p i : ( ∀ α ∈ u p ) η α ∈ <ω α, for some u p ∈ [ κ ] ≤ ω } . Definition 3.10.
We define the posets which we will need later.1) For S ⊆ S + • let P S be the countable supported product of Q θ -s ( θ ∈ S ), i.e. P S = { p is a function : dom( p ) ∈ [ S ] ≤ ω ∧ ( ∀ θ ∈ dom( p ) p ( θ ) ∈ Q θ ) } . With a slight abuse of notation for p ∈ P S and θ ∈ S \ dom( p ) we will mean1 Q θ under p ( θ ).2) For θ ∈ S + • , U ⊆ θ define its restriction from θ to U , i.e. Q θ,U = { p ∈ Q : u p ⊆ U } .
3) For S ⊆ S + • , U = h U θ : θ ∈ S i ∈ Q θ ∈ S P ( θ ) we define P S,U to be P -srestriction to coordinates in U θ -s, i.e. P S,U = { p ∈ P S : ( ∀ θ ∈ S ) p ( θ ) ∈ Q θ,U θ } .
4) For
S, S ′ ⊆ S + • , U = h U θ : θ ∈ S i ∈ Q θ ∈ S P ( θ ), U ′ = h U ′ θ : θ ∈ S i ∈ Q θ ∈ S ′ P ( θ ) we define • U + U ′ = h U θ ∪ U ′ θ : θ ∈ S ∪ S ′ i (where for θ ∈ S ′ \ S under U θ wemean the empty set, similarly for θ ∈ S \ S ′ , U ′ θ ), HARACTERIZING THE SPECTRA OF CARDINALITIES OF BRANCHES OF KUREPA TREES5 • U − U ′ = h U θ \ U ′ θ : θ ∈ S i (here we also mean the empty set under U ′ θ if θ ∈ S \ S ′ ), • id S = h θ : θ ∈ S i• for the set X if W α ∈ Q θ ∈ S P ( θ ) ( α ∈ X ) then X α ∈ X W α = * [ α ∈ X ( W α ) θ : θ ∈ S + .
5) Let P = P S + • .6) If p , p , . . . , p n ∈ P let V i ≤ n p i denote the greatest lower bound if exists.7) For p ∈ P , and S ⊆ S + • , U = h U θ : θ ∈ S i ∈ Q θ ∈ S P ( θ ) define p ↾ U ∈ P S to be the following restriction of p ↾ S in the obvious fashionfor each θ ∈ S : ( p ↾ U )( θ ) = h T p ( θ ) , u p θ ∩ U θ , η p ↾ U θ i . Definition 3.11.
For S ⊆ S + • define the notion of forcing P ∗ ( P ∗ S , P ∗ S,U , resp.) tobe the subposet of P ( P S , P S,U , resp.) consisting of elements p for that p ( θ ) ∈ Q ∗ θ holds for each θ ∈ S • ∩ supp( p ). Remark 3.12.
The notion of forcing P ∗ ( P ∗ S , P ∗ S,U , resp.) is a dense subposet of P ( P S , P S,U , resp.), therefore forcing with P ∗ ( P ∗ S , P ∗ S,U , resp.) yields the sameextensions as forcing with P ( P S , P S,U , resp.).
Claim 3.13.
Let S ⊆ S + • , U = h U θ : θ ∈ S i be fixed. Then the poset P S,U has the κ -cc property. Proof.
Suppose that { p α : α ∈ κ } ⊆ P S,U is an antichain. Working in V ′ , applyingthe ∆-system lemma (Lemma 2 .
5) for the system { dom( p α ) : α ∈ κ } of countablesets (1) from Definition 3 . A ∈ [ κ ] κ , such that the dom( p α )’s( α ∈ A ) form a ∆-system with kernel K ⊆ S . Since K is obviously countable, foreach α we have that h T p α ( θ ) : θ ∈ K i is a countable sequence of countable trees(by ( a ) from Definition 3 . CH we can assume that(3.1) h T p α ( θ ) : θ ∈ K i = h T p β ( θ ) : θ ∈ K i ( ∀ α, β ∈ A ) . Now applying the ∆-system lemma again for the system U α = [ θ ∈ S (cid:0) { θ } × u p α ( θ ) (cid:1) ( α ∈ κ )yields a set A ′ ∈ [ A ] κ such that the U α ’s ( α ∈ A ′ ) form a ∆-system with kernel I ⊆ S θ ∈ S { θ } × θ (of course, in fact, I ⊆ S θ ∈ K { θ } × θ ). Now by (3.1) it suffices toprove that(3.2) ∃ α = β ∈ A ′ such that (for each h θ, δ i ∈ I ) : η p α ( θ ) ,γ = η p β ( θ ) ,γ , for which it is enough to prove(3.3) (cid:12)(cid:12)(cid:8)(cid:10) η p α ( θ ) ,γ : h θ, γ i ∈ I (cid:11) : α ∈ A ′ (cid:9)(cid:12)(cid:12) < κ. Fix α ∈ A ′ . Now for each h θ, γ i ∈ I , if θ ∈ S • then η p α ( θ ) ,γ ∈ [ ω ] <ω (a branchthrough T p α ( θ ) ).This means that (using that I is countable)(3.4) (cid:8)(cid:10) η p α ( θ ) ,γ : h θ, γ i ∈ I, θ ∈ S • (cid:11) : α ∈ A ′ (cid:9) ⊆ Y h θ,γ i∈ I, θ ∈ S • [ ω ] <ω , MÁRK POÓR AND SAHARON SHELAH which latter set is of size ω by CH . Second, if θ = λ ∈ ( S + • \ S • ) ∩ S , then (cid:8)(cid:10) η p α ( θ ) ,γ : h θ, γ i ∈ I, θ = λ (cid:11) : α ∈ A ′ (cid:9) ⊆ Y h θ,γ i∈ I, θ = λ <ω . Finally we have to consider the coordinate θ = κ if κ ∈ S \ S • . Then letting δ = sup { γ : h κ, γ i ∈ I } we have δ < κ , because I is countable and κ is inaccessible.Then(3.5) {h η p α ( κ ) ,γ : h κ, γ i ∈ I } ⊆ Y h κ,γ i∈ I <ω δ, and since κ is inaccessible, this case | Q h κ,γ i∈ I <ω δ | < κ . We obtain (using ω < κ )that (cid:12)(cid:12) {h η p α ( θ ) ,γ : h θ, γ i ∈ I } (cid:12)(cid:12) ≤ ω · ω · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Y h κ,γ i∈ I <ω δ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < κ, therefore (3.3) holds. (cid:3) Now we make the intuition behind the easy idea of first adding the trees andsome branches, and then forcing over the extension precise.
Claim 3.14.
For each S ⊆ S + • , U = h U θ : θ ∈ S i we have P S,U ⋖ P S ⋖ P , i.e. P S,U completely embeds into P S , which completely embeds into P . Proof.
Since P ≃ P S × P S + • \ S , it is enough to prove that P S,U ⋖ P S .Assume that A ⊆ P S,U is a maximal antichain in P S,U , and let p ∈ P S \ P S,U .Then there exists a ∈ A , a ′ ∈ P S,U such that a ′ ≤ a , a ′ ≤ b ↾ U . But then it isstraightforward to check that also a ′ and b have a common lower bound. (cid:3) Definition 3.15.
Let S ⊆ S • , U = h U θ : θ ∈ S i , θ ∈ S , U ′ θ ⊆ θ \ U θ . Then Q ∼ ◦ θ ,U ′ θ = Q ∼ ◦ ( S,U ) ,θ ,U ′ θ denotes the P S,U -name for a notion of forcing which addsthe branches η ∼ θ ,α ( α ∈ U ′ θ ) to T θ ∼ in the following way1 (cid:13) P S,U Q ∼ ◦ θ ,U ′ θ = p = h η p , u p i : ( u p ∈ [ U ′ θ ] ≤ ω ) ∧ ( η p = h η p,α : α ∈ u p i ),such that each η p,α is a branch of T ∼ θ ,<δ α for some δ α ∈ { γ + 1 : γ < ω } . If it is clear from the context we will use Q ∼ ◦ θ ,U ′ θ not mentioning S and U . Definition 3.16.
Let S ⊆ S • , U = h U θ : θ ∈ S i , θ ∈ S .If θ ∈ S + • \ S • , and U ′ θ ⊆ θ \ U θ , then define the P S,U -name Q ∼ θ,U ′ θ = Q ∼ ◦ θ,U ′ θ to bethe name for Q θ,U ′ θ . Definition 3.17.
Let S ⊆ S + • , U = h U θ : θ ∈ S i , U ′ = h U ′ θ : θ ∈ S i ∈ Q θ ∈ S P ( θ ),where U θ ∩ U ′ θ = ∅ for each θ ∈ S . Then P ∼ ◦ U ′ = P ∼ ◦ ( S,U ) ,U ′ denotes the P S,U -name forthe countably supported product of Q ∼ ◦ θ,U ′ θ ’s ( θ ∈ S ), i.e. a notion of forcing which HARACTERIZING THE SPECTRA OF CARDINALITIES OF BRANCHES OF KUREPA TREES7 adds the branches η ∼ θ,α ( α ∈ U ′ θ ) to T ∼ θ for each θ ∈ S \ S • , and the sequences η ∼ κ,α ( α ∈ U ′ κ ) if κ ∈ S \ S • , η ∼ λ,α ( α ∈ U ′ λ ) if λ ∈ S \ S • :1 (cid:13) P S,U P ∼ ◦ U ′ = (cid:26) p is a function : dom( p ) ∈ [ S ] ≤ ω ∧ ( ∀ θ ∈ dom( p ) p ( θ ) ∈ Q ∼ ◦ θ,U ′ θ ) } (cid:27) . Again, as in Definition 3 .
15 if it does not cause any confusion we only use thenotation P ∼ ◦ U ′ not mentioning S and U .The following claim is an easy observation. Claim 3.18. If G is a P S,U -generic filter over V (where S ⊆ S + • , U = h U θ : θ ∈ S i , U ′ = h U ′ θ : θ ∈ S i ∈ Q θ ∈ S P ( θ ) , and U θ ∩ U ′ θ = ∅ for each θ ∈ S ), then with thenotation from [Kun13] P S,U + U ′ / G = { p ∈ P S,U + U ′ : ∀ q ∈ G p q } , the quotient poset P S,U + U ′ / G and the evaluation of P ∼ ◦ U ′ are isomorphic, i.e. V [ G ] | = P ∼ ◦ U ′ [ G ] ≃ P S,U + U ′ / G . Since P S,U completely embeds into P S,U + U ′ (by Claim 3 . Claim 3.19.
Let S ⊆ S + • , U = h U θ : θ ∈ S i , U ′ = h U ′ θ : θ ∈ S i ∈ Q θ ∈ S P ( θ ) ,where U θ ∩ U ′ θ = ∅ for each θ ∈ S . Then the canonical embedding from P S,U + U ′ tothe iteration P S,U ∗ ( P S,U + U ′ / G ) is a dense embedding. Now putting together Claims 3 .
18 and 3 .
19 we have the following, meaning thatinstead of forcing with P S,U + U ′ we can force with P S,U and then with (the evaluationof) P ∼ ◦ U ′ . Lemma 3.20.
Let S ⊆ S + • , U = h U θ : θ ∈ S i , U ′ = h U ′ θ : θ ∈ S i ∈ Q θ ∈ S P ( θ ) ,where U θ ∩ U ′ θ = ∅ for each θ ∈ S . Then forcing with P S,U + U ′ amounts to the sameas forcing with P S,U and then with P S,U + U ′ / G ≃ P ∼ ◦ U ′ . Definition 3.21. If S ⊆ S + • , U = h U θ : θ ∈ S i , U ′ = h U ′ θ : θ ∈ S i ∈ Q θ ∈ S P ( θ ).Now if G is generic over P = P S + • then we define • G S = G ∩ P S , • G S,U = G ∩ P S,U , • and G ◦ U ′ ⊆ P ◦ U ′ [ G S,U ] ∈ V [ G S,U ] to be the filter given by the canonicalmapping from Claims 3 .
18, 3 . ω -closed subset of a two-step iteration similarly as in [Kun78]. Claim 3.22. P ∗ (and in general each P ∗ S,U ) is ω -closed, i.e. for each decreasingsequence of type ω has a lower bound. In particular if G ∗ ⊆ P ∗ , (or in general G ∗ S,U ⊆ P ∗ S,U ) is generic over V , then there is no new sequence of ordinals of type ω . The last claim and Remark 3 .
12 obviously implies the following.
MÁRK POÓR AND SAHARON SHELAH
Claim 3.23.
Forcing with P (or P S,U ) doesn’t add new sequence of ordinals of type ω , and for a given generic filter G ⊆ P H ( ω ) V = H ( ω ) V [ G ] = H V [ G S,U ] . Lemma 3.24.
Let G ⊆ P S,U generic over V , B ∈ V [ G ] where B ⊆ H ( ω ) . Then(in V ) there exist S ∗ ⊆ S , | S ∗ | < κ and W ∗ = h W ∗ γ : γ ∈ S ∗ i ∈ Q γ ∈ S ∗ [ U γ ] <κ ,such that B ∈ V [ G S ∗ ,W ∗ ] . Proof.
Choose p ∈ G forcing that B ⊆ H ( ω ), and a nice P S,U -name for B ,obtaining for each x ∈ H ( ω ) an antichain A x ⊆ P S,U deciding about x ∈ B .Then by κ -cc we have that each | A x | < κ , the set S ∗ = S x ∈H ( ω ) S a ∈ A x dom( a )is of size less than κ (as κ is either inaccessible, or ω ). Also for θ ∈ S ∗ theset W ∗ θ = S x ∈H ( ω ) S a ∈ A x u a ( θ ) is smaller that κ . Now it is easy to see that W ∗ = h W ∗ γ : γ ∈ S ∗ i is as claimed. (cid:3) Then the following immediately follows from the ω -closedness, and κ -cc. Claim 3.25.
Forcing with P doesn’t collapse ω , and cardinals at least κ . Moreover,if G ⊆ P is generic, then V [ G ] | = " κ = ω " . Lemma 3.26.
Let T ∈ V [ G S,U ∗ ] be a Kurepa tree, S ′ ⊆ S ( S ′ ∈ V ). Then, if b ∈ V [ G S,U ∗ +id S ′ ] is a branch of T , then there exists a finite set S ′′ ⊆ S ′ , and U • = h U • θ : θ ∈ S ′′ i s.t. each U • θ is finite, and b is in the model obtained by addingthese finitely many η θ,α ’s ( θ ∈ S ′′ , α ∈ U • θ ) to V [ G S,U ∗ ] , i.e. b ∈ V [ G S,U ∗ + U • ] . Proof.
Let ˙ T ∈ V be a P S,U ∗ -name for T . Define(3.6) P ′ = P S,U ∗ +id S ′ . Suppose that p ∗ ∈ P ′ forces that ˙ b ∈ V is a P ′ -name for a counterexample (i.e.forcing that for no such U • there exists a P U ∗ + U • -name ˙ b ′ - which is of course alsoa P ′ -name - with ˙ b ′ = ˙ b ). Let χ be large enough, and let h N , ∈i ≺ hH ( χ ) , ∈i becountable s.t. p ∗ , ˙ b, ˙ T , S, S ′ , V , P S,U ∗ ∈ N .Let δ • = N ∩ ω . Define the countable set N to be such that N ∈ N , and h N , ∈i ≺ hH ( χ ) , ∈i . Let X be set of the indices of the new branches added to h T ∼ θ : θ ∈ S ′ i by G S,U ∗ +(id S ′ ) that are in V [ G S,U ∗ +id S ′ ] \ V [ G S,U ∗ ], and belong to N , i.e.(3.7) X = N ∩ {h θ, α i : ( θ ∈ S ′ ) ∧ ( α ∈ θ \ U ∗ θ ) } . We fix an enumeration of X and define also the sequence of the first n indices fromthis countable set, and as well for each n the one-length sequence consisting onlythe n ’th, that islet hh ̺ n , ξ n i : n ∈ ω, n > i enumerate X (starting from 1) , (3.8) W n = h W n,θ : θ ∈ S ′ ∩ N i , where W n,θ = { α : h θ, α i = h ̺ j , ξ j i for some j ≤ n } w n = h w n,θ : θ ∈ S ′ ∩ N i where w n,θ = { ξ n } if θ = ̺ n , w n,θ = ∅ otherwise . HARACTERIZING THE SPECTRA OF CARDINALITIES OF BRANCHES OF KUREPA TREES9
Observe that if p ∈ P ∩ N , then each θ ∈ dom( p ) is an element of N since dom( p )is countable (by Definition 3 . T p ( θ ) , u p ( θ ) ⊆ N (by Definitions3 . − . V we will construct an N -generic condition in P ′ , which will deriveus to a contradiction. It is enough to prove the following claim. Claim 3.27.
There exists a sequence h p n : n ∈ ω i ∈ V , p ′ ∈ P S,U ∗ and a sequence q = h q n : n ∈ ω i such that the following holds. ⊞ p = h p ,l : l ∈ ω i is such that(a) p , = p ∗ ↾ U ∗ ,(b) p ,l ∈ N ∩ P S,U ∗ for each l ∈ ω ,(c) h p ,l : l ∈ ω i is ≤ P -decreasing,(d) p ∈ N ,(e) letting G = { p ∈ P S,U ∗ ∩ N : ( ∃ l ) p ≥ p ,l } , the filter G is P S,U ∗ -generic over N . ⊞ p ′ ∈ P S,U ∗ satisfies the following(a) p ′ is a lower bound of p ,l for each l ∈ ω (hence forces a value to T ∼ θ,<δ • for each θ ∈ S ∩ N ),(b) p ′ forces a value to T ∼ θ, ≤ δ • for each θ ∈ S ∩ N such that for every δ • -branch B in T ∼ θ,<δ • the inclusion B ∈ N implies that B has anupper bound in T ∼ θ, ≤ δ • ,(c) p ′ forces a value to ˙ T ≤ δ • . ⊞ for every n > the sequence p n = h p n,l : l ∈ ω i has the following properties.(a) ∀ l ∈ ω p n,l ∈ N ∩ P S,U ∗ + w n ,(b) p n,l ↾ U ∗ ∈ G (c) h p n,l : l ∈ ω i is ≤ P -decreasing,(d) p n ∈ N ,(e) letting G n = { p ∈ P S,U ∗ + W n ∩ N : ( ∃ l , l , . . . , l n ) p ≥ n ^ j =0 p j,l j } , the filter G n is P S,U ∗ + W n -generic over N . ⊞ For the sequence q = h q n : n ∈ ω i (a) q n ∈ N ∩ P S,U ∗ +id S ′ for each n ∈ ω ,(b) q = p ∗ ,(c) h q n : n ∈ ω i is ≤ P -decreasing,(d) ∀ n : q n ↾ ( U ∗ + W n ) ∈ G n ,(e) Let h ˙ B n : n ∈ ω i enumerate the branches of ˙ T <δ • which has an upperbound in ˙ T ≤ δ • (forced by p ′ ). Then q n +1 ∧ p ′ forces that ˙ b = B n , whichwill be guaranteed by the following requirement:There exist δ < δ • , t = t ′ ∈ ˙ T ≤ δ \ ˙ T <δ , such that p ′ forces B n -s δ ’thlevel to be t ′ , and q n +1 forces t ∈ ˙ b , i.e. (3.9) p ′ (cid:13) ˙ B n ∩ ( ˙ T ≤ δ \ ˙ T <δ ) = { t ′ } and q n +1 (cid:13) ˙ b ∩ ( ˙ T ≤ δ \ ˙ T <δ ) = { t } . (Observe that the latter is a statement in N .) Before proving Claim 3 .
27 we argue why this claim implies Lemma 3 .
26. First,the claim gives the following condition in P S,U ∗ +id S ′ . For each n ∈ ω let η ̺ n ,ξ n bethe branch in T ∼ ̺ n ,<δ • represented by the sequence p n , i.e.(3.10) η ̺ n ,ξ n = ∪{ η p n,l ( ̺ n ) ,ξ n : l ∈ ω } , and note that η ̺ n ,ξ n ∈ N ( n ∈ ω ) by ⊞ / ( d ). Therefore by ⊞ / ( b ) we can extendeach η ̺ n ,ξ n to a branch η ′ ̺ n ,ξ n in ( T p ′ ( ̺ n ) ) <δ • +1 . Define the function p • to bethe extension of p ′ by the η ̺ n ,ξ n ’s in the obvious way: (Note that by ⊞ S ∩ N ⊆ dom( p ′ ) ⊆ S , and for each θ ∈ S ∩ N the inclusion U ∗ θ ∩ N ⊆ u p ′ ( θ ) ⊆ U ∗ θ .)Define p • to be function on dom( p ′ ) such that if θ / ∈ N ∩ S ′ , then p • ( θ ) = p ′ ( θ ),and for θ ∈ N ∩ S ′ define p • ( θ ) to be the following proper extension of p ′ ( θ ). Let u p • ( θ ) = u p ( θ ) ∪ ( θ ∩ N ), and if α / ∈ u p ′ ( θ ) (when necessarily α / ∈ U ∗ θ ) and (by(3.8)) choose n > h θ, α i = h ̺ n , ξ n i , and let η p • ( θ ) ,α = η ′ ̺ n ,ξ n , otherwise(3.12) η p • ( θ ) ,α = η p ( θ ) ,α (if α ∈ U ∗ θ ) . Observe that as η ′ ̺ n ,ξ n was a cofinal branch in ( T p • ( ̺ n ) ) <δ • +1 = ( T p ′ ( ̺ n ) ) <δ • +1 our function p • is indeed a condition in P S,U ∗ +id S ′ . Moreover, the following showsthat ∀ n ∈ ω p • ≤ q n . Fix n ∈ ω , then using ⊞ / ( d ) we have q n ↾ ( U ∗ + W n ) ∈ G n ,i.e. there exist l , l , . . . l n ∈ ω , such that V nj =0 p j,l j ≤ P q n ↾ ( U ∗ + W n ). This meansthat n ^ j =0 p j,l j ≤ q n ↾ ( U ∗ + W ) = q n ↾ ( U ∗ ) , and for each 0 < j ≤ n η q n ( ̺ j ) ,ξ j ⊆ η p j,lj ( ̺ j ) ,ξ j ⊆ η ′ ̺ j ,ξ j = η p • ( ̺ j ) ,ξ j . On the other hand, for j > n we have (recalling q = h q n : n ∈ ω i is ≤ P -decreasingby ⊞
4) that η q n ( ̺ j ) ,ξ j ⊆ η q j ( ̺ j ) ,ξ j ⊆ η ′ ̺ j ,ξ j = η p • ( ̺ j ) ,ξ j , therefore p • ≤ q n , indeed.Now assuming p • ∈ G S,U ∗ + id S ′ will easily yield a contradiction: First recall that p ∗ (and therefore as well q and p • ) forced that ˙ b is a branch through ˙ T . Then ⊞ / ( c ) implies that p ′ , thus p • as well determines ˙ T ≤ δ • , and p • forces (by ⊞ / ( e ))that each element of the δ • ’th level of ˙ T is the upper bound of B i for some i ∈ ω .This means that p • (cid:13) ( ∃ i ∈ ω ) ˙ b ∩ ˙ T <δ • = B i , while at the same time ( q i ∧ p ′ ) (cid:13) ˙ b = B i , since (3.9) holds.This together with p • ≤ q i , p ′ gives the contradiction. Now we can turn to theproof of the claim. Proof. (Claim 3 . p n and each q n we will work in N . Thiswill need a lot of preparation.Recall that X ⊆ N denoted the indices of branches added by forcing with P S,U ∗ +id S ′ ∩ N but missing from V [ G S,U ∗ ] (3.7), and that for each condition p , HARACTERIZING THE SPECTRA OF CARDINALITIES OF BRANCHES OF KUREPA TREES11 θ ∈ S • , and δ < ω the δ ’th level of T p ( θ ) is (a subset of) [ ω · δ, ω · ( δ + 1)). Define E ⊆ N as follows.(3.13) e ∈ E iff e ∈ N , and e = ( u e , η e ) , where u e ∈ [ X ] ≤ ω ,η e = h η e,θ,α : h θ, α i ∈ u e i , such that η e,θ,α ⊆ ω · ( δ θ,α + 1) for some δ θ,α < ω Definition 3.28.
For each n , p ∈ P S,U ∗ + W n , and e ∈ E , if for each h θ, α i ∈ u e wehave θ ∈ dom( p ), and for each i < n h ̺ i , ξ i i / ∈ u e holds then define p a e as(3.14) dom( p a e ) = dom( p ) ,u ( p a e )( θ ) = u p ( θ ) ∪ { α : h θ, α i ∈ u e } ) ( ∀ θ ∈ dom( p a e )) ,η ( p a e )( θ ) ,α = (cid:26) η p ( θ ) ,α , if α ∈ u p ( θ ) ,η e,θ,α , if h θ, α i ∈ u e , if this is a condition in P (i.e. for each h θ, α i ∈ u e η e,θ,α is a cofinal branch of ( T p ( θ ) ) <δ +1 for some δ ≤ ht( T p ( θ ) ) , otherwise p a e = ∅ . Let D denote the set of dense subsets of P S,U ∗ +id S ′ . Fix an enumeration hh J i , ε i i : i ∈ ω i ∈ N of ( D ∩ N ) × E, and let k ( D, e ) denote the index of the pair h D, e i (i.e.(3.15) J k ( D,e ) = D, ε k ( D,e ) = e ), then we also have k ∈ N , of course.Fix a function g ∈ N (3.16) g : P S,U ∗ +id S ′ × D → P S,U ∗ +id S ′ with ∀ p, D : • g ( p, D ) ∈ D, • g ( p, D ) ≤ p, (Then g ∈ N obviously implies ( p, D ∈ N ⇒ g ( p, D ) ∈ N ).)We will have to define also the auxiliary sequence r = h r l : l ∈ ω i with the followingproperty: ⊛ r ∈ N , ⊛ for each l r l ∈ P S,U ∗ ∩ N , ⊛ for each l p ,l +1 ≤ r l ≤ p ,l , ⊛ if there exists p ∈ P S,U ∗ such that p ≤ p ,l , and p a ε l is a conditionextending p ,l in P S,U ∗ +id S ′ , then r l is such that.Now we can construct the p ,i ’s (and r i ’s). Let p , = p ∗ ↾ U ∗ . For obtainingthe p ,l ’s proceed as follows. Assume we have defined p , , p , , . . . , p ,l − (andas well the r i ’s for i < l − p ∈ P S,U ∗ p ≤ p ,l − , s.t. p a ε l − = ∅ but a condition extending p ,l − , then let r l − ∈ N be such a p (recallthat ε l − ∈ E ⊆ N by (3.13)), otherwise define r l − = p ,l = p ,l − . Lastly, in theformer case define p ,l = g ( r l − , D l − ) ↾ U ∗ . It is clear from the construction andthe definition of g that p ,l − ≤ r l − ≤ p ,l , and r l − , p ,l ∈ N , and since everyobject as well as the series h ε i : i ∈ ω i are elements of N , we obtain p , r ∈ N ,too. Finally, it is straightforward to check that the filter G generated by the p ,l ’smeets every dense subset D ∈ N of P S,U ∗ . Fixing such a DD ′ = { p ∈ P S,U ∗ +id S ′ : p ↾ U ∗ ∈ D } is clearly a dense subset of P S,U ∗ +id S ′ belonging to N . This means that if e ∈ E is the empty sequence, then there exists i ∈ ω , such that J i = D ′ , and ε i = e ,therefore p ,i +1 ∈ D .For p ′ , first consider the condition p ′′ ∈ N consisting of only the generic treesgiven by G (for each θ ∈ dom( p ′′ ) = N ∩ S the tree T p ′ ( θ ) = ∪{ T p ( θ ) : p ∈ G } is of height δ • , but u p ′′ ( θ )= ∅ ). Then let p ′′′ ∈ P S,U ∗ , p ′′′ ≤ p ′′ be an extension sothat for each θ ∈ S ′ ∩ N the tree T p ′ ( θ ) satisfies that for each branch B through( T p ′′′ ( θ )) <δ • = T p ′′ ( θ ) , if B ∈ N , then there is an upper bound of B in T p ′′′ ( θ ) . Thiscan be done since N is countable. Moreover, we choose the other part of p ′′′ sothat for each θ, α ∈ N , if α ∈ U ∗ θ the chain η p ′′′ ( θ ) ,α (with a top element) containsthe chain ∪{ η p ( θ ) ,α : p ∈ G } which is given by G at this coordinate. This canbe done as ∪{ η p ( θ ) ,α : p ∈ G } ∈ N , since G , p ∈ N . Then clearly p ′′′ ≤ p ,l for each l ∈ ω .Finally, for the last item of ⊞ P ∗ S,U ∗ is an ω -closed densesubposet of P S,U ∗ by Remark 3 .
11. Then if a countable increasing sequence in P ∗ S,U ∗ (where a first element stronger than p ′′′ ) decides more and more about the δ • ’th level of ˙ T , then choosing p ′ to be an upper bound will work (e.g. choosean enumeration h ˙ t i : i ∈ ω } of the δ • ’th level of ˙ T , let h s i : i ∈ ω i enumerate˙ T <δ • in type ω , and let r j decide whether the j ’th ordered pair in the countable set { s i : i ∈ ω } × { ˙ t i : i ∈ ω } is in ≤ ˙ T ).The next step is to construct the p i ’s ( i >
0) and the q n ’s. This will be donesimultaneously by induction. The induction is carried out in V , but each step canbe done in N , which will guarantee that each p n ∈ N .It is straightforward to check that choosing q = p ∗ would satisfy our require-ments, as e.g. p , = p ∗ ↾ U ∗ . Then fixing n >
0, and assuming that p i , q i areconstructed for each i < n , first we construct q n . Recall that q n − ↾ ( U ∗ + W n − ) ∈ G n − (by ⊞ / ( d )).Recall the definition of the set E (3.13), and let E n − = { e ∈ E : ∀ i < n h ̺ i , ξ i i / ∈ e } . Using that p ∗ ∈ P S,U ∗ +id S ′ forced that ˙ b is not an element of V [ G S,U ∗ + W n − ], i.e.there is no P S,U ∗ + W n − -name of it, we argue that D = { p ∈ P S,U ∗ + W n − : ∃ e, e ′ ∈ E n − ( p a e ≤ q n − , p a e ′ ≤ q n − ) ∧ ( ∃ δ < ω , t = t ′ ∈ ˙ T ≤ δ \ ˙ T <δ : ( p a e (cid:13) t ∈ ˙ b ) ∧ ( p a e ′ (cid:13) t ′ ∈ ˙ b )) } is dense in P S,U ∗ + W n − under q n − ↾ ( U ∗ + W n − ). Indeed, assume on the contrarythat q ′ ∈ P S,U ∗ + W n − , q ′ ≤ q n − ↾ ( U ∗ + W n − ) is such that that D has no elementunder q ′ . Now for every δ < ω , consider the set D δ = { p ∈ P S,U ∗ + W n − : ( p ≤ q ′ ) ∧ ( ∃ e ∈ E n − : [ p a e ≤ q n − ] ∧∧ [ ∃ t p,e,δ ∈ ˙ T ≤ δ \ ˙ T <δ : p a e (cid:13) t p,e,δ ∈ ˙ b ]) } , HARACTERIZING THE SPECTRA OF CARDINALITIES OF BRANCHES OF KUREPA TREES13 which is dense under q ′ in P S,U ∗ +id S ′ . Now since for each δ < ω the sets D and D δ are disjoint, for p ∈ D δ the witnessing t p,e,δ doesn’t depend on e , therefore q ′ ∧ q n − forces that ˙ b is in V [ G S,U ∗ + W n − ] (i.e. forces that the P S,U ∗ + W n − -name {h p, t p,δ i : p ∈ D δ , δ < ω } and ˙ b are equal).Then as our set D ∈ N is indeed dense we have that there exists a condition q ′′ ∈ G n − ∩ D , witnessed by t = t ′ and e, e ′ . Finally, if t ∈ B n then define q n = q ′′ a e ′ , otherwise we can let q n = q ′′ a e , which are both stronger conditionsthan q n − by the definition of D . It is straightforward to check ⊞ q n is already defined (and so are p i , q i for each i < n ), we turn to the definitionof p n , which we will do similarly to that of p . Let p n, = q n ↾ ( U ∗ + w n ), assumethat p n, , p n, , . . . , p n,l − are already chosen.If ε l − / ∈ E n − , then p n,l = p n,l − , otherwise proceed as follows. Choose thesequence e = e ( n, l −
1) = h e i : 1 ≤ i ≤ n + 1 i ∈ E n +1 \{ } and the sequence m = m ( n, l −
1) = h m i : i ≤ n i ∈ ω n +1 with the property1) e n +1 = ε l − and m n = l − i < n + 1(3.17) J m i = D ∧ " e i = ( e i +1 plus ( η p i,mi ( ̺ i ) ,ξ i attained on h ̺ i , ξ i i ))" . Provided that the e j ’s are defined for j > i , and as well each m j for j ≥ i , let e i ∈ E be the element with u e i = u e i +1 ∪ {h ̺ i , ξ i i} , η e i ⊇ η e i +1 , η e,̺ i ,ξ i = η p i,mi ( ̺ i ) ,ξ i , andlet m i − = k ( D, e i ). Observe that by our procedure, and by the definition of thefunction k (3.15) we have e = ε m , and also(3.18) η e ,̺ n ,ξ n = η p n,l − ( ̺ n ) ,ξ n . At some point later we will use the following fact, hence it is worth to note that foreach i , 1 ≤ i ≤ n (3.19) e ( i, m i ) ⊆ e ( n, l − , and m ( i, m i ) ⊆ m ( n, l − . Finally consider the condition r m (from ⊛ − ⊛ r m a e is a not a conditionin P S,U ∗ +id ↾ S ′ , then let p n,l = p n,l − , otherwise first define the auxiliary condition(3.20) r • = g ( r m a e , D ) , and note that in this case η ( r m a e )( ̺ n ) ,ξ n = η p n,l − ( ̺ n ) ,ξ n by (3.18), therefore bythe properties of g we obtain(3.21) η r • ( ̺ n ) ,ξ n ⊇ η p n,l − ( ̺ n ) ,ξ n . Recall that p n,l − ↾ U ∗ ∈ G by our induction hypotheses ⊞
3, and it can be seenfrom the construction of p ,j ’s that in this case p ,m +1 = r • ↾ U ∗ ∈ G . Thereforeby (3.21) we have that ( r • ↾ U ∗ + w n ) ∧ p n,l − is a condition in P U ∗ + w n , and let p n,l = ( r • ↾ U ∗ + w n ) ∧ p n,l − . Then clearly p n,l ≤ p n,l − , and p n,l ↾ U ∗ ∈ G . From ⊞ d ) and ( e ) also hold. Since the whole construction of p n took place in N ( k ∈ N and so is the enumeration hh J i , ε i i : i ∈ ω i , g ∈ N ), p n ∈ N obviouslyfollows. Verifying the genericity of G n goes similarly as of G . Let D ⊆ P S,U ∗ + W n , D ∈ N be a fixed dense set, and e ′ ∈ E be the empty sequence. Now, if we choose l so that J l − = D ′ = { p ∈ P S,U ∗ +id S ′ : p ↾ U ∗ + W n ∈ D } , ε l − = e ′ , then it follows from the construction of p k,j ’s, that of m = m ( n, l −
1) and e = e ( n, l − p i,m i +1 = ( r • ↾ U ∗ + w i ) ∧ p i,m i if 1 ≤ i ≤ n ,and p ,m +1 = g ( r m a e ) ↾ U ∗ , therefore ^ i ≤ n p i,m i ≤ g ( r m a e ) ↾ ( U ∗ + W n ) ∈ D ′ . (cid:3) (Claim 3 . (cid:3) (Lemma 3 . Lemma 3.29.
Let T ∈ V [ G S,U ∗ ] be a Kurepa tree, S ′ ⊆ S ∩ S • ( S ′ ∈ V ), G ◦ id S ′ − U ∗ ⊆ P ◦ id S ′ − U ∗ be generic over V [ G S,U ∗ ] . Suppose that b ∈ V [ G S,U ∗ ][ G ◦ S ′ , (id S ′ − U ∗ ) ] \ V [ G S,U ∗ ] is a new branch of T , and suppose that γ ≥ κ is a cardinal, and for each θ ∈ S ′ the inequality | θ \ U ∗ θ | ≥ γ holds. Then the filter G ◦ id S ′ − U ∗ adds at least | γ | -many new branches to T . Proof.
W.l.o.g. we can assume that T ⊆ ω , and λ is a cardinal (in V [ G S,U ∗ ]).First we will choose a system W = h W ,θ : θ ∈ S ′ i ∈ Q θ ∈ S ′ P ( θ ) with ( ∀ θ ∈ S ′ ) | W ,θ | < κ , and b ∈ V [ G S,U ∗ ][ G ◦ W ]: since b ∈ V [ G S,U ∗ ][ G ◦ id S ′ − U ∗ ], S ′ ∈ V we canuse Lemma 3 .
20 and obtain that b ∈ V [ G S,U ∗ ][ G ◦ id S ′ − U ∗ ] = V [ G S,U ∗ +id S ′ ]. Andbecause b ⊆ H ( ω ) V , applying Lemma 3 .
24 with S , and U = U ∗ + id S ′ , there exists S ∗ ⊆ S , W ∗ ∈ Q S ∗ \ S ′ P ( U θ ) × Q θ ∈ S ∗ ∩ S ′ P ( θ ) with b ∈ V [ G S ∗ ,W ∗ ] ⊆ V [ G S,U ∗ + W ∗ ] = V [ G S,U ∗ ][ G ◦ W ∗ − U ∗ ] , where | S ∗ | < κ , and | W ∗ θ | < κ for each θ ∈ S ∗ . Then fixing W ∈ Q θ ∈ S ′ P ( θ )so that W , θ = W ∗ θ \ U ∗ θ if θ ∈ S ∗ , and W ,θ = ∅ for θ ∈ S \ S ∗ has the requiredproperties.Now, as | W ,θ | < κ ≤ γ , and γ ≤ | θ \ U ∗ θ | for each θ ∈ S ′ we can fix for each α < γ a system W α = h W α,θ : θ ∈ S ′ i ∈ Q θ ∈ S ′ P ( θ \ U ∗ θ ) such that for every θ ∈ S ′ (i) W α,θ ∩ W β,θ = ∅ for every α < β < γ ,(ii) | W ,θ | = | W α,θ | for each α < γ .For each 0 < α < γ define the bijections π α : [ θ ∈ S ′ { θ } × W ,θ → [ θ ∈ S ′ { θ } × W α,θ where π α ↾ { θ } × W ,θ is a bijection to { θ } × W α,θ . Then clearly each π α induces anautomorphism ˆ π α ∈ V [ G S,U ∗ ] of P ◦ W and P ◦ W α . Moreover, ˆ π α induces a naturaloperation ˆ π ∗ α from the class of P ◦ W -names to the class of P ◦ W α -names. Now fix a P ◦ W -name ˙ b ∈ V [ G S,U ∗ ] for our new branch b ∈ V [ G S,U ∗ ][ G ◦ W ], and choose anelement p • ∈ P ◦ W forcing that ˙ b is a new branch, i.e.(3.22) V [ G S,U ∗ ] | = p • (cid:13) ˙ b ∈ B ( T ) \ B V [ G S,U ∗ ] ( T ) . HARACTERIZING THE SPECTRA OF CARDINALITIES OF BRANCHES OF KUREPA TREES15
Let P ◦• = P ◦ P α<γ W α , i.e. adding the branches S α ∈ γ W α,θ to T ∼ θ for each θ ∈ S ′ ,which is of course equal to the countably supported product of P ◦ W α ’s ( α < γ ), andlet G ◦• denote the generic filter G ◦ id S ′ − U ∗ ∩ P ◦• .We will show that in V [ G S,U ∗ ][ G ◦• ] ⊆ V [ G S,U ∗ ][ G ◦ id S ′ − U ∗ ] there are at least γ -many new branches of T , i.e. (cid:12)(cid:12)(cid:12) B ( T ) ∩ (cid:16) V [ G S,U ∗ ][ G ◦• ] \ V [ G S,U ∗ ] (cid:17)(cid:12)(cid:12)(cid:12) ≥ λ, by arguing that ⊗ for any α < γ (in V [ G S,U ∗ ])ˆ π α ( p • ) (cid:13) P ◦• ˆ π ∗ α (˙ b ) / ∈ V [ G S,U ∗ ][ G ◦• ,<α ](where G ◦• ,<α stands for G ◦• ∩ P ◦ P β<α W β ), and ⊗ |{ α < γ : ˆ π α ( p • ) ∈ G ◦• }| = γ .This will complete the proof of Lemma 3 . ⊗
2, for which recall that we assumed that γ is a cardinal,and choose a system of uncountable regular cardinals { ρ β : β < χ < γ } , and apartition h I β : β < χ i of γ with otp( I β ) = ρ β for each β < χ (i.e. I β ∩ I δ = ∅ for β < δ < ρ , and S β<ρ I β = γ ). Then it is enough to verify(3.23) ( ∀ β < χ ) |{ α ∈ I β : ˆ π α ( p • ) ∈ G ◦• }| = ρ β , which can be seen by a standard density argument: Fix β < ̺ , α ∈ I β , then itsuffices to show that D β,α = { p ∈ P ◦• : p ≤ ˆ π δ ( p • ) for some δ > α, δ ∈ I β } is dense,which obviously holds by the regularity of the uncountable ρ β = | I β | (since for δ ∈ I β we have ˆ π δ ( p • ) ∈ P ◦ W δ , P ◦• is the countably supported product of P ◦ W α ’s( α < γ ), and I β ⊆ γ ).For ⊗ P ◦• as the product of P ◦ P β<γ,β = α W β and P ◦ W α . We will needthe following claim. Claim 3.30.
For each p ∈ P ◦ W α , p ≤ ˆ π α ( p • ) there exist q , q ∈ P ◦ W α q , q ≤ p ,and the incomparable elements t , t of the tree T such that V [ G S,U ∗ ][ G ◦• ,γ \{ α } ] | = ( q i (cid:13) P ◦ Wα t i ∈ ˆ π ∗ α (˙ b )) for each i ∈ { , } , where G ◦• ,γ \{ α } = G ◦• ∩ P ◦ P β<γ,β = α W β . Before proving the claim we verify that ⊗ π α ( p • ) (cid:13) P ◦• ˆ π ∗ α (˙ b ) / ∈ V [ G S,U ∗ ][ G ◦• ,γ \{ α } ] . Since G ◦• ⊆ P ◦• is generic over V [ G S,U ∗ ], and P ◦• can be identified with (cid:18) P ◦ P β<γ,β = α W β (cid:19) × P ◦ W α , by [Kun13, Lemma V.1.1] G ◦• ,γ \{ α } = G ◦• ∩ P ◦ P β<γ,β = α W β is generic over V [ G S,U ∗ ],and G ◦• ,α = G ◦• ∩ P ◦ W α is generic over V [ G S,U ∗ ][ G ◦• ,γ \{ α } ]. For each branch c ∈ V [ G S,U ∗ ][ G ◦• ,γ \{ α } ] of T define (in V [ G S,U ∗ ][ G ◦• ,γ \{ α } ]) D c = { q ∈ P ◦ W α : ∃ t ∈ T \ c such that q (cid:13) P ◦ Wα t ∈ ˆ π ∗ α (˙ b ) } , which is dense under ˆ π α ( p • ) by Claim 3 .
30, since for a fixed p ∈ P ◦ W α at most one t i can be in the branch c . Proof. (Claim 3 .
30) First we argue that the statement holds in V [ G S,U ∗ ], i.e. foreach p ∈ P ◦ W α , p ≤ ˆ π α ( p • ) there exist q , q ∈ P ◦ W α q , q ≤ p , and the incomparableelements t , t of the tree T such that(3.24) V [ G S,U ∗ ] | = ( q i (cid:13) P ◦ Wα t i ∈ ˆ π ∗ α (˙ b )) for each i ∈ { , } . Now (3.22) implies that V [ G S,U ∗ ] | = ˆ π α ( p • ) (cid:13) P ◦ Wα ˆ π ∗ α (˙ b ) ∈ (cid:16) B ( T ) \ B V [ G S,U ∗ ] ( T ) (cid:17) since ˙ b ∈ V [ G S,U ∗ ] is a P ◦ W -name and T ∈ V [ G S,U ∗ ]. Suppose that p ≤ ˆ π α ( p • ) isa counterexample, but then for the set b ′ = { t ∈ T : ∃ q ∈ P ◦ W α , q ≤ p s.t. q (cid:13) t ∈ ˆ π ∗ α (˙ b ) } ∈ V [ G S,U ∗ ]we have p (cid:13) ˆ π ∗ α (˙ b ) = b ′ (since ˆ π α ( p • ) forced that ˆ π ∗ α (˙ b ) is a cofinal branch in T ), a contradiction. Finally, fixing p ≤ ˆ π α ( p • ), if q , q ∈ P ◦ W α q , q ≤ p , and theincomparable elements t , t ∈ T are such that (3.24) holds, then V [ G S,U ∗ ][ G ◦• ,γ \{ α } ] | = ( q i (cid:13) P ◦ Wα t i ∈ ˆ π ∗ α (˙ b )) for each i ∈ { , } , since if q i ∈ H ⊆ P ◦ W α is generic over V [ G S,U ∗ ][ G ◦• ,γ \{ α } ], and t i / ∈ ˆ π ∗ α (˙ b )[ H ](for some i ∈ { , } ), then H is generic over V [ G S,U ∗ ] too, and the same holds in V [ G S,U ∗ ][ H ]. (cid:3) It is left to argue why Lemma 3 .
26 and Lemma 3 .
29 complete the proof ofTheorem 3 . . T ∈ V [ G ] is a Kurepa tree (where G ⊆ P = P S + • , id S + • is generic), and assume on the contrary that |B V [ G ] ( T ) | / ∈ S • .We can also assume that T ⊆ H ( ω ) V , and by Lemma 3 .
24 there exists S ∗ ⊆ S + • , | S ∗ | < κ , W ∗ = h W ∗ θ : θ ∈ S ∗ i ∈ Q θ ∈ S ∗ [ θ ] <κ such that T ∈ V [ G S ∗ ,W ∗ ]. Forestimating (2 ω ) V [ G S ∗ ,W ∗ ] first a straightforward calculation yields that | P S ∗ ,W ∗ | <κ : Since | P S ∗ , h∅ : θ ∈ S ∗ i | = ( | S ∗ || ω | ) ω which is either ( ω · ω ) ω = ω < ω (if κ = ω , by CH ), or γ ω < κ (for some γ < κ , if κ is inaccessible). Thus recallingthe definition of Q θ,W ∗ θ ’s, the fact P θ ∈ S ∗ | W ∗ θ | < κ as κ is regular, and sup W ∗ κ < κ (if κ ∈ S ∗ ) we have the following (in both cases regardless of whether κ = ( ω ) V ,or an inaccessible) | P S ∗ ,W ∗ | = | P S ∗ , h∅ : θ ∈ S ∗ i | · ( ω ) · X θ ∈ S ∗ \{ κ } | W ∗ θ | ω · ( | W ∗ κ | · sup W ∗ κ ) ω < κ. At this point we have to discuss the two cases (i.e. whether κ ∈ S • ) differently,arguing that in both cases there are branches outside V [ G S ∗ ,W ∗ ].If κ = ω ∈ S • , then as V | = | P S ∗ ,W ∗ | ω ·| P S ∗ ,W ∗ | = ω , HARACTERIZING THE SPECTRA OF CARDINALITIES OF BRANCHES OF KUREPA TREES17 we have V [ G S ∗ ,W ∗ ] | = 2 ω = ω , therefore as |B V [ G ] ( T ) | / ∈ S • , there are branches of T in V [ G ] not in V [ G S ∗ ,W ∗ ].On the other hand, if κ / ∈ S • is inaccessible, then we obtain that V [ G S ∗ ,W ∗ ] | = |B ( T ) | ≤ ω < κ, and as κ remains a cardinal in V [ G ] (by Claim 3 . V [ G ] | = |B ( T ) ∩ V [ G S ∗ ,W ∗ ] | = ω , we conclude that this case there also must be branches of T not in V [ G S ∗ ,W ∗ ] as T is a Kurepa tree in V [ G ]. Now let R ∈ Q θ ∈ S + • \ S • P ( θ ), R θ = θ \ W ∗ θ , then P = P S + • , id S + • ≃ ( P S ∗ , id S ∗ − R ) × ( P S ∗ ∩ ( S + • \ S • ) ,R ) × ( P S + • \ S ∗ , id S + • \ S ∗ ) , and there are no new sequences of type ω in V [ G ] (by Claim 3 . ω -closed, the third component has an ω -closed dense subset (whichthus remain ω -closed in V [ G S ∗ , id S ∗ − R ]) we obtain that each branch of T is addedby G S ∗ , id S ∗ − R = G ∩ P S ∗ , id S ∗ − R (since an ω -closed forcing do not add new branchesto Kurepa trees [Kun13, Lemma V.2.26]). We only have to derive a contradictionfrom V [ G S ∗ , id S ∗ − R ] | = |B ( T ) | / ∈ S • . Now letting ∂ = |B V [ G S ∗ , id S ∗− R ] ( T ) | / ∈ S • , S −∗ = S ∗ ∩ S • ∩ ∂ , S + ∗ = ( S ∗ ∩ S • ) \ S −∗ by Lemma 3 .
20 we have V [ G S ∗ , id S ∗ − R ] = V [ G S ∗ ,W ∗ +id S −∗ ][ G ◦ id S + ∗ − W ∗ ] . As ∂ / ∈ S −∗ , S + ∗ , it is enough to prove that in V [ G S ∗ ,W ∗ +id S −∗ ] there are lessthan ∂ -many branches of T , because if G ◦ id S + ∗ − W ∗ adds new branches, then addsmin( S + ∗ )-many new branches by Lemma 3 .
29 (since each | W ∗ θ | < κ ≤ min( S • ) ≤ min( S + ∗ )).Now if ∂ = κ , then S −∗ = ∅ , we are done, so we can assume that ∂ > κ , andsup S −∗ ≥ κ . As | S ∗ | < κ (in V ), and our conditions (Case1 / ( iii ), or Case2 / ( ii ))states that then sup( S ∗ ∩ S • ∩ ∂ ) ∈ S • implying sup S −∗ < ∂ . Therefore using that W ∗ θ ⊆ θ we get P θ ∈ S −∗ | W ∗ θ | ≤ | sup S −∗ | < ∂ . Now by Lemma 3 .
26 for each branch b of T in V [ G S ∗ ,W ∗ +id S −∗ ] = V [ G S ∗ ,W ∗ ][ G ◦ (id S −∗ ) − W ∗ ] there exist θ , θ , . . . , θ n − , U • θ , U • θ , . . . , U • θ n − finite such that b ∈ V [ G S ∗ ,W ∗ ][ G ◦ U • ]. Therefore, as | P ◦ U • | = ω n = ω , counting the nice P ◦ U • -names of subsets T for each possible n , sequenceof θ ’s, and U • B ( T ) ∩ ( V [ G S ∗ ,W ∗ ][ G ◦ (id S −∗ ) − W ∗ ] \ V [ G S ∗ ,W ∗ ]) ≤ ( | sup S −∗ | <ω · ω ω ) V [ G S ∗ ,W ∗ ] ≤ sup S −∗ , which is smaller than ∂ , a contradiction.For V [ G ] | = 2 ω = λ we only need to show that 2 ω ≤ λ . But a similarstraightforward calculation yields that P = P S + • , id S + • is of cardinality λ , and then(using κ -cc and the equality λ <κ = λ ) by counting the possible nice names forsubsets of ω we obtain the desired inequality. Remark 3.31. If S • also satisfies(3.25) ∀ µ ∈ S • : cf( µ ) < κ → µ + ∈ S • , and GCH holds in V then S • \{ λ } is the spectrum for the Jech-Kunen trees in V [ G ].(A tree T of height ω and power ω is a Jech-Kunen tree if ω < |B ( T ) | < ω .)For more on Jech-Kunen trees see also [JS93], [JS92], [JS94]. Note that CH inthe final model implies that the product of countably many Jech-Kunen trees is aJech-Kunen tree, so is the diagonal product of ω -many Jech Kunen trees, hence(3.25) cannot be dropped.One can obtain similar cardinal arithmetic conditions for Sp µ with µ large.4. The necessity of the inaccessible cardinal
In this section we prove that if ω is not an element of the spectrum, then ω is inaccessible in L . The idea of using transitive collapses of elementary submodelsof constructible sets as nodes of a tree goes back to Solovay’s original unpublishedargument for the consistency strength of the negation of the Kurepa Hypothesis.Although the next proof is deemed to be well-known, for the sake of completenesswe include the proof as there is probably no known source to cite. Theorem 4.1.
Suppose that ω V is a successor in L . Then there exists a Kurepatree T with B V ( T ) = ω . Proof.
We will use an extension of L , an inner model between L and V , what servesas the motivation for the following definition of relative constructibility, which canbe found in e.g. [Kan03]. Definition 4.2.
For a set A define L [ A ] = S α ∈ ON L α [ A ] by transfinite recursionas follows. L [ A ] = ∅ , L α +1 [ A ] = def A ( L α [ A ]), and α limit L α [ A ] = S β<α L β [ A ](where def Y ( X ) are the subsets of X that can be defined in the structure ( X, ∈ ↾ ( X × X ) , Y ∩ X ) by parameters from X , see [Kan03, Chapter 1, §3].The following is standard easy exercise, but for the sake of completeness weinclude the proof. Claim 4.3.
There exists a set A ⊆ ω such that ω L [ A ]1 = ω , ω L [ A ]2 = ω . Proof. If ω V = ( λ + ) L , where | λ | = ω , then in a single subset A of ω we cancode a well-ordering of ω in type λ , and also for each α < ω a well-ordering of ω in type α in the obvious fashion, and such that L can read this coding (implying ω L [ A ]1 = ω , ω L [ A ]2 = ω ): First let h X α : α ≤ ω i ∈ L be a set of pairwise disjointsets of ω with | X α | L = ω for each α < ω , and | X ω | L = ω , then for each α < ω we can code the well ordering X α in order type α , and the well ordering of X ω intype λ in a subset A ′ of S α ≤ ω X α ⊆ ω . Finally, taking the preimage of this setunder a bijection f ∈ L between ω and ω , i.e. A = f − ( A ′ ) works. (cid:3) We have to recall a classical Lemma [Kan03, Theorem 3.3]. Recall that L ∈ ( R A )stands for the (first-order) language of set theory extended by the unary predicate R A . Lemma 4.4.
There is a sentence σ ∈ L ∈ ( R A ) such that for every transitive set N ( N, ∈ , X ∩ N ) | = σ implies N = L γ [ X ] for some limit γ. HARACTERIZING THE SPECTRA OF CARDINALITIES OF BRANCHES OF KUREPA TREES19
In particular, if M ≺ ( L β [ X ] , ∈ , X ∩ L β [ X ]) , where β is a limit ordinal and π is thecollapsing isomorphism from M onto the transitive set ran( π ) , then the Mostowskicollapse ran( π ) = L γ [ { π ( x ) : x ∈ M ∩ X } ] for some γ ≤ β . The following is immediate.
Claim 4.5.
For each infinite ordinal β and Y ⊆ L β [ X ] , if Y ∈ L [ X ] and X ⊆ L β [ X ] , then µ = ( | β | + ) L [ X ] implies Y ∈ L µ [ X ] . (Working in L [ X ], if Y ∈ L γ [ X ], then let M ≺ L γ [ X ] with { Y } ∪ L β [ X ] ⊆ M , | M | = | L β [ X ] | , and apply the lemma recalling that π ↾ L β [ X ] is the identity.)Now we can turn to the definition of the tree T , which will be defined by itsbranches.Recall that there exists a definable well-order on L [ A ], which is downward abso-lute to almost every initial segment of L [ A ] (to the ones indexed by limit ordinals)[Kan03, Theorem 3.3]: Lemma 4.6.
There exists a formula ϕ ∈ L ∈ ( R A ) (i.e. in the language of settheory extended with the unary relation symbol A ) which define a well-ordering on ( L [ A ] , ∈ , A ) , moreover if δ is a limit ordinal, x, y ∈ L δ [ A ] , then ( L [ A ] , ∈ , A ) | = ϕ ( x, y ) ⇐⇒ ( L δ [ A ] , ∈ , A ∩ L δ [ A ]) | = ϕ ( x, y ) . From now on ’ x < L [ A ] y ’ abbreviates ϕ ( x, y ).We will take Skolem hulls many times, thus we need to introduce the followingvariant of this standard notion. Definition 4.7.
Let ( M, ∈ , X, ∂ ), M ⊆ L [ A ] be a set model of the language L ∈ ( R A , c ∂ ) with ∅ ∈ M , M ′ ⊆ M such that the well-ordering formula ϕ ∈ L ∈ ( R A )from Lemma 4 . M , i.e.(4.1) ( ∀ x, y ∈ M ) : ( L [ A ] , ∈ , A ) | = ϕ ( x, y ) iff ( M, ∈ , X ) | = ϕ ( x, y ) , e.g. when ( M, ∈ , X ) = ( L ζ [ A ] , ∈ , A ∩ L ζ [ A ]) for some limit ordinal ζ . Then theSkolem-hull of M ′ in ( M, ∈ , X, ∂ ) (in symbols, H ( M, ∈ ,X,∂ ) ( M ′ )) is the closure of M ′ under the functions f ( M, ∈ ,X,∂ ) ψ for each formula ψ ( v , v , . . . , v n ψ ) ∈ L ∈ ( R A , c ∂ )with n ψ + 1 free variables, where the function f ( M, ∈ ,X,∂ ) ψ satisfies the following. f ( M, ∈ ,X,∂ ) ψ : M n ψ → M is defined so that for every h x , x , . . . , x n ψ i ∈ M n ψ :if ∃ y ! ∈ M s.t. ( M, ∈ , X, ∂ ) | = ψ ( y, x , x , . . . , x n ψ ) , then let f ( M, ∈ ,X,∂ ) ψ ( x , x , . . . , x n ψ ) be the unique such y, otherwise let f ( M, ∈ ,X,∂ ) ψ ( x , x , . . . , x n ψ ) = ∅ . Then the fact that for each formula ψ ′ we can define the formula saying that y is theleast y (w.r.t. the well-order given by ϕ ) satisfying ψ ′ ( y, x , x , . . . x n ψ ′ ) togetherwith the Tarski-Vaught criterion implies that the closure is an elementary submodelof M , in symbols, M ′ ≺ ( M, ∈ , X, ∂ ). Observe that this closure only depends on the isomorphism class of ( M, ∈ , X, ∂ )by the absoluteness of the well-ordering formula ϕ (4.1).Choose ξ < ω such that(4.2) ξ is the minimal ordinal ( ∀ α < ω ) ∃ f α ∈ L ξ [ A ] bijection between ω and α (which can be done due to Corollary 4 .
5, in fact ξ = ω , but we won’t use thisequality, hence we don’t argue that).Now we will define an operation which assigns for each δ ∈ [ ξ, ω ) the ordinal δ ′ < ω in the following way. We would like to choose δ ′ so that in L δ ′ [ A ] it istrue that for each set x there exists a surjection from ω to x , and for δ ′′ = δ ′ thestructures ( L δ ′ [ A ] , ∈ , A, δ ) and ( L δ ′′ [ A ] , ∈ , A, δ ) cannot be elementarily equivalent. Definition 4.8.
Fix δ ∈ [ ξ, ω ), and define δ ′ to be the least ordinal such thata) δ ∈ L δ ′ [ A ],b) for each x ∈ L δ ′ [ A ] there is a bijection f ∈ L δ ′ [ A ] between ω and x ,c) taking the sentence σ from Lemma 4 . L δ ′ [ A ] , ∈ , A ) | = σ .(Using Claim 4 . | L α [ A ] | = | α | ) L [ A ] for α ≥ ω it is easy to see that we cando this closure operation, and there is such a δ ′ < ω .) Then we have(4.3) ( δ ′ is a limit ) ^ ( L δ ′ [ A ] | = ’ ω is the largest cardinal’) , and also the desired uniqueness by our next claim. Claim 4.9.
There is a statement σ ′ ∈ L ∈ ( R A , c ∂ ) such that for each δ ∈ [ ξ, ω )( L δ ′ [ A ] , ∈ , A, δ ) | = σ ′ , moreover, for each δ > ω and δ ′′ > δ (( L δ ′′ [ A ] , ∈ , A, δ ) | = σ ′ ) ⇒ ( δ ′′ = δ ′ ) . Proof.
First define σ ′′ = σ ∧ ( ∀ y ∃ f : ω → y bijection) and let σ ′ be the followingsentence σ ′ = σ ′′ ∧ (cid:0) ¬ ( ∃ X ) ( X is transitive) ∧ ( σ ′′ ) X ∧ ( δ ∈ X ) (cid:1) (where under ψ X we always mean the formula ψ ∈ L ∈ ( R A , c ∂ ) relativized to X ,and σ is from Lemma 4 . (cid:3) Now fix δ ∈ [ ξ, ω ), and for each ordinal 0 < α < ω define M δ,α to be theSkolem-hull(4.4) M δ,α = H ( L δ ′ [ A ] , ∈ ,A,δ ) ( α ) (for each α < ω ) , Also define(4.5) M δ, = ∅ . Then(4.6) M δ,α ≺ ( L δ ′ [ A ] , ∈ , A, δ ) (for each α > . Observe that whenever M ∗ ≺ ( L δ ′ [ A ] , ∈ , A, δ ) we have for the Skolem functionsfrom Definition 4 . f ( L δ ′ [ A ] , ∈ ,A,δ ) ψ ↾ ( M ∗ ) n ψ = f ( M ∗ , ∈ ,A ∩ M ∗ ,δ ) ψ , hence(4.7) ∀ M ′ ⊆ M ∗ ≺ ( L δ ′ [ A ] , ∈ , A, δ ) : H ( L δ ′ [ A ] , ∈ ,A,δ ) ( M ′ ) = H ( M ∗ , ∈ ,A ∩ M ∗ ,δ ) ( M ′ ) . Now as we defined h M δ,α : α < ω i note that(4.8) ( M ≺ ( L δ ′ [ A ] , ∈ , A, δ )) ∧ ( | M | = ω ) → ( M ∩ ω ∈ ω ) , in particular(4.9) M δ,α ∩ ω ∈ ω , HARACTERIZING THE SPECTRA OF CARDINALITIES OF BRANCHES OF KUREPA TREES21 since (4.2) together with ξ ≤ δ < δ ′ implies that in L δ ′ [ A ] there is an enumerationof each ordinal less than ω (and M δ,α is countable). This implies that( C δ = { α < ω : M δ,α ∩ ω = α } is a club in ω ) ∧ (0 ∈ C δ ) . It is easy to see that(4.10) ∀ α < ω : M δ,α = M δ, min( C δ \ α ) . For later use we verify the following statement.
Claim 4.10. [ α<ω M δ,α = L δ ′ [ A ] . Proof.
Since the union of an increasing chain of elementary submodels is anelementary submodel, we have M ω = S α<ω M δ,α ≺ ( L δ ′ [ A ] , ∈ , A, δ ). Now recall,that in L δ ′ [ A ] every set x admits a surjection from ω onto x , therefore ω ⊆ M ω implies that M ω is transitive. Then by Lemma 4 . M ω | = σ we have M ω = L δ ′′ [ A ] for some δ ′′ > δ . But then either M ω ∈ L δ ′ [ A ], or M ω = L δ ′ [ A ], andbecause the former would contradict Claim 4 .
9, we arrive at our conclusion. (cid:3)
For each α ∈ C δ and β < ω , if α = max( C δ ∩ ( β + 1)), then let N δ,β,α bethe range of the Mostowski-collapse π δ,α of ( M δ,α , ∈ ), and let A δ,β,α = π δ,α ( A ), ∂ δ,β,α = π δ,α ( δ ):(4.11) π δ,α : M δ,α → N δ,β,α , which is of course not only an isomorphism between ( M δ,α , ∈ ) and ( N δ,β,α , ∈ ), butwitnesses(4.12) ( M δ,α , ∈ , A ∩ M δ,α , δ ) ≃ ( N δ,β,α , ∈ , A δ,β,α , ∂ δ,β,α ) . Now we are ready to construct the tree T . For a fixed δ ∈ [ ξ, ω ), α ∈ C δ , β < ω , if 0 < α = max( C δ ∩ ( β + 1)) holds then we define(4.13) t δ,β,α = ( N δ,β,α , ∈ , A δ,β,α , ∂ δ,β,α ) , i.e. the structure ( N δ,β,α , ∈ ) extended by the one-place relation for the image of A ∈ M δ,α under the collapsing isomorphism, and the constant symbol for ∂ δ,β,α .For max( C δ ∩ ( β + 1)) = 0 let t δ,β, = ∅ .Observe that given t = t δ,β,α we can decode α from t , as α is the first uncountableordinal of t . Definition 4.11.
Define T = { ( β, t δ,β,α ) : δ ∈ [ ξ, ω ) , β < ω , α = max( C δ ∩ ( β + 1)) } , with the partial order ( β , t δ ,β ,α ) ≤ T ( β , t δ ,β ,α ) iff either α = 0 (thus t δ ,β ,α is the empty structure), or(i) β ≤ β , and(ii) taking the Skolem-hull M of α in t δ ,β ,α = ( N δ ,β ,α , ∈ , A δ ,β ,α , ∂ δ ,β ,α )(i.e. M = H t δ ,β ,α ( α ) is isomorphic to t δ ,β ,α :( M, ∈ , A δ ,β ,α ∩ M, ∂ δ ,β ,α ) ≃ ( N δ ,β ,α , ∈ , A δ ,β ,α , ∂ δ ,β ,α ) , and (iii) if α < α , then there is no proper elementary submodel M ≺ ( N δ ,β ,α , ∈ , A δ ,β ,α , ∂ δ ,β ,α ) with α ∪ { α } ⊆ M, and M ∩ α ⊆ β . Roughly speaking, in level β we have (isomorphism types of) initial segments M of models of the form ( L ∆ ′ [ A ] , ∈ , A, ∆) (for some ∆ ∈ [ ξ, ω )), such that M ∩ ω ≤ β ,and M is maximal w.r.t. this condition. We need to check that T is a tree, itslevels are countable, and that it has only ω -many branches even in V .The following claim is a standard calculation, but for the sake of completenesswe include the proof. Claim 4.12.
Let δ ∈ [ ξ, ω ) be fixed, β ≤ β < ω , let α = max( C δ ∩ ( β + 1)) , α = max( C δ ∩ ( β + 1)) . Then ( β , t δ,β ,α ) ≤ T ( β , t δ,β ,α ) .Moreover, the embedding π β ,β : N δ,β ,α → N δ,β ,α is unique. Proof.
First observe that by (4.4) and (4.7) for δ ∈ [ ξ, ω ), α < α H ( M δ,α , ∈ ,A,δ ) ( α ) = H ( L δ ′ [ A ] , ∈ ,A,δ ) ( α ) = M δ,α , therefore since β < ω is such that α = max( C δ ∩ ( β + 1)), then applying (therestriction of) the collapsing isomorphism π δ,α to the left side, we obtain( H ( N δ,β ,α , ∈ ,A δ,β ,α ,∂ δ,β ,α ) ( α ) , ∈ ) ≃ ( M δ,α , ∈ )and because β < β is such that α = max( C δ ∩ ( β + 1)), then applying theisomorphism π δ,α to the right side (which fixes α ) we obtain( H ( N δ,β ,α , ∈ ,A δ,β ,α ,∂ δ,β ,α ) ( α ) , ∈ ) ≃ ( N δ,α ,β , ∈ ) . Finally, since π δ,α ( A ) = A δ,β ,α , π δ,α ( A ) = A δ,β ,α , and π δ,α ( δ ) = ∂ δ,β ,α , π δ,α ( δ ) = ∂ δ,β ,α , we have( H N δ,β ,α ( α ) , ∈ A δ,β ,α , ∂ δ,β ,α )is isomorphic to ( N δ,β ,α , ∈ , A δ,β ,α , ∂ δ,β ,α ) , therefore ( ii ) holds. The uniqueness easily follows from the facts that the embeddingof ( N δ,β ,α , ∈ , A δ,β ,α , ∂ δ,β ,α ) has to fix the ordinals less than α , and elementaryembeddings uniquely extend to Skolem-hulls.For ( iii ) suppose that α < α , and note that( N δ,β ,α , ∈ ) | = ’ α is the least uncountable ordinal, α is countable’,and for M ≺ ( N δ,β ,α , ∈ , A δ,β ,α , ∂ δ,β ,α ) if α ∪ { α } ⊆ M then consider thecorresponding submodel M ′ ≺ ( M δ,α , ∈ , A, δ ), for which M ′ ⊇ M δ,α +1 . But(recalling (4.8)) since max( C δ ∩ ( β + 1)) = α we obtain β ∪ { β } ⊆ M ′ ⊆ M δ,α ,that can happen only if β is smaller than the least uncountable ordinal in N δ,β ,α , α . But then β ∈ M ∩ α . (cid:3) The next claim will verify that T is a tree of height ω (for the transitivity of ≤ T use the claim two times). Claim 4.13.
For a fixed δ ∈ [ ξ, ω ) , β ≤ β < ω , let α = max( C δ ∩ ( β + 1) ,and fix arbitrary α ∈ ω , δ ∈ [ ξ, ω ) . Then ( β , t δ ,β ,α ) ≤ T ( β , t δ ,β ,α ) iff t δ ,β ,α = t δ ,β , max( C δ ∩ ( β ) . HARACTERIZING THE SPECTRA OF CARDINALITIES OF BRANCHES OF KUREPA TREES23
Proof.
We only have to check the ’only if’ part, but first observe that Defi-nition 4 .
11 clearly implies that up to isomorphism there exists only one t forwhich ( β , t ) ≤ ( β , t δ ,β ,α ). Now the claim is the consequence of the fact that t δ ∗ ,β ,α ∗ = t δ ∗∗ ,β ,α ∗∗ implies that they are not isomorphic as structures of thelanguage L ∈ ( R A , c ∂ ): For transitive sets N and N ′ with X, ∂ ∈ N , X ′ , ∂ ′ ∈ N ′ the structures ( N, ∈ , X, ∂ ), ( N ′ , ∈ , X ′ , ∂ ′ ) are isomorphic if and only if N = N ′ , X = X ′ and ∂ = ∂ ′ (since by the uniqueness of the Mostowski collapse we knowthat ( N, ∈ ) ≃ ( N ′ , ∈ ) iff N = N ′ ). (cid:3) Lemma 4.14.
For each β < ω the β ’th level of T is countable. Proof.
By Claim 4 .
13 we have that the β ’th level of T is T ≤ β \ T <β = { ( β, t δ,β,α ) : δ ∈ [ ξ, ω ) , α = max( C δ ∩ ( β + 1)) }} . For a fixed δ ∈ [ ξ, ω ) fix α = max( C δ ∩ ( β + 1)) too, and consider the structure t δ,β,α = ( N δ,β,α , ∈ , A δ,β,α , ∂ δ,β,α ) , where N δ,β,α is the Mostowski collapse of ( M δ,α , ∈ ) (by the isomorphism π δ,α ), and A δ,β,α = A ∩ α . Now (4.6) states M δ,α ≺ ( L δ ′ , ∈ , A ) then (recalling M δ,α ∩ ω = α ,and π δ,α ↾ α = id α ) by Lemma 4 . N δ,β,α = L γ [ A ∩ α ]for some γ = γ ( δ, α ) ∈ ( α, ω ). Now we determine an upper bound γ α for the set { γ ( δ, α ) : δ ∈ [ ξ, ω ) ∧ α ∈ C δ } . If we have such a bound for each possible α ≤ β ,then letting γ ∞ denote sup { γ α : α ≤ β } , we get { t δ,β,α ) : δ ∈ [ ξ, ω ) , α = max( C δ ∩ ( β + 1)) }} ⊆{ ( L γ [ A ∩ α ] , ∈ , A ∩ α, ∂ ) : γ ≤ γ ∞ , α ≤ β, ∂ < γ } , which latter set is obviously countable, this will finish the proof of the lemma.So fix α ≤ β and δ ∈ [ ξ, ω ) such that α ∈ C δ . Now we have two cases dependingon whether there is any (cardinal) L [ A ∩ α ] in ( α, ω ). If λ ∈ ( α, ω ) is a cardinalin the inner model L [ A ∩ α ], then for each δ if α = max( C δ ∩ ( β + 1)), then thetransitive set N δ,β,α cannot contain λ , as M δ,α sees ω as the largest cardinal, and π δ,α ( ω ) = α . This case choosing γ α = λ works.On the other hand, if ( | α | + ) L [ A ∩ α ] = ω , then we first prove that α ∈ C δ implies( | α | = ω ) L [ A ∩ α ] : otherwise in M δ,α , and in N δ,β,α each ordinal less than α arecountable, thus as well in L [ A ∩ α ]. Then it is easy to see that the condition( λ is the unique cardinal in ( ω, ω V )) L [ A ∩ λ ] cannot hold for two different λ ’s, therefore α can be defined in L [ A ]. But then usingClaim 4 . X = A ∩ α we have that for each ζ ∈ ( α, ω ) there is a bijection f ζ ∈ L ω [ A ∩ α ] between α and ζ , therefore α can be defined also in L δ ′ [ A ], and M ≺ ( L δ ′ [ A ] , ∈ ) implies α ∈ M , contradicting that M δ,α ∩ ω = α (which holdsby α ∈ C δ ). Then ( | α | = ω ) L [ A ∩ α ] and Claim 4 . λ < ω such that there exists a bijection between α and ω in L λ [ A ∩ α ], implying N δ,β,α = L γ ( δ,α ) [ A ∩ α ] ( L λ [ A ∩ α ] , since α is uncountable in N δ,β,α . This case { γ ( δ, α ) : δ ∈ [ ξ, ω ) ∧ α ∈ C δ } ⊆ γ α = λ, which completes the proof of Lemma 4 . (cid:3) Now T is obviously a Kurepa tree by the following fact and lemma. Fact 4.15.
The sequence h B δ : δ ∈ [ ξ, ω ) i lists pairwise distinct cofinal branchesin T , where B δ = { ( β, t δ,β, max( C δ ∩ ( β +1)) ) : β < ω } . Proof.
We only need to prove that B δ = B γ if δ = γ . But according to thesecond statement of Claim 4 .
12 for each β < β ′ < ω there is a unique elementaryembedding of t δ,β ′ , max( C δ ∩ ( β ′ +1)) to t δ,β, max( C δ ∩ ( β +1)) , therefore there is a uniquedirect-limit of this elementary chain, isomorphic to S α ∈ C δ M δ,α , which is ( L δ ′ [ A ] , ∈ , A, δ ) by Claim 4 . (cid:3) It is only left to prove that each branch of T is of the form B δ for some δ ∈ [ ξ, ω )(even in V ). The following lemma will complete the proof of Theorem 4 . Lemma 4.16.
Let B ⊆ T a cofinal branch in T , B ∈ V . Then B = B δ • for aunique δ • ∈ [ ξ, ω ) . Proof.
Let t δ β ,β,α β = ( N δ β ,β,α β , ∈ , A δ β ,β,α β , ∂ δ β ,β,α β ) denote the element in B ∩ ( T ≤ β \ T <β ). Working in V first we define the following bonding maps: for γ ≤ β <ω let π γ,β : N δ γ ,γ,α γ → N δ β ,β,α β be the unique elementary embedding (combining Claim 4 .
13, and the second state-ment of Claim 4 . π β ′ ,β ◦ π β ′′ ,β ′ is an elementary embedding for each β ′′ ≤ β ′ ≤ β < ω , therefore by the uniqueness(4.14) ( ∀ β ′′ ≤ β ′ ≤ β < ω ) : π β ′ ,β ◦ π β ′′ ,β ′ = π β ′′ ,β . This elementary chain allows us to define the limit D = ( N ω , E , A ω , ∂ ω ) of thedirected system { t δ β ,β,α β , π β ′ ,β : β ′ ≤ β < ω } .Let π β : N δ β ,β,α β → N ω be the embedding, N β = ran( π β ) (hence N ω = S β<ω N β ).First note that ( N ω , E ) is well-founded, otherwise there would be an infinite E -decreasing chain in the embedded image of N δ β ,β,α β for some (in fact, everylarge enough) β , contradicting that ( N δ β ,β,α β , ∈ ) is well-founded. Now (by the E -extensionality in N ω ) we can assume that N ω is a Mostowski collapse, i.e.( N ω , E ) = ( N ω , ∈ ). Then it is easy to see that if β < ω for the elementaryembedding π β : N δ β ,β,α β → N ω we have π β ↾ α β = id α β , and π β ( α β ) = ω , thus(recalling that A δ β ,β,α β = A ∩ α β ) we obtain ( N ω , E , A ω , ∂ ω ) = ( N ω , ∈ , A, δ • ) forsome δ • ∈ ( ω , ω ). Now we can use Lemma 4 . N δ β ,β,α β , ∈ , A δ β ,β,α β ) | = σ ),there exists ζ > δ • such that N ω = L ζ [ A ] , and then ( N ω , ∈ , A, δ • ) = ( L ζ [ A ] , ∈ , A, δ • ) . Now because the formula σ ′ ∈ L ∈ ( R A , c ∂ ) from Claim 4 . L δ ′ [ A ] , ∈ , A, δ )(for each δ ∈ [ ξ, ω )) (for our mapping δ δ ′ from Definition 4 .
8) and thereforealso in M δ,α ’s, N δ,β,α ’s ( δ ∈ [ ξ, ω )), so it must hold in ( N ω , ∈ , A, δ • ), which meansthat δ • ≥ ξ , and ζ = δ ′• , i.e.( N ω , ∈ , A, δ • ) = ( L δ ′• [ A ] , ∈ , A, δ • ) , HARACTERIZING THE SPECTRA OF CARDINALITIES OF BRANCHES OF KUREPA TREES25
Finally, we have to prove that for each β < ω t δ β ,β,α β = ( N δ β ,β,α β , ∈ , A δ β ,β,α β , ∂ δ β ,β,α β ) = t δ • ,β, max( C δ • ∩ ( β +1)) by arguing (having β fixed) that for a large enough γ ( β, t δ • ,β, max( C δ • ∩ ( β +1)) ) ≤ T ( γ, t δ γ ,γ,α γ ) . Let α = max( C δ • ∩ ( β + 1)), α ′ = min( C δ • \ ( β + 1)), β ′ = α ′ , and considerthe models M δ • ,α , M δ • ,α ′ ≺ ( L δ ′• [ A ] , ∈ , A, δ • ). Choose γ ≥ β ′ , γ < ω so that N γ = π γ [ N δ γ ,γ,α γ ] ⊇ M δ • ,α ′ . Then(4.15) α γ ≥ α ′ > β + 1 , and α ′ ∪ { ω } ⊆ N γ ≺ ( L δ ′• [ A ] , ∈ , A, δ • ) with (4.7) imply H ( N γ , ∈ ,A ∩ N γ ,δ • ) ( α ) = H ( L δ ′• [ A ] , ∈ ,A,δ • ) ( α ) = M δ • ,α . Therefore in ( N γ , ∈ , A ∩ N γ , δ • ) ≃ ( N δ γ ,γ,α γ , ∈ , A δ γ ,γ,α γ , ∂ δ γ ,γ,α γ ) there is an elemen-tary submodel isomorphic to ( M δ • ,α , ∈ , A ∩ M δ • ,α , δ • ), which latter is isomorphicto ( N δ • ,β,α , ∈ , A ∩ α, ∂ δ • ,β,α ), thus ( ii ) from Definition 4 .
11 holds.Similarly, using also (4.10) and the definitions of α , α ′ H ( N γ , ∈ ,A ∩ N γ ,δ • ) ( α + 1) = M δ • ,α +1 = M δ • ,α ′ ⊇ α ′ ⊇ β ∪ { β } , ans since the isomorphism between ( N γ , ∈ , A ∩ N γ , δ • ) and ( N δ γ ,γ,α γ , ∈ , A δ γ ,γ,α γ , ∂ δ γ ,γ,α γ ) fixes the ordinals less than or equal to α ′ we obtain H ( N δγ,γ,αγ , ∈ ,A δγ,γ,αγ ,∂ δγ,γ,αγ )( α + 1) ⊇ β ∪ { β } . Therefore recalling (4.15) we obtain that ( iii ) (of Definition 4 .
11) holds as well. (cid:3)
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