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Thomas Jech and Jiˇr´i Witzany

arXiv:math/9302202v1 [math.LO] 25 Feb 1993FULL REFLECTION AT AMEASURABLE CARDINALThomas Jech and Jiˇr´i WitzanyThe Pennsylvania State University and Charles University (Prague)Abstract. A stationary subset S of a regular uncountable cardinal κ reflects fullyat regular cardinals if for every stationary set T ⊆κ of higher order consisting ofregular cardinals there exists an α ∈T such that S ∩α is a stationary subset of α.Full Reflection states that every stationary set reflects fully at regular cardinals.

Wewill prove that under a slightly weaker assumption than κ having Mitchell order κ++it is consistent that Full Reflection holds at every λ ≤κ and κ is measurable.1. Definitions and results.It has been proved in [M82] that reflection of stationary sets is a large cardinalproperty.

Reflection of stationary subsets of ωn (n ≥2) and ωω+1 has been inves-tigated in [M82] and [JS90] and consistency strength of Full Reflection at regularcardinals at a Mahlo cardinal has been characterized in [JS92]. In this paper weaddress the question of Full Reflection at a measurable cardinal.If S is a stationary subset of a regular uncountable cardinal κ then the trace ofS is the setTr(S) = {α < κ; S ∩α is stationary in α}(and we say that S reflects at α).

If S and T are both stationary, we defineS < T if for almost all α ∈T, α ∈Tr(S)and say that S reflects fully in T. (Throughout the paper, “for almost all” means“except for a nonstationary set of points”).Lemma 1.1. ([J84]) The relation < is well founded.Proof.

By contradiction suppose there is a sequence of stationary sets such thatA1 > A2 > A3 > · · ·1991 Mathematics Subject Classification. 03E35, 03E55.Key words and phrases.

Stationary sets, full reflection, measurable cardinals, repeat points.The first author was supported by NSF grant number DMS-8918299. The second author waspartially supported by the first author’s NSF grant.

Both authors wish to thank S. Baldwin, W.Mitchell and H. Woodin for their comments on the subject of this paper.Tt bAMS T X

2THOMAS JECH AND JIˇR´I WITZANYIt means that there are clubs Cn such thatAn ∩Cn ⊆Tr(An+1) for n = 1, 2, . .

.If C ⊆κ is a club let us denote C′ = {α < κ; C ∩α is unbounded in α}, ( C′ isagain a club) and put˜An = An ∩Cn ∩C′n+1 ∩C′′n+2 ∩· · ·for n = 1, 2, . .

.Then all ˜An are stationary.Observe that α ∈Tr(S) implies cf(α) > ω andTr(S ∩C) = Tr(S) ∩C′ where C is any club. Now it is easy to verify that˜An ⊆Tr( ˜An+1)for n = 1, 2, .

. .Let αn = min( ˜An).

Since ˜An+1 ∩αn is stationary in αn, the ordinal αn+1 must beless then αn. We obtain a sequence of ordinalsα1 > α2 > α3 > · · ·- a contradiction.The order o(S) of a stationary set of regular cardinals is defined as the rank ofS in relation <:o(S) = sup{o(T) + 1; T ⊆Reg(κ) stationary and T < S}.For a stationary set T such that T ∩Sing(κ) is stationary we define o(T) = −1.The order of κ is then defined aso(κ) = sup{o(S) + 1 ; S ⊆κ is stationary}.Note that if Tr(S), where S ⊆Reg(κ), is stationary then o(S) < o(Tr(S)) becauseS < Tr(S).

It follows from [J84] that the order o(κ) provides a natural general-ization of the Mahlo hierarchy: κ is exactly o(κ)-Mahlo if o(κ) < κ+ and greatlyMahlo if o(κ) ≥κ+.We say that a stationary set S reflects fully at regular cardinals if for any sta-tionary set T of regular cardinals o(S) < o(T) implies S < T.Axiom of Full Reflection at κ. Every stationary subset of κ reflects fully atregular cardinals.Following [J84] we say that a stationary set E is canonical of order ν if E ishereditarily of order ν (i.e.

o(X) = ν for every stationary X ⊆E) and E meetsevery stationary set of order ν.The existence of canonical stationary sets of order less then κ+ (if a set of suchorder exists) is proved in [BTW76] and [J84]. In the model constructed in Section3 we get a sequence of stationary sets with the following properties:Lemma 1.2.

Let ⟨Eδ; −1 ≤δ < θ⟩be a maximal antichain of stationary subsetsof λ such that(i) E−1 = Sing(λ), Eδ ⊆Reg(λ) for δ ≥0,(ii) for any δ ≥0 the set Tr(Eδ) ∩Eδ is nonstationary,(iii) if S ⊆Eδ is stationary and −1 ≤δ < δ′ then S < Eδ′.Then each Eδ is a canonical stationary set of order δ, o(λ) = θ and Full Reflectionholds at λ.ProofWe will prove the lemma in several steps Obviously E < Eif δ < δ′

FULL REFLECTION AT A MEASURABLE CARDINAL3Claim 1. Let T ⊆Reg(λ) be a stationary set such that T ∩Eδ′ is nonstationaryfor δ′ ≤δ and S ⊆Eδ stationary.

Then S < T.Proof. We need to prove that T \ Tr(S) is nonstationary.

But (T \ Tr(S)) ∩Eδ′ isnonstationary for δ′ ≤δ because T ∩Eδ′ is nonstationary, and for δ′ > δ the set(T\Tr(S))∩Eδ′ is nonstationary because Eδ′\Tr(S) is nonstationary. ConsequentlyT \ Tr(S) is nonstationary.Claim 2.

If S ⊆Eδ is stationary then o(S) = o(Eδ).Proof. Suppose the claim holds for δ′ < δ and that for some S ⊆Eδ stationaryo(S) > o(Eδ).

Then there is T < S such that o(T) = o(Eδ). By the inductionhypothesis T ∩Eδ′ must be nonstationary for δ′ < δ (T1 ⊆T2 stationary implieso(T1) ≥o(T2)).

Moreover Eδ ∩Tr(T ∩Eδ) is nonstationary. Thus S \ Tr(T \ Eδ)must be nonstationary because S \Tr(T) = (S \Tr(T \Eδ))∩(λ\(Eδ ∩Tr(T ∩Eδ))is nonstationary.

It means that (T \ Eδ) < S but by claim 1 S < (T \ Eδ) which isa contradiction with well-foundedness of <.Claim 3. o(Eδ) = δ for δ < θλ.proof. Suppose by induction that o(Eδ′) = δ′ for δ′ < δ.

Certainly o(Eδ) ≥δ,suppose by contradiction that there is a set T < Eδ such that o(T) = δ. As in theproof of claim 2 we can suppose that T ∩Eδ′ is nonstationary for δ′ ≤δ.

But itimplies by claim 1 that Eδ < T - a contradiction.It follows from these claims that each Eδ is a canonical stationary set of order δ.Any S ⊆Reg(λ) stationary must have a nonstationary intersection with some Eδwhich means o(S) ≤δ and so o(λ) = θλ, actuallyo(S) = min{δ < θλ; Eδ ∩S is stationary}.Finally let S ⊆λ, T ⊆Reg(λ) be stationary and δ = o(S) < o(T) then by claim 1S ∩Eδ < T which implies S < T.To state our result we need to review the definition of Mitchell order and of acoherent sequence.IfU, V are two measures on κ then U ⊳V is defined as U ∈V κ/V. Thetransitive relation ⊳is known to be well-founded (see [Mi74]).

The Mitchell orderof κ is then defined as the rank of this relation on measures over κ.A coherent sequence of measures is a function −→U with domain of the form{(α, β); α < l(U) and β < oU(α)} for some ordinal l(U) and a function oU(·)such that(i) For all (α, β) ∈dom−→UUαβ = −→U (α, β) is a measure on α,(ii) if j is the canonical embedding j : V →V α/Uαβ thenj(−→U ) ↾(α + 1) = −→U ↾(α, β) where−→U ↾(α, β) = −→U ↾{(α′, β′); α′ < α or α′ = α and β′ < β} and−→U ↾(α + 1) = −→U ↾(α + 1, 0).Observe that in particular Uα0 ⊳Uα1 ⊳· · · ⊳Uαβ ⊳· · ·(β < oU(α)). Thefollowing is proved in [Mi83]

4THOMAS JECH AND JIˇR´I WITZANYProposition 1.3. There is a class sequence −→U such that L[−→U ] |= “ For every α−→U ↾(α + 1) is a coherent sequence, every measure on α is equal to some Uαβ andoU(α) = min{(Mitchell order)V (α), α++}” .

Moreover, L[−→U ] satisfies GCH.We say that κ has a repeat point (see [Mi82]) if there is a coherent sequence −→Uup to κ and an ordinal θ < oU(κ) such that∀X ∈Uκθ ∃α < θ : X ∈Uκα .It can be proved that such θ must be greater than κ+. Suppose we have a coherentsequence −→U such that oU(κ) = κ++; then using a simple counting argument we canprove the existence of a repeat point for κ. Consequently, if Mitchell order of (κ)is κ++ then there is an inner model satisfying GCH where κ has a repeat point.Our result is the following:Theorem.

If κ has a repeat point in the ground model V satisfying GCH thenthere is a generic extension of V preserving cardinalities, cofinalities and GCH inwhich Full Reflection holds at all λ ≤κ and κ is measurable.Actually if we start with V = L[−→U ] from proposition 1.3 then our constructionprovides a class generic extension V [G] preserving cardinalities, cofinalities andGCH such that for any cardinal λ V [G] satisfies Full Reflection at λ, (Mitchellorder)V (λ) = oV [G](λ) and if λ has a repeat in V then λ is measurable in V [G].2. The Forcing Pκ+1.From now on we work in a ground model V satisfying GCH with a coherentsequence −→U up to κ and a repeat point at κ.

For λ ≤κ let θλ be oU(λ) if λ doesnot have a repeat point, or otherwise the least θ such that Uλθ is a repeat point.As usual, if P is a forcing notion then V (P) denotes either the Boolean valuedmodel or a generic extension by a P-generic filter over V .Pκ+1 will be an Easton support iteration of ⟨Qλ; λ ≤κ⟩, Qλ will be nontrivialonly for λ Mahlo. Qλ (for λ Mahlo) is defined in V (Pλ), where Pλ denotes theiteration below λ, as an iteration of length λ+ with < λ-support of forcing notionsshooting clubs through certain sets X ⊆λ (we will denote this standard forcingnotion CU(X)), always with the property that X ⊇Sing(λ).

This condition willguarantee Qλ to be essentially < λ-closed (i.e. for any γ < λ there is a dense γ-closed subset of Qλ).

Qλ will also satisfy the λ+-chain condition. Consequently Pλwill satisfy λ-c.c.

and will have size λ. Cardinalities, cofinalities and GCH will bepreserved, stationary subsets of λ can be made nonstationary only by the forcing atλ, not below λ, and not after the stage λ - after stage λ no subsets of λ are added.We use the λ+-chain condition of Qλ to get a canonical enumeration of lengthλ+ of all the λ+ Qλ-names for subsets of λ so that the βth name appears inV (Pλ ∗Qλ|β). Moreover for δ < θλ we will define filters Fδ = F λδ in V (Pλ ∗Qλ|β).Their definition will not be absolute, however the filters will extend the V -measuresUλδ and will increase coherently during the iteration.Definition.

An iteration Q of ⟨CU(Bα); α < α0⟩with < λ-support and lengthα0 < λ+ is called an iteration of order δ0 if for all α < α0,V (P ∗Q|α) |B∈Ffor any δ < δand Sing(λ) ⊆B

FULL REFLECTION AT A MEASURABLE CARDINAL5(Note that an iteration of order δ0 is also an iteration of order δ, for all δ < δ0. )Qλ is then defined as an iteration of ⟨CU(Bα); α < λ+⟩with < λ-support andlength λ+ so that every Qλ|α is an iteration of order θλ and all potential names˙X ⊆λ are used cofinally many times in the iteration as some Bα.Observe that Qλ can be represented in V (Pλ) as a set of sequences of closedbounded subsets of λ in V (Pλ) rather than in V (Pλ ∗Qλ|α).

Moreover if ˙q is aPλ-name such that 1∥– Pλ ˙q ∈Qλ then using the λ-chain condition of Pλ there is aset A ⊆λ+ (in V ) of cardinality < λ and γ0 < λ so that1∥– Pλ supp ˙q ⊆A and ∀α ∈A : ˙q(α) ⊆γ0 .Consequently, Qλ can be represented as a set of functions g : A × γ0 →[Pλ]<λwhere A ⊆λ+, |A| < λ and γ0 < λ. In this sense Qλ has cardinality λ+ and anyQλ|α has cardinality at most λ.We will need to lift various elementary embeddings to generic extensions.

Fora review of basic methods see [WoC92]. We will often use the following simplefact: Let N be a submodel of M such that M ∩κN ⊆N and let G be a filterP-generic/M where P satisfies κ+-c.c.

ThenM[G] ∩κN[G] ⊆N[G] .Moreover if Q is κ-closed and H is Q-generic/M[G] thenM[G ∗H] ∩κN[G ∗H] ⊆N[G ∗H] .Definition of filters Fδ.The filters Fδ in V (Pλ ∗Q), where Q is any iteration of order δ + 1, are definedby induction so that the following is satisfied:Proposition 2.1. Let Q, Q′ = Q ∗R be two iterations of order δ′ + 1 thenF V (Pλ∗Q)δ′= F V (Pλ∗Q′)δ′∩V (Pλ ∗Q) .Moreover F V (Pλ)δ′∩V = Uλδ′.Proposition 2.2.

Let j = jδ′ be the canonical embedding from V into V λ/Uλδ′ = Mand Q an iteration of order δ′ +1. Then j can be lifted to an elementary embeddingfrom the generic extension V (Pλ ∗Q) of V to a generic extension M(jPλ ∗jQ) ofM.Lemma 2.3.

Let N = V λ/Uλβ for some β > δ′ and Q be an iteration of orderδ′ + 1. ThenF V (Pλ∗Q)δ′= F N(Pλ∗Q)δ′.Note that it also means that the definition of Fδ′ relativized to N(Pλ ∗Q) makessense.Lemma 2.4.

Let j = jδ′ : V →M.Then any iteration Q of order δ′ is ansubiteration of (jPλ)λ, where (jPλ)λ is the factor of jPλ = Pλ ∗(jPλ)λ ∗(jPλ)>λ.Consequently for any G∗jPλ-generic/V and any q ∈Q there is an H ∈M[G∗] Q-generic/V [G] containing q given by an embedding of Q as a subiteration of (jPλ)λ,where G = G∗↾Pλ.

6THOMAS JECH AND JIˇR´I WITZANYDefinition. Let j, Q, G∗, G be as in the lemma.

Then Genj(Q, G∗) is the set ofall filters H ∈M[G∗] Q-generic/V [G] given by an embedding of Q as a subiterationof (jPλ)λ.Lemma 2.5. Let j be as above, Q an iteration of order δ′ + 1, G∗jPλ-generic/V ,H ∈Genj(Q, G∗).

For every β < l(Q) let Cβ ⊂λ be the club S{rβ; r ∈H}, andlet [H]j denote the j(l(Q))-sequence given by[H]jγ = Cλ ∪{λ},if γ = j(β)∅otherwise.Then [H]j ∈jQ/G∗.Propositions 2.1 and 2.2 will be essential to prove Full Reflection in the genericextension. We will later prove that if lemma 2.3 holds for δ′ < δ then lemma 2.4holds for all δ′ ≤δ.Now suppose that the filters F λ′δ′ were defined for all λ′ < λ and δ′ < θλ′ andfor λ′ = λ and δ′ < δ so that 2.1-2.5 holds.

Moreover let α < λ+ and Fδ = F λδ bedefined for all iterations of order δ + 1 of length < α so that lemma 2.5 holds for δand iterations of length ≤α. Let j = jδ : V →M = V λ/Uλδ .

Then we can defineFδ for iterations of order δ + 1 and length α.Definition. Let Q be an iteration of order δ + 1 and length α, j = jδ : V →M.For a Pλ ∗Q-name ˙X of a subset of λ and (p, q) ∈Pλ ∗Q we define(p, q)∥– Pλ∗Q ˙X ∈Fδif the following holds in V:j(p)∥– jPλ “For any H ∈Genj(Q, G∗) containing q :[H]j∥– jQ ˇλ ∈j ˙X”.The definition says that (p, q)∥– ˙X ∈Fδ if λ ∈j∗X whenever j∗: V [G ∗H] →M[G∗∗H∗] is a lifting of j of certain kind and (p, q) ∈G ∗H.

To verify soundnessof the definition let us first prove lemma 2.4 for δ.Proof of lemma 2.4. Let j = jδ : V →M = V λ/Uδλ, Q be an iteration of order δ. Weassume that lemma 2.3 holds for δ′ < δ.

Observe that jPλ = Pλ∗(jPλ)λ∗(jPλ)>λ isan Easton support iteration (in M) below jλ and (jPλ)λ is an iteration of length λ+with < λ-support such that for any α < λ+ (jPλ)λ|α is an iteration of order δ = θMλand all potential names ˙X ⊆λ are used cofinally many times in the iteration. Thatis true in M(Pλ) as in V (Pλ).Let us now define what it means for Q to be a subiteration of (jPλ)λ. Supposethat P, Q are iterations of legths l(P) ≤l(Q) of ⟨˙Rγ; γ < l(P)⟩and ⟨˙Sα; α < l(Q)⟩with < λ-support essentially < λ-closed.

Then we say that P is a subiteration of Qif there is an increasing sequence ⟨αγ; γ < l(P)⟩of ordinals below l(Q) such that1)Qα0∥– ˇR0 = ˙Sα0consequently for β > α0 Qβ ≃P1 ∗Q′β ,2)and by induction Qαγ∥– ˇ˙Rγ = ˙Sαγ ,using the inductive assumption that Qαγ ≃Pγ ∗Q′αγ ,consequently again for β > αwe have Q≃P∗Q′

FULL REFLECTION AT A MEASURABLE CARDINAL7It is now obvious that in this sense Q is a subiteration of (jPλ)λ. Moreover for anyα < λ+ there is a sequence ⟨αγ; γ < l(Q)⟩determining an embedding of Q into(jPλ)λ such that α0 > α.Finally let G∗be jPλ-generic/V and q ∈Q.

Then (G∗)λ is (jPλ)λ-generic/V [G].Note that the setD = {r ∈(jPλ)λ; there is a sequence ⟨αγ; γ < l(Q)⟩determining an embedding of Q into (jPλ)λsuch that q corresponds to r ↾⟨αγ; γ < l(Q)⟩}is dense in (jPλ)λ. Thus let r ∈D ∩(G∗)λ and ⟨αγ; γ < l(Q)⟩be the sequence.Then (G∗)λ ↾⟨αγ; γ < l(Q)⟩gives the Q-generic/V [G] filter H ∋q.We have defined Fδ for iterations of length ≤α.

Let us now prove lemma 2.5for iterations of length α + 1.Proof of lemma 2.5. Let j = jδ : V →M, Q be an iteration of order δ + 1of ⟨CU(Bβ); β < α + 1⟩, G∗jPλ-generic/V , H ∈Genj(Q, G∗), [H]j as in thelemma.

Note that [H]j is a j(α + 1)-sequence of closed bounded subsets of jλ,supp [H]j = j”(α + 1). Since |j”(α + 1)| = λ < jλ in M, [H]j has a small supportand [H]j ∈M[G∗].

It follows from the induction hypothesis that [H]j ↾j(α) ∈j(Q ↾α). We only have to verify that in M[G∗][H]j ↾j(α)∥– j(Q)↾j(α) [H]jj(α) ∈j(⟨CU(Bβ); β < α + 1⟩)j(α)which means just that[H]j ↾j(α)∥– [H]jj(α) ⊆j(Bα)where [H]jj(α) = ∪{rα; r ∈H} ∪{λ}.

If γ ∈[H]jj(α), γ < λ, then γ ∈rα for somer ∈H andV [G] |= r ↾α∥– Q|αrα ⊆BαsoM[G∗] |= j(r ↾α)∥– j(Q|α)rα = jrα ⊆jBα .Since [H]j ↾j(α) ≤j(r ↾α) we get [H]j ↾j(α)∥– γ ∈jBα . So we only have toprove that in M[G∗](*)[H]j ↾j(α)∥– λ ∈jBαHere we use the fact that Q is an iteration of order δ + 1.

It implies that Q ↾α∥– Bα ∈Fδ and that exactly gives (*) in V [G∗] and so in M[G∗] by the definitionof Fδ in V (Pλ ∗Q|α).Fδ is now well defined in V (Pλ ∗Q) for any iteration Q of order δ + 1. We haveto verify Proposition 2.1, 2.2 and Lemma 2.3 for Fδ.Proof of Lemma 2.3.Let β > δ, N = V λ/Uλβ , j = jδ : V →M, Q an iteration of order δ +1.

We wantto prove thatF V (Pλ∗Q)F N(Pλ∗Q)

8THOMAS JECH AND JIˇR´I WITZANYLet j′ : N →N λ/Uλδ = N ′, observe that j′ = j ↾N. We need to prove that thefollowing two conditions are equivalent:V |= jp∥– jPλ∀H ∈Genj(Q, G∗), H ∋q(1)[H]j∥– jQˇλ ∈j ˙X ,N |= jp∥– jPλ∀H ∈Genj(Q, G∗), H ∋q(2)[H]j∥– jQˇλ ∈j ˙X .Observe that V [G∗] ∩λN[G∗] ⊆N[G∗], and soGenj(Q, G∗)V [G∗] = Genj(Q, G∗)N[G∗].From that the equivalence of (1) and (2) easily follows.Proof of Proposition 2.2.

Let j = jδ : V →M, Q be an iteration of order δ + 1.Let G∗be jPλ-generic/V, G = G∗↾Pλ. Then j is lifted to j∗: V [G] →M[G∗] .Using Lemma 2.4 find H ∈M[G∗] Q-generic/V [G].

By Lemma 2.5 [H]j ∈j∗Q andr ∈H implies j∗r ≥[H]j. Thus let H∗be jQ-generic/V [G∗] containing [H]j. Thenr ∈H implies j∗r ∈H∗and j∗is lifted to j∗∗: V [G ∗H] →M[G∗∗H∗] .Proof of Proposition 2.1.

Let Q, Q′ = Q ∗R be iterations of order δ + 1, we wantto proveF V (Pλ∗Q)δ= F V (Pλ∗Q′)δ∩V (Pλ ∗Q) .Firstly let us prove the easy direction:˙X ⊆λ a Pλ ∗Q-name, (p, q) ∈Pλ ∗Q,(p, q)∥– Pλ∗Q ˙X ∈Fδ . Then it is straightforward that (p, q⌢1)∥– Pλ∗Q′ ˙X ∈Fδ .Now let ˙X ⊆λ be a Pλ ∗Q-name, (p, q′) ∈Pλ ∗Q′ and (p, q′)∥– Pλ∗Q′ ˙X ∈Fδ .We prove that (p, q)∥– Pλ∗Q ˙X ∈Fδ where q = q′ ↾l(Q).

Let G∗∋p be jPλ-generic/V, j = jδ : V →M, H ∈Genj(Q, G∗) and q ∈H. We want to prove that[H]j∥– j(Q)ˇλ ∈j ˙X .

Suppose not, then there is ˜q ≤[H]j such that ˜q∥– j(Q)ˇλ /∈j ˙X .Obviously there exists ˜H ∈Genj(Q′, G∗) such that ˜H ↾Q = H and ˜H ∋q′. Notethat ˜q⌢1 and [ ˜H]j are compatible, ˜q⌢1 ∪[ ˜H]j ∈jQ′.

But [ ˜H]j∥– jQ′ˇλ ∈j ˙X and˜q⌢1∥– jQ′ˇλ /∈j ˙X - a contradiction.Finally let us prove that Uλδ = F V (Pλ)δ∩V. The inclusion Uλδ ⊆Fδ is obvious.Now let X ∈V, X ⊆λ and p∥– Pλ ˇX ∈Fδ.

Then it means that j(p)∥– jPλˇλ ∈j( ˇX)which can be true only if λ ∈j(X). So X ∈Uλδ .3.

Full Reflection in V (Pκ+1).To prove that Full Reflection holds in V (Pκ+1) at some λ ≤κ it is enough toprove that in V (Pλ+1). Fix λ ≤κ.Firstly let us prove the existence of sets Eδ (δ < θλ) separating the measures Uλδin the sense that Eδ ∈Uλδ′ iffδ = δ′.Proposition 3.1.

Suppose that Uλδ is not a repeat point.Then there is a setE ∈Uλδ such that E /∈Uλδ′ for any δ′ ̸= δ, δ′ < oU(λ).Proof. Since Uλδ is not a repeat point there is a set X ∈Uλδ such that X /∈Uλδ′ forall δ′ < δ. Define for ξ < λfδ(ξ) = sup{η; η ≤oU(ξ)&∀η′ < η : X ∩ξ /∈Uξη′}andY = {ξ < λ; fδ(ξ) = oU(ξ)}.

FULL REFLECTION AT A MEASURABLE CARDINAL9Claim. Y ∈Uλδ but Y /∈Uλδ′ for any δ′ such that δ < δ′ < oU(λ).Proof.

1) Let j : V →N = V λ/Uλδ . ThenY ∈Uλδ iffλ ∈jY = {ξ < jλ; N |= ojU(ξ) = j(fδ)(ξ)}iffN |= ojU(λ) = j(fδ)(λ)By the definition of a coherent sequence ojU(λ) = δ. MoreoverN |= j(fδ)(λ) = sup{η; η ≤δ & ∀η′ < η : jX ∩λ /∈jUλη′}.By coherence jUλη′ = Uλη′ for η′ < δ, and also jX∩λ = X. Consequently j(fδ)(λ) = δand Y ∈Uλδ .2) Let δ < δ′ < oU(λ) and j : V →N = V λ/Uλδ′.

Again Y ∈Uλδ′ iffN |= ojU(λ) =j(fδ)(λ). As above ojU(λ) = δ′ but j(fδ)(λ) = δ since X /∈Uλδ .

Thus Y /∈Uλδ′ andthe claim is proved.Finally put E = Y ∩X.So we have a separating sequence. We can suppose that the Eα are sets of regularcardinals because Reg(λ) ∈Uλδ for any δ.

We are going to prove the following:Proposition 3.2. In V (Pλ+1) the sets ⟨Eα; α < θλ⟩form a maximal antichainof stationary subsets of Reg(λ).

Moreover if S ⊆E−1 = Sing(λ) or S ⊆Eα isstationary then S reflects in any Eβ for β > α and Tr(Eα)∩Eα is nonstationary ifα > −1. Consequently each Eα is a canonical stationary set of order α, o(λ) = θλand Full Reflection at λ holds.To prove the proposition we need the following lemmas:Lemma 3.3.

V (Pλ ∗Qλ|α) |= Club(λ) ⊆Fδ for any δ < θλ.Proof. Let δ < θλ and (p, q)∥– Pλ∗Qλ|α “ ˙X ⊆λ is a club”.

If (p, q) does not force˙X ∈Fδ then by the definition of Fδ there is a G∗jPλ-generic/V, G∗∋jp, wherej : V →V λ/Uλδ = M, and H ∈Genj(Qλ|α, G∗), H ∋q, such that in V [G∗][H]j ̸ ∥– j(Qλ|α) ˇλ ∈j ˙X. So there is H∗∋[H]j j(Qλ|α)-generic /V [G∗] so thatλ /∈j ˙X/G∗∗H∗.

The embeding j is by the proof of Propostion 2.2 lifted to theelementary embeding j∗∗: V [G ∗H] →M[G∗∗H∗], X =˙X/G∗H is a club inV [G ∗H], thus j∗∗X = j ˙X/G∗∗H∗is a club in M[G∗∗H∗]. Since j∗∗X ∩λ = Xnecessarily λ ∈j∗∗X - a contradiction.Lemma 3.4.

If S in V (Pλ ∗Qλ|α) is Fδ-positive then Tr(S) ∈Fδ′ for any δ′ > δ.If S ⊆Sing(λ) is stationary then Tr(S) ∈Fδ for any δ. Moreover V (Pλ) |=Eδ \ Tr(Eδ) ∈Fδ.Proof.

Suppose (p, q)∥– Pλ∗Qλ|α “ ˙S is Fδ-positive” but (p, q) ̸ ∥– Pλ∗Qλ|α “Tr( ˙S) ∈Fδ′” for some δ′ > δ. As usual denote j : V →M = V λ/Uλδ′.

Then there is a filterG∗jPλ-generic/V, G∗∋jp, and H ∈Genj(Qλ|α, G∗), H ∋q, and a filter H∗j(Qλ|α)-generic/V [G∗], H∗∋[H]j, so that λ /∈jTr( ˙S)/G∗∗H∗. As above j is liftedtoj∗∗: V [G ∗H] →M[G∗∗H∗]

10THOMAS JECH AND JIˇR´I WITZANYS = ˙S/G∗H is Fδ-positive in V [G ∗H] andjTr( ˙S)/G∗∗H∗= j∗∗(Tr(S)) = TrM[G∗∗H∗](j∗∗S) .Thus λ /∈TrM[G∗∗H∗](j∗∗S) which means thatM[G∗∗H∗] |= “ S is not stationary in λ”because S = j∗∗S ∩λ. Consequently alsoV [G∗∗H∗] |= “S is not stationary in λ”.Observe that j(Qλ|α) has a dense subset λ-closed in M[G∗] and thus also in V [G∗].Moreover jPλ = Pλ ∗(jPλ)λ ∗R where R is essentially λ-closed in V [G∗|λ + 1].

Itimplies that already V [G∗|λ + 1] |= “S is not stationary in λ”. Let us now considerthe isomorphism (jPλ)λ ≃(Qλ|α)∗˜Q from the proof of Lemma 2.4 giving the filterH = G∗↾(Qλ|α), let ˜H = G∗↾˜Q.

Since every subset of λ in V [G ∗H ∗˜H] isalready in some V [G ∗H ∗˜H|β] there is a β < λ+ so thatV [G ∗H ∗˜H|β] |= “S is not stationary in λ”.But since (Qλ|α)∗( ˜Q|β) is an iteration of order δ′ ≥δ+1 it follows from Proposition2.1 thatV [G ∗H ∗˜H|β] |= “S is Fδ-positive”which contradicts Lemma 3.3.The proof for S ⊆Sing(λ) is the same using the following fact instead of Propo-sition 2.1.Claim. Stationary subsets of Sing(λ) are preserved by iterations of order 0.Proof.

For simplicity assume that R = CU(X) where X ⊇Sing(λ); the general-ization for an iteration of order 0 is straightforward. We closely follow the proof of7.38 in [J86].Let S ⊆Sing(λ) be stationary, ˙C an R-name and p∥– R ˙C ⊆λ is a club.

Weneed a ˜q ≤p and β ∈S so that ˜q∥– β ∈˙C. Put A0 = {p}, γ0 = max(p), andinductively for q ∈Aα find r(q) ≤q and β(q) > γα so that max(r(q)) > γα andr(q)∥– β(q) ∈˙C.

PutAα+1 = Aα ∪{r(q); q ∈Aα} andγα+1 = sup({max(q); q ∈Aα+1} ∪{β(q); q ∈Aα}).For β limit putAβ =[α<βAα ∪{unions of all decreasing sequences⊆[α<βAα that are in R} andγsup{γ ; α < β}

FULL REFLECTION AT A MEASURABLE CARDINAL11Find a β ∈S such that γβ = β. Observe that cf(β) < β and all unions of increasingsequences ⊆Scf(β)<α<β Aα of length ≤cf(β) are in R. Now it is easy to find anincreasing sequences βα ր β and decreasing qα ց ˜q ∈R (α < cf(β)) so thatqα∥– βα ∈˙C.

Consequently ˜q∥– β ∈˙C.Let us now prove that V (Pλ) |= Eδ \ Tr(Eδ) ∈Fδ . Let j = jδ : V →M then(jPλ)λ is an iteration of length λ+ such that (jPλ)λ|α is always an iteration oforder δ and every potential name is used cofinally many times.

Thus a club is shotthrough λ \ Eδ in the iteration. It implies thatV [G∗] |= Eδ ⊆λ is nonstationaryand consequentlyV [G∗] |= λ ∈j(Eδ \ Tr(Eδ)).Proof of Proposition 3.2.

That each Eδ is stationary in V (Pλ+1) folows easily fromProposition 2.1 and Lemma 3.3. Let δ ̸= δ′ < θλ, then λ\ (Eδ ∩Eδ′) ∈Uλη for anyη < θλ and λ \ (Eδ ∩Eδ′) ⊇Sing(λ), so λ \ (Eδ ∩Eδ′) contains a club in V (Pλ+1),and Eδ ∩Eδ′ is nonstationary.

Let now A ⊆Reg(λ), A ∈V (Pλ+1) be such thatA∩Eδ is nonstationary in V (Pλ+1) for any δ < θλ. We know that A ∈V (Pλ∗Qλ|β)for some β < λ+.Claim.

V (Pλ ∗Qλ|β) |= λ \ A ∈Fδ for any δ < θλ.Proof. If A was Fδ-positive then Eδ ∩A would be Fδ-positive in V (Pλ ∗Qλ|α) forα ≥β.

Therefore Eδ ∩A would be stationary in V (Pλ+1).Since also Sing ⊆λ \ A there is a club C ⊆λ \ A in V (Pλ+1), and so A isnonstationary.We have proved that ⟨Eδ; δ < θλ⟩forms a maximal antichain of stationarysubsets of Reg(λ) in V (Pλ+1).Now let S ⊆Eδ be stationary, δ′ > δ, S ∈V (Pλ ∗Qλ|α). S is Fδ-positive (orjust stationary if δ = −1) in V (Pλ ∗Qλ|α) and so by Lemma 3.4 Tr(S) ∈Fδ′.Consequently Eδ′ \ Tr(S) is nonstationary in V (Pλ+1) - (λ \ Eδ′) ∪(Eδ′ ∩Tr(S))contains a club - which exactly means that S < Eδ′.

Since by Lemma 3.4 Eδ \Tr(Eδ) ∈Fδ the set Tr(Eδ) ∩Eδ is nonstationary in V (Pλ+1) - (λ \ Eδ) ∪(Eδ \Tr(Eδ)) contains a club.The following easy observation tells us more about the properties of the algebraP(κ)/NS in the resulting model.Proposition 3.5. Let −1 ≤α < β < θλ, then the sum of the sets {Eδ; α < δ ≤β}in the algebra P(κ)/NS exists.

Moreover for any normal measure over κ this sumhas measure zero.Proof. It follows immediately from proposition 3.2 that Tr(Eα) is the sum of{Eδ; α < δ < θλ}.

Hence the desired sum is just Tr(Eα) \ Tr(Eβ). For any nor-mal measure over κ the measure of Tr(Eβ) is one, consequently the measure ofTr(Eα) \ Tr(Eβ) must be zero.

12THOMAS JECH AND JIˇR´I WITZANY4. Measurability of κ in V (Pκ+1).Let Uκθ be the first repeat point of κ and j = jθ : V →M = V κ/Uκθ .

Then(j−→U ) ↾κ + 1=−→U ↾(κ, θ) and it follows from Lemma 2.3 that (jPκ)κ is aniteration of length κ+ with < κ-support such that any initial segment is an iterationof order θ and any potential name is used cofinally many times in M(Pκ) as wellas in V (Pκ). Consequently we can suppose that Qκ = (jPκ)κ.Using methods for extending elementary embeddings (see [WoC92] and [JWo85])we will prove that κ is actually measurable in V (Pκ+1).

Let G be a Pκ-genericfilter/V , Gκ a Qκ-generic/V [G]. We know that jPκ = Pκ ∗Qκ ∗R, where the factorR is κ-closed in M[G][Gκ] and consequently in V [G][Gκ].Lemma 4.1.

There is a filter H ∈V [G][Gκ] R-generic/M[G][Gκ].Proof. Since V |= |Pκ| = κ we have M |= |jPκ| = jκ and therefore the factor R hascardinality jκ in M[G][Gκ].

Thus M[G][Gκ] |= |P(R)| = (jκ)+ because of GCH.PutD = {D ∈M[G][Gκ]; D is a dense subset of R}then the cardinality of D in V [G][Gκ] is same as the cardinality of (jκ)+M whichis κ+. Now use the fact that R is κ-closed to get a generic filter H ∈V [G][Gκ].Consequently j can be lifted toj∗: V [G] →M[G][Gκ][H]where j∗is defined in V [G][Gκ].

Next we need to prove the following importantlemma:Lemma 4.2. For any α < κ+[Gκ ↾α]j ∈j∗(Qκ|α)where [Gκ ↾α]j is defined as in Lemma 2.5.Proof.

Let jδ denote the elementary embedding jδ : V →Mδ = V κ/Uκδ for δ < θ.It follows from Lemma 2.5 and the proof of Lemma 2.3 that for any δ < θMδ |= 1∥– jδPκ ∀H ∈Mδ[G∗] Qκ|α-generic/Mδ[G](*)[H]j ∈jδ(Qκ|α)/G∗Denote this formula ϕ(jδPκ, Qκ|α, jδ(Qκ|α)).Now we need to introduce the notion of a canonical name. We say that f ∈V κis a canonical name for x ∈V ifffor any measure U over κ the set x belongs to thetransitive collapse V κ/U and is equal to [f]U. LetC = {x ∈V ; x has a canonical name}.Obviously Vκ ⊆C and C≤κ ⊆C.

Since Pα ∈Vκ for α < κ we get that Pκ ∈C andQκ|α ∈C. Let f be the canonical name for Qκ|α.

Then by the Lo´s Theorem (*) isequivalent to{β < κ; V |ϕ(Pf(β) Q |α)} ∈Uκ

FULL REFLECTION AT A MEASURABLE CARDINAL13Since this is true for any δ < θ and Uκθ is a repeat point it follows{β < κ; V |= ϕ(Pκ, f(β), Qκ|α)} ∈Uκθwhich (again by the Lo´s Theorem) means thatM |= ϕ(jPκ, Qκ|α, j(Qκ|α))In particular for G∗= G ∗Gκ ∗H and Gκ ↾α ∈M[G∗] it says that[Gκ ↾α]j ∈j(Qκ|α)/G∗= j∗(Qκ|α).Lemma 4.3. There is a j∗Qκ-generic/M[G∗Gκ ∗H] filter H∗such that for everyα < κ+ the condition [Gκ ↾α]j is in H∗.Proof.

Put˜Q = {q ∈j∗Qκ; ∀β < j(κ+) : qβ = ∅or max(qβ) ≥κ}andD = {a ∈M[G ∗Gκ ∗H]; a ⊆˜Q is a maximal antichain}.It follows from the κ+-c.c. of Qκ thatV [G] |= ∀a ⊆Qκ : if a is an antichain then ∃α < κ+ : a ⊆Qκ|αsoM[G∗Gκ ∗H] |= ∀a ⊆j∗Qκ : if a is an antichain then ∃α < j(κ+) : a ⊆j∗Qκ|α .Moreover the cardinality of the power set of j∗Qκ|α in M[G ∗Gκ ∗H] is at mostj(κ+).

Thus the cardinality of D in M[G ∗Gκ ∗H] is j(κ+) and in V [G ∗Gκ]the cardinality is κ+. Let ⟨aα; α < κ+⟩be an enumeration of D in which eachmaximal antichain occurs cofinally many times.Observe that ˜Q is κ-closed inV [G ∗Gκ].

Now it is easy to construct in V [G ∗Gκ] a descending sequence ofconditions ⟨qα; α < κ+⟩⊆˜Q with the following properties:(i) qα ∈j∗(Qκ|α),(ii) qα ≤[Gκ ↾α]j,(iii) if aα ⊆j∗(Qκ|α) then qα strenghtens a condition in aα.The sequence ⟨qα; α < κ+⟩generates a j∗Qκ-generic/M[G ∗Gκ ∗H] filter H∗suchthat each [Gκ ↾α]j is in H∗.It means that p ∈Gκ implies j∗(p) ∈H∗and consequently j∗is lifted toj∗∗: V [G ∗Gκ] →M[G ∗Gκ ∗H ∗H∗]in V [G ∗G ] We have proved that κ is measurable in V [G ∗G ]

14THOMAS JECH AND JIˇR´I WITZANY5. Generalizations and questions.We say that S = ⟨Sλ; λ ≤κ⟩is a generalized coherent sequence of measures iffor any λ ≤κ the set Sλ is a set of measures over λ and for any U ∈SλjU(S)(λ) = Sλ ↾U = {V ∈Sλ; V ⊳U}.For example if each Sλ is the set of all measures over λ then the sequence is coherent.Suppose now that GCH holds, S is a generalized coherent sequence and moreoverthere are separating sets of regular cardinals ⟨XU; U ∈Sκ⟩, i.e.

XU ∈V iffU = Vfor U, V ∈Sκ. By a straightforward modification of our construction we get ageneric extension V (Pκ+1) preserving cardinalities, cofinalities and GCH with thefollowing properties:1.

⟨XU; U ∈Sκ⟩forms a maximal antichain of stationary subsets of Reg inV (Pκ+1).2. If U, V ∈Sκ and S ⊆XU, T ⊆XV are stationary then S < T (in V (Pκ+1)) iffU ⊳V.

If U ∈Sκ and S ⊆Sing, T ⊆XU are stationary then S < T. Consequently(Mitchell order)V (κ) = oV (Pκ+1)(κ).Moreover if κ has a repeat point (in a generalized sense) then κ is measurablein the generic extension. It is shown in [Ba85] that any prewellordering P with|P| < κ can be represented as the set of all measures over κ.

That does not giveus anything new - in that case Full Reflection again holds in the resulting model.However recent papers of Cummings ([Cu92a], [Cu92b]) provide models with arather complex structure of the Mitchell order. Using the model of [Cu92a] thatsatisfies GCH we can for example construct a generalized coherent sequence S suchthat Sκ is isomorphic to the four element poset of the type◦◦↓ց ↓◦◦Thus in the resulting model κ is 2-Mahlo and we get two disjoint sets of inacces-sible non-Mahlo cardinals X1, X2 ⊂E0 and two disjoint sets of 1-Mahlo cardinalsY1, Y2 ⊂E1 so that for any stationary S1 ⊆X1, S2 ⊆X2 the following holds:S1 < Y1 but S1 ≮Y2, Tr(S1) = Y1 (mod NS)S2 < Y1 and S2 < Y2, Tr(S2) = E1 (mod NS).The following question immediately comes to mind:Question 1.

Does the consistency of Full Reflection at a measurable cardinal implythe consistency of a cardinal with a repeat point?Let V (Pκ+1) be our generic extension, let U denote the measure on κ and C[Eλδ ]the filters of subsets of λ generated in V (Pκ+1) by closed unbounded sets and thecanonical stationary set Eλδ . Let F code all these filters, then in L[F, U] we getback the original measures and U becomes a repeat point of κ.

Hence we can ask amore specific question and conjecture that the answer is yes

FULL REFLECTION AT A MEASURABLE CARDINAL15Question 2. Suppose that Full Reflection holds at all λ ≤κ, κ is measurable andcanonical stationary sets Eλδ of all orders exist.

Let C[Eλδ ], F, U be as above. Is itthen true that all C[Eλδ ] ∩L[F, U] are measures in L[F, U] and U ∩L[F, U] is arepeat point of κ?Another way to state an equiconsistency result would be to improve our con-struction so that the filters C[Eλδ ] are λ+-saturated.

If we add this property of thefilters to the assumptions of question 2 then using a method of [J84] or [JWo85]we can prove that the answer is yes. Unfortunately if we analyze our construc-tion we find out that already the filters F λδ are not λ+-saturated.We can tryto use the ideas of [JWo85] and instead of extensions of j = jδ : V →M intoj∗: V (Pκ ∗Q) →V (j(Pκ ∗Q)) constructed in V (j(Pκ ∗Q)) work only with ex-tensions constructed in V (Pκ ∗(jPκ)κ).

We can get the construction to work butthe filters still will not be saturated. Hence we conjecture that the answer of thefollowing question is no.Question 3.

Is it consistent that Full Reflection holds at κ measurable, all canon-ical stationary sets Eκδ exist and the filters C[Eκδ ] are κ+-saturated?References[Ba85] S.Baldwin, The ⊳-ordering on normal ultrafilters, JSL 51 (1985), 936 – 952. [BTW76] J.Baumgartner, A.Taylor and S.Wagon, On splitting stationary subsets of large cardi-nals, JSL 42 (1976), 203–214.

[Cu92a] J.Cummings, Possible behaviors for the Mitchell ordering, preliminary version. [Cu92b] J.Cummings, Possible behaviors for the Mitchell ordering II, preliminary version.

[HS85] L.Harrington and S.Shelah, Some exact equiconsistency results in set theory, Notre DameJ.Formal logic 26 (1985), 178–188. [J84] T.Jech, Stationary subsets of inaccessible cardinals, Contemporary Mathematics 31 (1984),115–141.

[J86] T.Jech, Multiple Forcing, Cambridge University Press, 1986. [JMa..80] T.Jech, M.Magidor, W.Mitchell and K.Prikry, Precipitous ideals, JSL 45 (1980), 1–8.

[JS90] T.Jech, S.Shelah, Full reflection of stationary sets below ℵω, JSL 55 (1990), 822–829. [JS92] T.Jech, S.Shelah, Full reflection of stationary sets at regular cardinals, American Journalof Mathematics, to appear.

[JWo85] T.Jech, W.H.Woodin, Saturation of the closed unbounded filter on the set of regularcardinals, Transactions of AMS 292 (1985), 345–356. [M82] M.Magidor, Reflecting stationary sets, JSL 47 (1982), 755–771.

[Mi74] W.J.Mitchell, Sets constructible from sequences of ultrafilters, JSL 39 (1974), 57–66. [Mi80] W.J.Mitchell, How weak is a closed unbounded filter?, Logic Colloquium ’80, ed.

by D.vanDalen, D.Lascar and J.Smiley, N.Holland, 1982. [Mi83] W.J.Mitchell, Sets constructible from sequences of measures: revisited, JSL 48 (1983),600–609.

[Ra82] L.B.Radin, Adding closed cofinal sequences to large cardinals, Annals of Math. Logic 22(1982), 243–261.

[Wo92] H.Woodin, personal communication. [WoC92] H.Woodin, J.Cummings, Generalised Prikry Forcings, in preparation.Department of Mathematics, The Pennsylvania State University, University Park,PA 16802E-mail address: jech@math.psu.edu, witzany@math.psu.edu

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