SUCCESSORS OF SINGULAR CARDINALS AND COLORING THEOREMS I TODD EISWORTH AND SAHARON SHELAH

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1 SUCCESSORS OF SINGULAR CARDINALS AND COLORING THEOREMS I TODD EISWORTH AND SAHARON SHELAH Abstract. We investigate the existence of strong colorings on successors of singular cardinals. This work continues Section 2 of [], but now our emphasis is on finding colorings of pairs of ordinals, rather than colorings of finite sets of ordinals.. Introduction The theme of this paper is that strong coloring theorems hold at successors of singular cardinals of uncountable cofinality, except possibly in the case where the singular cardinal is a limit of regular cardinals that are Jonsson in a strong sense. Our general framework is that λ = µ +, where µ is singular of uncountable cofinality. We will be searching for colorings of pairs of ordinals < λ that exhibit quite complicated behaviour. The following definition (taken from [2]) explains what complicated means in the previous sentence. Definition.. Let λ be an infinite cardinal, and suppose κ+θ µ λ. Pr (λ, µ, κ, θ) means that there is a symmetric two place function c from λ to κ such that if ξ < θ and for i < µ, α i,ζ : ζ < ξ is a strictly increasing sequence of ordinals < λ with all α i,ζ s distinct, then for every γ < κ there are i < j < µ such that (.) ζ < ξ and ζ 2 < ξ = c(α i,ζ, α i,ζ2 ) = γ. Just as in [], one of our main tools is a game that measures how Jonsson a given cardinal is. Recall that a cardinal λ is a Jonsson cardinal if for every c : [λ] <ω λ, we can find a subset I λ of cardinality λ such that the range of c I is a proper subset of λ. A reader seeking more background should investigate [4] and [3] in [5]. Date: October 6, Key words and phrases. Jonsson cardinals, coloring theorms, successors of singular cardinals. This is publication number 535 of the second author.

2 2 TODD EISWORTH AND SAHARON SHELAH Definition.2. Assume µ λ are cardinals, γ is an ordinal, n ω, and J is an ideal on λ. We define the game Gm n J [λ, µ, γ] as follows: A play lasts γ moves. In the α th move, the first player chooses a function F α : [λ] <n µ, and the second player responds by choosing (if possible) a subset A α λ such that A α β<α A β A α J + ran(f α [A α ] <n ) is a proper subset of µ. The second player loses if he has no legal move for some α < γ, and he wins otherwise. In the previous definition, if J = Jλ bd then we may omit it. Note that it causes no harm if we use a set E of cardinality λ instead of λ itself; in this case, we write Gm n J [E, µ, γ]. Note that λ is a Jonsson cardinal if and only if Player I does not have a winning strategy in the game Gm ω [λ, λ, ]. One may view the lack of a winning strategy for Player I in games of longer length as a strong version of Jonsson-ness or a weak version of measurability if λ is measurable, then Player II can make sure her moves are elements of some λ complete ultrafilter. The following claim investigates how the existence of winning strategies is affected by modifications to the game; the proof is left to the reader. Claim.3. () If µ µ and the first player has a winning strategy in Gm n J [λ, µ, γ], then she has a winning strategy in Gm n J[λ, µ, γ]. (2) Suppose we weaken the demand on the second player to (.2) ( ζ < λ)[ran(f α [A α \ ζ] <n ) is a proper subset of µ]. If cf(λ) γ and J Jλ bd, then the first player has a winning strategy in the revised game if and only if she has a winning strategy in the original game. (3) If J is γ complete, then the same applies to the case where we weaken the demand on the second player to (.3) ( Y J)[ran(F α [A α \ Y ] <n ) is a proper subset of µ]. (4) We can allow the second player to pass, i.e., to let A α = β<α A β (even if this is not a legal move) as long as we declare that the

3 COLORING THEOREMS 3 second player loses if the order type of the set of moves where he did not pass is < γ. (5) If Player I has a winning strategy in Gm n J[λ, µ, γ] for every µ < µ where µ is singular and γ > cf(µ ) is regular, then Player I has a winning strategy in Gm n J [λ, µ, γ]. We can weaken the requirement that γ is regular and instead require that cf(γ) > cf(µ ) and ω γ = γ. In Section 2 of [], the existence of winning strategies for Player I in variants of the game is investigated. We will prove one such result here; the reader should look in [] for others. Claim.4. If 2 χ < λ < ℶ (2 χ ) +(χ) then Player I has a winning strategy in Gm ω [λ, χ, (2 χ ) + ]. Proof. At a stage i, Player I will select a function F i : [λ] <ω χ coding the Skolem functions of some model M i. For the initial move, we let the model M 0 have universe λ, and include in our language all relations on λ and all functions from λ to λ of any finite arity that are first order definable in the structure H(λ + ),, < λ with the parameters χ and λ. + For subsequent moves, M i is an expansion of M 0 with universe λ that has all relations on λ and all functions from λ to λ of any finite arity that are first order definable in the structure H(λ + ),, < λ from the + parameters χ, λ, M 0, and A j : j < i. To obtain the function F i, we let Fn i : n < ω list the Skolem functions of M i in such a way that Fn i has m i(n) n places. Let h : ω ω be such that for all n, h(n) n and h ({n}) is infinite. We then define { Fh( u ) i (.4) F i (u) = ({α u : u α < m i(n)}) if this is < χ 0 otherwise The point of doing this is that whenever Player II chooses A i, we know that ran(f i [A i ] <ω ) will look like the result of intersecting an elementary submodel of M i with χ; in particular, this range will be closed under the functions from M i. Note that M 0 (and all expansions of it) has definable Skolem functions and so for any i and A λ, the Skolem hull of A in M i (denoted by Sk Mi (A)) is well defined. Let (F i, A i ) : i < (2 χ ) + be a play of the game in which Player I uses this strategy (with M i the model corresponding to F i ). For each i, define (.5) α i = min{α : Sk M0 (A i ) ℶ α (χ) > χ}.

4 4 TODD EISWORTH AND SAHARON SHELAH By the choice of M 0 and M i, clearly α(i) is a successor ordinal or a limit ordinal of cofinality χ +, and (.6) Sk M0 (A i ) ℶ αi (χ) 2 χ. Since A i A j for i > j, we know the sequence α i : i < (2 χ ) + is non decreasing. Furthermore, for each i we know (.7) α i < min{β : λ ℶ β (χ)} < (2 χ ) +. This means that the sequence α i : i < (2 χ ) + is eventually constant, say with value α. Let i be the least ordinal < (2 χ ) + such that α i = α for i i. Proposition.5. If i i < (2 χ ) +, then Sk M0 (A i+ ) ℶ α (χ) is a proper subset of Sk M0 (A i ) ℶ α (χ). Proof. Note that i, α, and ℶ α (χ) are all elements of M i+ as they are definable in H(λ + ),, < λ + from the parameters M 0 and A j : j i. Furthermore, (.8) γ := min{γ < λ : Sk M0 (A i ) γ = χ} is also definable in M i+ (and < (2 χ ) + ). Thus the language of M i+ includes a bijection between Sk M0 (A i ) γ and χ. If Player I has not won the game at this stage, after Player I selects A i+ we will be able to find an ordinal β < χ such that β / ran(f i+ [A i+ ] <ω ). By definition of h, we know β := h (β) is an element of Sk M0 (A i ) ℶ α (χ). However, β is not an element of Sk Mi+ (A i+ ) since F i+ codes the Skolem functions of M i+, the range of F i+ [A i+ ] <ω is Sk Mi+ (A i+ ) χ. Since Sk Mi+ (A i+ ) is closed under h, this contradicts our choice of β. Since Sk M0 (A i+ ) Sk Mi+ (A i+ ), we have established the proposition. Note that the preceding proposition finishes the proof of the claim if play of the game continues for all (2 χ ) + steps, then Sk M0 (A i ) ℶ α (χ) : i < (2 χ ) + is a strictly decreasing family of subsets of Sk M0 (A i ), contradicting (.6). 2. Club guessing technology In this section, we prove that if λ = µ +, where µ is singular, then under certain circumstances we can find a complicated library of colorings of smaller cardinals. In the next section, we will use this library of colorings to get a complicated coloring of λ. The basics of club guessing are explained in [4], but we will take a few minutes to recall some of the definitions.

5 COLORING THEOREMS 5 Let us recall that if S is a stationary subset of λ, then an S club system is a sequence C = C δ : δ S such that for (limit) δ S, C δ is closed unbounded in δ. In this section, we will be concerned with the case where λ is the successor of a singular cardinal, i.e., λ = µ + where cf(µ) < µ. In this context, if C is an S club system, then for δ S we define an ideal J b[µ] δ γ < δ, on C δ by A J b[µ] δ if and only if A C δ, and for some θ < µ and β A nacc(c δ ) [β < γ or cf(β) < θ]. Note that it is a bit easier to understand the definition of J b[µ] δ by looking at the contrapositive a subset A of C δ is large, i.e., not in, if and only if A nacc(c δ ) is cofinal in δ, and the cofinalities of members of any end segment of A nacc(c δ ) are unbounded below µ. J b[µ] δ Claim 2.. Let λ = µ +, where µ is a singular cardinal of cofinality κ < µ. Let S λ be stationary, and assume that sup{cf(δ) : δ S} = µ < µ. Let C be an S club system, and for each δ S, let J δ be the ideal J b[µ] δ. Let κ i : i < κ be a non decreasing sequence of cardinals such that (2.) κ = i<κ κ i µ, and let γ < µ. Assume we are given a λ club system ē and a sequence of ideals Ī = I α : α < λ such that () I α is an ideal on e α extending J bd e α (2) if δ S, then for each i < κ, {α nacc(c δ ) : Player I wins Gm ω I α [e α, κ i, γ ]} = nacc(c δ ) mod J δ (3) for any club E λ, for stationarily many δ S, where {α nacc(c δ ) : B 0 [E, e α ] / I α } / J δ, B 0 [E, e α ] = {β nacc(e α) : E meets the interval (sup(β e α ), β)}. Then there is a function h : λ (κ + ) and a sequence F = F δ : δ < λ, δ a limit such that F δ : [e δ ] <ω κ h(δ) (where κ := κ κ ) and

6 6 TODD EISWORTH AND SAHARON SHELAH 2 for every club E λ, for each i < κ there are stationarily many δ S such that the set of β nacc(c δ ) satisfying the following h(β) i B 0 [E, e β ] / I β for all γ < β, κ h(β) ran(f β [ B 0 [E, e β ] \ γ ] <ω ) is not in J δ. Now admittedly the previous claim is quite a lot to digest, so we will take a little time to illuminate the basic situation we have in mind. Claim 2.2. The assumptions of Claim 2. are satisfied if () λ = µ + where κ = cf(µ) < µ (2) S {δ < λ : cf(δ) = κ} (3) δ S δ = µ (i.e., S λ \ µ) (4) C is an S club system (5) J = J δ : δ S where J δ = J b[µ] C δ (6) id p ( C, J) is a proper ideal (7) κ i : i < κ is a non decreasing sequence of cardinals with supremum κ µ (8) γ < µ, and for each i < κ, Player I wins the game Gm ω [θ, κ i, γ ] for all large enough regular θ < µ (9) ē is a λ club system such that e β < µ (0) for α < λ, I α = J bd e α Proof of Claim 2.2. We need only check items (2) and (3) in the statement of Claim 2. everything else is trivially satisfied. Concerning (2), given δ S and i < κ, we need to show {α nacc(c δ ) : Player I wins Gm ω [e α, κ i, γ ]} = nacc(c δ ) mod J δ. Let A consist of those α nacc(c δ ) for which Player I does not win the game Gm ω [e α, κ i, γ ]. By our assumptions, there is a θ < µ such that e α < θ for all α A, and therefore A is in the ideal J b[µ] C δ = J δ and we have what we need. Concerning (3), given E λ club, we must find stationarily many δ S such that {α nacc(c δ ) : B 0 [E, e α ] / I α } / J δ. Let E = {ξ E : otp(e ξ) = ξand µ divides ξ}. Clearly E is a closed unbounded subset of E, and since id p ( C, J) is a proper ideal,

7 the set COLORING THEOREMS 7 S := {δ S E : E nacc(c δ ) / J δ } is stationary. Fix δ S, and suppose we are given θ < µ and ξ < δ. Since E nacc(c δ ) / J δ, we can find α E nacc(c δ ) such that α > max{ξ, µ} and cf(α) > θ. Since the order type of E α is α µ > e α, we know that B 0 [E, e α ] is unbounded in e α hence a member of I α. This shows that the set of such α is in J + δ, as required. Now we return to the proof of Claim 2.. Proof of Claim 2.. Let σ = cf(σ) be a regular cardinal < µ that is greater than µ and γ. For each limit β < λ, if there is an i κ such that Player I wins the version of Gm ω I β [e β, κ i, σ + ] where we allow Player II to pass, then we let h(β) be the maximal such i note that i exists by (5) of Claim.3 and let Str β be a strategy that witnesses this. Note that since γ < σ + and J δ = J b[µ] δ for δ S, we have that for δ S and i < κ that {β nacc(c δ ) : Str β is defined and i h(β)} = nacc(c δ ) mod J δ. We will make σ + attempts to build F witnessing the conclusion. In stage ζ < σ +, we assume that our prior work has furnished us with a decreasing sequence E ξ : ξ < ζ of clubs in λ, and, for each β < λ where Str β is defined, an initial segment F ξ β, Aξ β : ξ < ζ of a play of Gm ω I β [e β, κ h(β), σ+ ] in which Player I uses Str β. (Note that our convention is that if Player II chooses to pass at a stage, we let A ξ β be undefined.) For each such β, let F ζ β : [e β] <ω κ h(β) be given by Str β, and for those β for which Str β is undefined, we let F β ζ be any such function. Now if F ζ β : β < λ := F ζ is as required then we are done. Otherwise, there is a club E λ and i ζ < κ exemplifying the failure of F ζ, and without loss of generality, (2.2) ( δ S) [ B iζ [E ζ, C δ, Ī, ē, F ζ ] ] J δ. Now let E ζ = acc(e ζ ξ<ζ E ξ). For each β where Str β is defined, we let Player II respond to F ζ β by playing the set B 0[E ζ, e β ] if it is a legal move, otherwise we let him pass. We then proceed to stage ζ +. Assuming that this construction continues for all σ + stages, we will arrive at a contradiction. Let E = ζ<σ E + ζ. By assumption (3) there

8 8 TODD EISWORTH AND SAHARON SHELAH is a δ( ) S for which A := {β nacc(c δ( ) ) : B 0 [E, e β ] / I β } / J δ( ). By assumption (2), we have A 2 := {β A : Str β is defined } / J δ( ). For β A 2, look at the play F ζ β, Aζ β : ζ < σ+. Since Player I wins, there is a ζ β < σ + such that Player II passed at stage ζ for all ζ ζ β. Since σ > µ and J δ( ) is µ based, for some ζ < σ +, A 3 = {β A : Str β is defined and ζ β ζ } / J δ( ). Now E ζ was defined so that for some i ζ, for all δ S, (2.3) B iζ [E ζ, C δ, Ī, ē, F ζ ] J δ, but (again by assumption (2)) A 4 = {β A : Str β is defined, ζ β ζ, and i ζ h(β)} / J δ( ). For β A 4, we know that at stage ζ of our play of Gm ω I β [e β, κ h(β), σ + ] the set B 0 [E ζ, e β ] was not a legal move. Since our sequence of clubs is decreasing, we know that B 0 [E ζ, e β ] is a subset of B 0 [E ξ, e β ] for all ξ < ζ ], so we have B 0 [E ζ, e β ] ξ<ζ A ξ β. Since β A, we know that B 0 [E ζ, e β ] / I β. Thus the reason for B 0 [E ζ, e β ] being an illegal move must be that for all γ < β, κ h(β) ran(f ζ β [B 0[E ζ, e β ] \ γ] <ω ). All of these facts combine to tells us that β B iζ [E ζ, C δ, Ī, ē, F ζ ], and thus contradicting (2.3). A 4 B iζ [E ζ, C δ, Ī, ē, F ζ ] / J δ( ), The proofs in this section (and the next) can be considerably simplified if we are willing to restrict ourselves to the case κ < µ, as we can dispense with the sequence κ i : i < κ.

9 COLORING THEOREMS 9 3. Building the Coloring We now come to the main point of this paper; we dedicate this section and the next to proving the following theorem. Theorem. Assume λ = µ +, where µ is a singular cardinal of uncountable cofinality, say ℵ 0 < κ = cf(µ) < µ. Assume κ i : i < κ is non decreasing with supremum κ µ, and there is a γ < µ such that for each i, for every large enough regular θ < µ, Player I has a winning strategy in the game Gm ω [θ, κ i, γ ]. Then Pr (λ, λ, κ, κ) holds. Let S i : i < κ be a sequence of pairwise disjoint stationary subsets of {δ < λ : cf(δ) = κ}. For i < κ, let C i be an S i club system such that λ / id p ( C i, J i ), where J i = J b[µ] : δ S Cδ i i for δ S i, otp(cδ i ) = cf(δ) = κ = cf(µ) Such ladder systems can be found by Claim 2.6 (and Remark 2.6A (6)) of [2] for the second statement to hold, we need that µ has uncountable cofinality. Claim 3.. There is a λ club system ē such that e β cf(β) + cf(µ), and ē swallows each C i, i.e., if δ S i (e β {β}), then C i δ e β. Proof. Let S = i<κ S i, and let β < λ be a limit ordinal. Let e 0 β be a closed cofinal subset of β of order type cf(β). We will construct the required ladder e β in ω stages, with e n β denoting the result of the first n stages of our procedure. The construction is straightforward, but it is worthwhile to note that we need to use the fact that each member of S has uncountable cofinality. Given e n β, let us define (3.) B n = S (e n β {β}). Now we let e n+ β be the closure in β of (3.2) e n β {C δ : δ B n }. Note that e n+ cf(µ) + cf(β) as C δ = cf(µ) = κ for each δ S. Finally, we let e β be the closure of n<ω e n β in β. Clearly e β cf(µ) + cf(β). Also, since each element of S has uncountable cofinality, if δ S e β, then there is an n such that δ e n β, and therefore (3.3) C δ e n+ β e β, as required.

10 0 TODD EISWORTH AND SAHARON SHELAH For each i < κ, there are h i and F i = Fδ i : δ < λ, δ limit as in the conclusion of Claim 2. applied to C i and ē; note that we satisfy the assumptions of Claim 2. by way of Claim 2.2. Let λ i : i < κ be a strictly increasing sequence of regular cardinals > κ and cofinal in µ such that (3.4) λ = tcf ( ) λ i /Jκ bd, i<κ and let f α : α < λ exemplify this. Finally, let h 0 : κ ω and h : κ κ be such that (3.5) ( n)( i < κ)( κ j < κ)[h 0(j) = n and h (j) = i]. Before we can define our coloring, we must recall some of the terminology of [2]. Definition 3.2. Let 0 < α < β < λ, and define We also define (by induction on l) γ(α, β) = min{γ e β : γ α}. γ 0 (α, β) = β, γ l+ (α, β) = γ(α, γ l (α, β)) (if defined). We let k(α, β) be the first l for which γ l (α, β) = α. The sequence γ i (α, β) : i k(α, β) will be referred to as the walk from β to α along the ladder system ē. We now define the coloring c that will witness Pr (λ, λ, κ, κ). Recall that c must be a symmetric two place function from λ to κ. Given α < β, we let i = i(α, β) be the maximal j < κ such that f β (j) < f α (j) (if such an j exists). Next, we walk from β down to α along ē until we reach an ordinal ν(α, β) such that f α (i) < f ν(α,β) (i), (again, if such an ordinal exists.) After this, we walk along ē from α toward the ordinal max(α e ν(α,β) ) until we reach an ordinal η(α, β) for which f ν(α,β) (i) < f η(α,β) (i). The idea now is to look at how the ladders e ν(α,β) and e η(α,β) intertwine. Let us make a temporary definition by calling an ordinal ξ e ν(α,β) relevant if e η(α,β) meets the interval (sup(ξ e ν(α,β) ), ξ). If it makes sense, we let w(α, β) e ν(α,β) be the last h 0 (i(α, β)) relevant ordinals in e ν(α,β) (so we need that the relevant ordinals have order type γ + h 0(i(α, β)) for some γ).

11 COLORING THEOREMS Finally, we define our coloring by (3.6) c(α, β) = F h (i(α,β)) ν(α,β) (w(α, β)). If the attempt to define c(α, β) breaks down at some point for some specific α < β, then we set c(α, β) = 0. We now prove that this coloring works, so suppose t α : α < λ are pairwise disjoint subsets of λ such that t α = θ < κ and j < κ, and without loss of generality α < min t α and θ ω. We need to find δ 0 and δ such that (3.7) α t δ0 and β t δ α < β and c(α, β) = j. Let j be the least j such that j < κ j, and let S, C, and F denote S j, Cj, and F j respectively. Given δ < λ, we define the envelope of t δ (denoted env(t δ )) by the formula (3.8) env(t δ ) = {γ l (δ, ζ) : l k(δ, ζ)}. ζ t δ The envelope of t δ is the set of all ordinals obtained by walking down to δ from some ζ t δ using the ladder system ē. This makes sense as we have arranged that δ < min t δ. Note also that env(t δ ) t δ = θ. Next we define functions g min δ and g max δ in i<κ λ i by (3.9) g min δ (i) = min{f γ (i) : γ env(t δ )}, and (3.0) gδ max (i) = sup{f γ (i) + : γ env(t δ )}. Note that gδ max is well defined as we assume that κ < min{λ i : i < κ}. The following claim is quite easy, and the proof is left to the reader. Claim 3.3. () f δ = J bd g min κ δ (2) gδ min (i) gδ max (i) for all i < κ (3) There is a δ > δ such that gδ max J bd κ g min δ. Now let χ = (2 λ ) +, and let M α : α < λ be a sequence of elementary submodels of H(χ ),, < χ that is increasing and continuous in α and such that each M α λ is an ordinal, M β : β α M α+, and f α : α < λ, g, c, ē, S, C, tα : α < λ all belong to M 0. Note that µ + M 0. The set E = {α < λ : M α λ = α} is closed unbounded in λ, and furthermore, (3.) α < δ E sup t α < δ.

12 2 TODD EISWORTH AND SAHARON SHELAH By the choice of C and F, for some δ S E we have the set (3.2) A = {β nacc(c δ ) : ( γ < β) ran(f β [ B 0 [E, e β ] \ γ ] <ω ) κ j } is not in J b[µ] C δ. Note that A acc(e), as B 0 [E, e β ] is unbounded in β for β A. For β t δ, if l < k(δ, β) then e γl (δ,β) δ is bounded in δ, and since it is closed it has a well defined maximum. Since t δ < κ = cf(δ), this means the ordinal γ := sup{max[e γl (δ,β) δ] : β t δ and l < k(δ, β)} is strictly less than δ. For β t δ, let us define (3.3) A β := {β A : ( l k(β, δ))[cf(β ) e γl (δ,β) ]}. Since the cardinality of each ladder in ē is less than µ, each set A β is an element of J b[µ] C δ. The ideal J b[µ] C δ is κ complete, so the fact that t δ < κ and k(β, δ) is finite for each β t δ together imply that (3.4) A β J b[µ] C δ. β t δ By the definition of A and our choice of δ, this means it is possible to choose β A \ (γ + ) that is not in any A β, i.e., (3.5) β t δ and l < k(δ, β) = cf(β ) > e γl (δ,β). Claim 3.4. () If ɛ t δ, and l = k(δ, ɛ), then β nacc(e γl (δ,ɛ)). (2) If ɛ t δ and γ < γ β, then γ l (δ, ɛ) = γ l (γ, ɛ) for l < k(δ, ɛ), and γ k(δ,ɛ) (γ, ɛ) = β Proof. For the first clause, note that δ is an element of e γl (δ,ɛ) and hence by our choice of ē, C δ e γl (δ,ɛ). Thus β e γl (δ,ɛ), and since cf(β ) > e γl (δ,ɛ), we know that β cannot be an accumulation point of e γl (δ,ɛ). The first part of the second statement follows because of the definition of γ. As far as the second part of the second statement goes, it is best visualized as follows: We walk down the ladder system ē from ɛ to γ, we eventually hit a ladder that contains δ this happens at stage k(δ, ɛ). Since C δ is a subset of this ladder, the next step in our walk from ɛ to γ must be down to β because γ < γ < β.

13 COLORING THEOREMS 3 We can visualize the preceding claim in the following manner: β is chosen so that for all sufficiently large γ < β, all the walks from some element of t δ to γ are funnelled through β β acts as a bottleneck. This will be key when want to prove that our coloring works. Since β A, we can choose a finite increasing sequence ξ 0 < ξ < < ξ n of ordinals in acc(e) nacc(e β ) \ (γ + ) such that F j β ({ξ 0,..., ξ n }) = j, the color we are aiming for. For each l n, we can find ζ l E \ (γ + ) such that sup(e β ξ l ) < ζ l < ξ l. Now we let φ(x 0, y 0, x, y,..., x n, y n, z 0, z ) be the formula (with parameters γ, f, λ i : i < κ, C, ē, t α : α < λ, h, h 0, j ) that describes our current situation with x l, y l standing for ζ l, ξ l, and z 0, z standing for β, δ, i.e., φ states γ < x 0 < y 0 < < x n < y n < z 0 < z are ordinals < λ z S and z 0 nacc(c z ) γ = sup{max[e γl (z,ζ) z ] : l < k(z, ζ) and ζ t z } z 0 nacc(e γk(z,ɛ)(z,ɛ)) for all ɛ t z F j z 0 ({y 0,..., y n }) = j Now clearly we have (3.6) H(χ) = φ[ζ 0, ξ 0,..., ζ n, ξ n, β, δ]. Recall that all the parameters needed in φ are in M 0, except possibly for γ, so the model M γ + contains all the parameters we need. Also, {ζ 0, ξ 0,..., ζ n, ξ n } M β, β M δ \ M β, and since δ λ \ M δ, we have (recalling that z < λ means for unboundedly many z < λ) (3.7) M δ = ( z < λ)φ(ζ 0, ξ 0,..., ζ n, ξ n, β, z ). Therefore, this formula is true in H(χ) because of elementarity. Similarly, we have H(χ) = ( z 0 < λ)( z < λ)φ(ζ 0, ξ 0,..., ζ n, ξ n, z 0, z ). Now each of the intervals [γ +, ζ 0 ), [ζ 0, ξ 0 ),..., contains a member of E, so (by the definition of E) similar considerations give us H(χ) = ( x 0 < λ)... ( y n < λ)( z 0 < λ)( z < λ)φ(x 0, y 0,..., z 0, z ). Now we can choose (in order) (3.8) ζ a 0 < ζ b 0 < ξ a 0 < ζ a < ξ b 0 < ζ b < < ζ a n < ξ b n < ζ b n < ξ a n

14 4 TODD EISWORTH AND SAHARON SHELAH such that (3.9) ( z 0 < λ)( z < λ)[φ(ζ a 0,..., ξa n, ζ a n, ξa n, z 0, z )], and (3.20) ( y n < λ)( z 0 < λ)( z < λ)[φ(ζ b 0,..., ξ b n, ζ b n, y n, z 0, z )], Our goal is to show that for all sufficiently large i < κ, it is possible to choose objects β a, δ a, ξ b n, βb, and δ b such that () ζ b n < β a < δ a < min(t δ a) max(t δ a) < ξ b n < β b < δ b (2) φ(ζ a 0,..., ξa n, βa, δ a ) (3) φ(ζ b 0,..., ξ b n, β b, δ b ) (4) for all ɛ env(t δ a), g min δ a (5) for all ɛ env(t δ b), g min δ b (6) g max δ b (7) g max δ a [i, κ) f ɛ [i, κ) gδ max a [i, κ) [i, κ) f ɛ [i, κ) g max δ b [i, κ) (i) < g min (i) gmax (i) < f β b(i) < f β a(i) δ a δ a [i +, κ) < gmin [i +, κ) δ b Table Claim 3.5. If for all sufficiently large i < κ it is possible to find objects satisfying the requirements of Table, then we can find δ a < δ b such that c(ɛ a, ɛ b ) = j for all ɛ a t δ a and ɛ b t δ b. Proof. Let us choose i < κ such that suitable objects (as above) can be found, and h (i ) = j and h 0 (i ) = n Choose ɛ a t δ a and ɛ b t δ b; we verify that c(ɛ a, ɛ b ) = j. Subclaim. i(ɛ a, ɛ b ) = i. Proof. Immediate by (4)-(7) in the table. Subclaim 2. ν(ɛ a, ɛ b ) = β b. Proof. Note that γ < ɛ a < β b. Clause (3) of the table implies that the assumptions of Claim 3.4 hold. Thus by Claim 3.4, for l < k(δ b, ɛ b ) we have γ l (ɛ a, ɛ b ) = γ l (δ b, ɛ b ),

15 COLORING THEOREMS 5 hence γ l (ɛ a, ɛ b ) env(t δ b) and (by (6) of the table and the definitions involved) (3.2) f γl (ɛ a,ɛ )(i ) g max b δ (i ) < g min b δ a (i ) f ɛ a(i ). For l = k(δ b, ɛ b ), Claim 3.4 tells us and we have arranged that γ l (ɛ a, ɛ b ) = β b, (3.22) f ɛ a(i ) g max δ a (i ) < f β b(i ). This establishes β b = ν(ɛ a, ɛ b ). Subclaim 3. η(ɛ a, ɛ b ) = β a. Proof. Let α = max(e β b ɛ a ). We have arranged that ζ b n < β a < δ a < ɛ a < ξ b n and γ < max(e β b δ a ), hence γ < α < β a. For l < k(δ a, ɛ a ), Claim 3.4 implies γ l (α, ɛ a ) = γ l (δ a, ɛ a ) env(t δ a). By our choice of i, we have (3.23) f γl (α,ɛ a )(i ) g max δ a (i ) < f β b(i ). For l = k(δ a, ɛ a ), Claim 3.4 implies γ l (α, ɛ a ) = β a, and we have ensured (3.24) f β b(i ) < f β a(i ). Thus β a is the first ordinal η in the walk from ɛ a to max(e β b ɛ a ) for which f η (i ) > f β b(i ), and therefore η(ɛ a, ɛ b ) = β a. Subclaim 4. w(ɛ a, ɛ b ) = {ξ b 0,... ξb n }. Proof. Our previous subclaims imply that an ordinal ξ e β b is relevant if and only if the ladder e β a meets the interval (sup(e β b ξ), ξ). Since h 0 (i ) = n +, we know that w(ɛ a, ɛ b ) consists of the last n + relevant ordinals in e β b. For i n, clearly ξi b e β and b sup(ξb i e β b) ζb n. We have made sure that e β a (ζi b, ξi b ) (for example, ξi a is an element in this intersection) and so each ξi b is relevant. Since β a < ξn, b it is clear that there are no relevant ordinals larger than ξn. b Given i < n, if ξ e β b (ξi b, ξb i+ ), then ξ b i sup(ξ e β b) ξ ζ b i+.

16 6 TODD EISWORTH AND SAHARON SHELAH Since ζi+ a < ξb i < ζ i+ b < ξa i+, it follows that [sup(ξ e β b), ξ) [ζ a i+, ξa i+ ), and so ξ is not relevant. Thus {ξ0, b..., ξn} b are the last n + relevant elements of e β b, as was required. To finish the proof of Claim 3.5, we note that as h (i ) = j, we have (3.25) c(ɛ a, ɛ b ) = F j β b ({ξ b 0,..., ξ b n}) = j. 4. Finding the required ordinals The whole of this section will be occupied with showing that for all sufficiently large i < κ, it is possible to find objects satisfying the requirements of Table. We begin with some notation intended to simplify the presentation. φ a (z 0, z ) abbreviates the formula φ(ζ a 0,..., ξa n, z 0, z ) φ b (y n, z 0, z ) abbreviates the formula φ(ζ b 0, ζ n b, y n, z 0, z ) For i < κ, ψ(i, z ) abbreviates the formula (4.) ( ɛ env(t z ))[g min z [i, κ) f ɛ [i, κ) g max z We have arranged things so that the sentence (4.2) ( z a 0 < λ)( z a < λ)( y b n < λ) [i, κ)] ( z b 0 < λ)( z b < λ)[φa (z a 0, za ) φb (y b n, zb 0, zb )] holds. There are far too many alternations of quantifiers in the above formula for most people to deal with comfortably; the best way to view them is as a single quantifier that asserts the existence of a tree of 5 tuples with the property that every node of the tree has λ successors, and every branch through the tree gives us five objects satisfying φ a φ b. Let Φ(i, z0 a,..., zb ) abbreviate the formula φ a (z0, a z) a φ b (yn, b z0, b z) b ψ(i, z) a ψ(i, z) b ( ) gz max [i +, κ) < g min a z [i +, κ). b

17 COLORING THEOREMS 7 By pruning the tree so that every branch through it is a strictly increasing 5 tuple, we get (4.3) ( z a 0 < λ)( z a < λ)( y b n < λ) ( z b 0 < λ)( z b < λ)( i < κ)[φ(i, z a 0,..., z b )]. We now make a rather ad hoc definition of another quantifier in an attempt to make the arguments that follow a little bit clearer. Given i < κ, let the quantifier,i z0 b < λ mean that not only are there unboundedly many z0 b s below λ satisfying whatever property, but also that for each α < λ i, we can find unboundedly many suitable z0 b s for which f z b 0 (i) is greater than α. Claim 4.. If we choose β a < δ a < ξn b such that (4.4) ( z b 0 < λ)( z b < λ)( i < κ)[φ(i, β a, δ a, ξ b n, zb 0, zb )], then (4.5) ( i < κ)(,i z b 0 < λ)( z b < λ)[φ(i, βa, δ a, ξ b n, zb 0, zb )]. Proof. Suppose that we have β a < δ a < ξn b such that (4.4) holds but (4.5) fails. Then there is an unbounded I κ such that for each i I, (4.6) (,i z b 0 < λ)( z b < λ)[φ(i, βa, δ a, ξ b n, zb 0, zb )]. In (4.4), we can move the quantifier i < κ past the quantifiers to its left, i.e., (4.7) ( i < κ)( z b 0 < λ)( z b < λ)[φ(i, β a, δ a, ξ b n, z b 0, z b )], so without loss of generality, for all i I, (4.8) ( z b 0 < λ)( z b < λ)[φ(i, β a, δ a, ξ b n, z b 0, z b )]. Since (4.6) holds for all i I, it must be the case that for each i I, there is a value g(i) < λ i such that for all sufficiently large β < λ, if (4.9) ( z b < λ)[φ(i, β a, δ a, ξ b n, β, z b )], then (4.0) f β (i) g(i). Since {f α : α < λ} witnesses that the true cofinality of i<κ λ i is λ, we know (4.) ( x < λ)( i I)[g(i) < f x (i)]. When we combine this with (4.4), we see that it is possible to choose β b < λ such that (4.2) ( i I)[g(i) < f β b(i)],

18 8 TODD EISWORTH AND SAHARON SHELAH and (4.3) ( z b < λ)( j < κ)[φ(j, β a, δ a, ξ b n, β b, z b )]. (Note that we have quietly used the fact that I < λ = cf(λ) to get a β b that is large enough so that (4.9) implies (4.0) for all i I for this particular β b.) This last equation implies ( j < κ)( z b < λ)[φ(j, βa, δ a, ξ b n, βb, z b )], so it is possible to choose i I large enough so that g(i) < f β b(i) and ( z b < λ)[φ(i, βa, δ a, ξn b, βb, z b )]. This is a contradiction, as (4.9) holds for our choice of i and β = β b, yet (4.0) fails. Notice that an immediate corollary of the preceding claim is (4.4) ( z a 0 < λ)( z a < λ)( y b n < λ)( i < κ) Claim 4.2. If β a < λ is chosen so that (4.5) ( z a < λ)( y b n < λ)( i < κ) then where and (,i z b 0 < λ)( z b < λ)[φ(i, βa, δ a, ξ b n, zb 0, zb )]. (,i z b 0 < λ)( z b < λ)[φ(i, βa, z a, yb n, zb 0, zb )], ( i < κ)( v < λ i )( z a < λ)[ψ ψ ] ψ := gz max (i) < v, a ψ := ( y b n < λ)( z b 0 < λ) [ v < f z b 0 (i) and ( z b < λ)[φ(i, βa, z a, yb n, zb 0, zb )] ]. Proof. In (4.5), we can move the quantifier ( i < κ) past the other quantifiers to its left, so (4.6) ( i < κ)( z a < λ)( y b n < λ) (,i z b 0 < λ)( z b < λ)[φ(i, βa, z a, yb n, zb 0, zb )] holds. The claim will be established if we show that for each i < κ for which (4.7) ( z a < λ)( y b n < λ) (,i z b 0 < λ)( z b < λ)[φ(i, β a, z a, y b n, z b 0, z b )]

19 holds, it is possible to find v < λ i such that [ (4.8) ( z a < λ) gz max (i) < v and a COLORING THEOREMS 9 ( y b n < λ)( z b 0 < λ) [v < f z b 0 (i) and ( z b < λ)[φ(i, βa, z a, yb n, zb 0, zb )] ] ]. Despite the lengths of the formulas involved, this is not that hard to accomplish. Since λ i < λ = cf(λ), we can find v < λ i such that ( z a < λ) [ gz max (i) < v and a ( yn b < λ)(,i z0 b < λ)( z b < λ)[φ(i, βa, z a, yb n, zb 0, zb )]], and now the result follows from of the definition of,i z b < λ. Thus there are unboundedly many z0 a < λ for which there is a function g i<κ λ i such that for all sufficiently large i < κ, [ (4.9) ( z a < λ) g max (i) g(i) and z a ( y b n < λ)( z b 0 < λ)[ g(i) < f z b 0 (i) and ( z b < λ)[φ(i, za 0, za, yb n, zb 0, zb )]]]. Now this is logically equivalent to the statement (4.20) ( z a < λ)( yn b < λ)( z0 b < λ) [ g max z (i) g(i) < f a z b 0 (i) and ( z b < λ)[φ(i, za 0, za, yb n, zb 0, zb )]]. Suppose we are given a particular z0 a < λ for which a function g as above can be found, and let us fix i < κ large enough so that (4.9) holds. Also fix ordinals δ a < λ and ξn b < λ that serve as suitable z a and yn b. Just to be clear, this means that for these choices we have ( z b 0 < λ) [ g max δ a (i) g(i) < f z b 0 (i) and ( z b < λ)[φ(i, β a, δ a, ξ b n, z b 0, z b )] ]. Since λ i < λ = cf(λ), there must be some value w satisfying ( z b 0 < λ)[ g(i) < f z b 0 (i) < w and ( z b < λ)[φ(i, βa, δ a, ξ b n, zb 0, zb )]]. This implies for our particular β a, g, i, δ a, and ξ b n that (4.2) ( w < λ i )( z b 0 < λ) [ g max δ a (i) g(i) < f z b (i) < w and ( z b < λ)[φ(i, β a, δ a, y b n, z b 0, z b )] ].

20 20 TODD EISWORTH AND SAHARON SHELAH Since λ i < λ = cf(λ), the quantifier ( w < λ i ) can move to the left past the quantifiers ( z a < λ)( yn b < λ). This tells us that for our βa and g, (4.22) ( i < κ)( w < λ i )( z a < λ)( yn b < λ)( z0 b < λ) [ g max z (i) g(i) < f a z b 0 (i) < w and ( z b < λ)[φ(i, βa, z a, yb n, zb 0, zb )]]. When we put all this together, we end up with the statement (4.23) ( z a 0 < λ)( i < κ)( v < λ i )( w < λ i )( z a < λ) ( yn b < λ)( z0 b < λ)[ gz max (i) v < f a z b 0 (i) < w and ( z b < λ)[φ(i, β a, z, a yn, b z0, b z)] ] b. Since both κ and λ i are less than λ = cf(λ), we can move some quantifiers around and achieve (4.24) ( i < κ)( w < λ i )( z a 0 < λ)( v < λ i )( z a < λ) ( yn b < λ)( z0 b < λ)[ gz max (i) v < f a z b 0 (i) < w and ( z b < λ)[φ(i, β a, z, a yn, b z0, b z)] ] b. Thus there is a function h i<κ λ i such that (4.25) ( i < κ)( z a 0 < λ)( v < λ i )( z a < λ) ( yn b < λ)( z0 b < λ)[ gz max (i) v < f a z b 0 (i) < h(i) and ( z b < λ)[φ(i, β a, z, a yn, b z0, b z)] ] b. After all this work, it is finally time to prove that we can select objects β a < δ a < ξ b n < βb < δ b that satisfy all of our requirements. Clearly, for every unbounded Λ λ, ( i < κ)( x Λ)(h [i, κ) < f x [i, κ). Thus we can choose i < κ such that h (i ) = j and h 0(i ) = n, and ( z a 0 < λ) [h [i, κ) < f za 0 [i, κ) and ( v < λ i )( z a < λ)( y b n < λ) ( z0 b < λ)[ gz max (i ) v < f a z b 0 (i ) < h(i ) and ( z b < λ)[φ(i, z a 0,..., zb )]]].

21 COLORING THEOREMS 2 So now we choose β a such that h(i ) < f β a(i ) and for some α < λ i, ( z a < λ)( yn b < λ)( z0 b < λ)[ gz max (i ) α < f a z b 0 (i ) < h(i ) and ( z b < λ)[φ(i, z0, a..., z)] ] b. Now we choose δ a, ξ b n, βb, and δ b such that β a < δ a < ξ b n < βb g max δ a (i ) α < f β b(i ) < h(i ) < f β a(i ) Φ(i, β a, δ a, ξ b n, βb, δ b ) It is straightforward to check that these objects satisfy all the requirements listed in Table, so by Claim 3.5, we are done. 5. Conclusions In this final section, we will deduce some conclusions in a few concrete cases. Theorem 2. If µ is a singular cardinal of uncountable cofinality that is not a limit of regular Jonsson cardinals, then Pr (µ +, µ +, µ +, cf(µ)) holds. Proof. The proof of this theorem occurs in two stages we first show that Pr (µ +, µ +, µ, cf(µ)) holds, and then we show that this result can be upgraded to obtain Pr (µ +, µ +, µ +, cf(µ). Let µ be as hypothesized, and let us define λ = µ + and κ = cf(µ). Claim 5.. Pr (λ, λ, µ, κ) holds. Proof. Let κ i : i < κ be a strictly increasing continuous sequence cofinal in µ. Let S {δ [µ, λ) : cf(δ) = κ} be stationary. Standard club guessing results tell us that there is an S club system C such that id p ( C, J) is a proper ideal, where J δ is the ideal J b[µ] C δ for δ S, and furthermore, satisfying C δ = κ. (Note that this last requires that κ = cf(µ) is uncountable.) At this point, we have satisfied all of the assumptions of Claim 2.2 except possibly for clause (8). It suffices to show that for each i < κ, for all sufficiently large regular θ < µ, Player I has a winning strategy in the game Gm ω [θ, κ i, ]. Since µ is not a limit of regular Jonsson cardinals, it follows that for all sufficiently large regular θ < µ, Player I has a winning strategy in Gm ω [θ, θ, ]. This implies, by Lemma.3 (), that for all sufficiently large regular θ, Player I has a winning strategy in Gm ω [θ, κ i, ], and so clause (8) of Claim 2.2 is satisfied.

22 22 TODD EISWORTH AND SAHARON SHELAH To finish the proof of Theorem 2, it remains to show that we can increase the number of colors from µ to λ = µ + we need Pr (λ, λ, λ, κ) instead of Pr (λ, λ, µ, κ). Lemma 5.2. There is a coloring c : [λ] 2 λ such that whenever we are given θ < κ, t α : α < λ a sequence of pairwise disjoint elements of [λ] θ, ζ α t α for α < λ, and Υ < λ, we can find α < β such that t α min(t β ) and (5.) ( ζ t α )[c (ζ, ζ β ) = Υ]. Proof. Let c : [λ] 2 µ be a coloring that witnesses Pr (λ, λ, µ, κ). For each α < λ, let g α be a one to one function from α into µ. We define (5.2) c (α, β) = g (c(α, β)). β Suppose now that we are given objects θ, t α : α < λ, ζ α : α < λ, and Υ as in the statement of the lemma. Clearly we may assume that min(t α ) > α. For i < µ, we define X i := {α [γ, λ) : g ζα (Υ) = i}. Since λ is a regular cardinal, it is clear that there is i < µ for which X i = λ. Since c exemplifies Pr (λ, λ, µ, κ), for some α < β in X i we have t α min(t β ) and (5.3) ( ζ t α )[c(ζ, ζ β ) = i ]. By definition, this means (5.4) ( ζ t α )[c (ζ, ζ β ) = g (c(α, β)) = g (i ) = Υ], hence α and β are as required. To continue the proof of Theorem 2, we define a coloring c 2 : [λ] 2 λ by (5.5) c 2 (α, β) = c (α, ν(α, β)), where ν(α, β) is as in the proof of Theorem. It remains to check that c 2 witnesses Pr (λ, λ, λ, κ). Toward this end, suppose we are given θ < κ, t α : α < λ a sequence of pairwise disjoint members of [λ] θ, and Υ < λ. We need to find δ a and δ b less than λ such that (5.6) ɛ a t δ a ɛ b t δ b = c 2 (ɛ a, ɛ b ) = Υ.

23 COLORING THEOREMS 23 Lemma 5.3. There is a stationary set of γ < λ such that for some γ 0 < γ and β [γ, λ), if γ 0 α < γ, then the function ν is constant on t α t β. Proof. Let E be an arbitrary closed unbounded subset of λ, and let W be the set of ordinals < λ satisfying the properties of γ. In the proof of Theorem, without loss of generality we can have E M 0. This means that the ordinal β found in the course of that proof will be in E, so we finish by observing that β W. An application of Fodor s Lemma gives us a single ordinal γ 0 and a stationary W W such that for all γ W, there is a β γ [γ, λ) such that for all α [γ 0, γ), ν (t α t β ) is constant. Using properties of the coloring c, we can find α and γ such that γ 0 α < λ γ W \ (sup(t α ) + ), and ζ t α = c (ζ, γ) = Υ. Now given ɛ a t α and ɛ b t βγ, we find (5.7) c 2 (ɛ a, ɛ b ) = c (ɛ a, γ) = Υ, and therefore c 2 exemplifies Pr(λ, λ, λ, κ). Theorem 2 strengthens results in [] as clearly Pr (µ +, µ +, µ +, cf(µ)) implies that µ + has a Jonsson algebra (i.e., µ + is not a Jonsson cardinal). The question of whether the successor of a singular cardinal can be a Jonsson cardinal is a well known open question. We note that many of the results from Section 2 of [] dealing with the existence of winning strategies for Player I in Gm ω [λ, µ, γ] can be combined with Theorem to give new results. For example, we have the following result from []. Proposition 5.4. If τ 2 κ but ( θ < κ)[2 θ < τ], then Player I has a winning strategy in the game Gm ω (τ, κ, κ + ). Proof. See Claim 2.3() and Claim 2.4() of []. Armed with this, the following claim is straightforward. Claim 5.5. Let µ be a singular cardinal of uncountable cofinality. Further assume that χ is a cardinal such that 2 <χ µ < 2 χ. Then Pr (µ +, µ +, χ, cf(µ)) holds. Proof. If 2 <χ < µ, then Claims 2.3() and 2.4() of [] imply that for every sufficiently large θ < µ, Player I has a winning strategy in the game Gm ω (θ, χ, χ + ).

24 24 TODD EISWORTH AND SAHARON SHELAH If µ = 2 <χ, then cf(µ) = cf(χ). Let κ i : i < cf(µ) be a strictly increasing continuous sequence of cardinals cofinal in χ. Given i < cf(µ), we claim that for all sufficiently large regular τ < µ, Player I has a winning strategy in Gm ω (τ, κ i, χ). Once we have established this, Pr (µ +, µ +, χ, cf(µ)) follows by Theorem. Given τ = cf(τ) satisfying 2 κ i < τ < µ, let η be the least cardinal such that τ 2 η. Clearly κ i < η < χ. By Proposition 5.4, Player I wins the game Gm ω (τ, η, η + ). This implies (since η + < χ and κ i < η) that Player I wins the game Gm ω (τ, κ i, χ) as required. We can also use Claim.4 to prove similar results. For example we have the following. Claim 5.6. Let µ be a singular cardinal of uncountable cofinality. Further assume that χ < µ satisfies 2 χ < µ < ℶ (2 χ ) +(χ). Then Pr (µ +, µ +, χ, cf(µ)) holds. Proof. Again, the main point is that for all sufficiently large regular θ < µ, Player I has a winning strategy in the game Gm ω [θ, χ, (2 χ ) + ]. This follows immediately from Claim.4. Since (2 χ ) + < µ, Theorem is applicable. In a sequel to this paper, we will address the situation where λ is the successor of a singular cardinal of countable cofinality. Similar results hold, but the combinatorics involved are trickier. References [] Saharon Shelah, More Jonsson algebras, Archive for Mathematical Logic accepted. [2], Cardinal arithmetic, Oxford Logic Guides, vol. 29, Oxford University Press, 994. [3], Jonsson Algebras in an inaccessible λ not λ-mahlo, Cardinal Arithmetic, Oxford Logic Guides, vol. 29, Oxford University Press, 994. [4], There are Jonsson algebras in many inaccessible cardinals, Cardinal Arithmetic, Oxford Logic Guides, vol. 29, Oxford University Press, 994. [5], Proper and improper forcing, Perspectives in Mathematical Logic, Springer, 998. Department of Mathematics, University of Northern Iowa, Cedar Falls, IA, address: eisworth@uni.edu Institute of Mathematics, Hebrew University of Jerusalem, Jerusalem, Israel address: shlhetal@math.huji.ac.il