saharon shelah- categoricity of theories in l-kappa^-omega when kappa^ is a measurable cardinal....

8/3/2019 Saharon Shelah- Categoricity of Theories in L-kappa^*-omega When kappa^* is a Measurable Cardinal. Part 2

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CATEGORICITY OF THEORIES IN L, WHEN IS A

MEASURABLE CARDINAL. PART II

SAHARON SHELAH

Abstract. We continue the work of [2] and prove that for successor, a -categorical theory T in L, is -categorical for every , which is above

the (2LS(T))+-beth cardinal.

0. Introduction

We deal here with the categoricity spectrum of theories T in the logic: L,with measurable and more generally, continued the attempts develop classifica-tion theory of non elementary classes in particular non forking. Makkai and Shelah[3] dealt with the case a compact cardinal. So measurable is too high com-pared with the hope of dealing with T L1, (or any L,) but seems quite smallcompared to the compact cardinal in [3]. Model theoretically a compact cardinalensures many cases of amalgamation, whereas measurable cardinal ensures no max-imal model. We continue [11], Makkai and Shelah [3], Kolman and Shelah [2]; tryto imitate [3]; a parallel line of research is [14]. Earlier works are [6], [8], [9]; forlater works on the upward Los conjecture, look at [15] and [4].

On the situation generally see more [15].

This paper continues the tasks begun in Kolman and Shelah [2]. We use theresults obtained there in to advance our knowledge of the categoricity spectrum oftheories in L,, when

is a measurable cardinal.The main theorems are proved in section three; section one treats of types and

section two describes some constructions.Note that we may expect to be able to develop better, more informative clas-

sification theory, in particular stability theory, for T L, measurable thanwithout the measurables assumption, and less informative then the case com-pact.

The notation follows [2], except in two important details: we reserve for thefixed measurable cardinal and T for the fixed -categorical theory in L, in a givenvocabulary L; is any infinite cardinal and T is usually some kind of tree. To recap

briefly: T is a - categorical theory in L,, LS(T) def= + |T|, K = K, Fis the class of models of T, where F is a fragment of L, satisfying T F,|F| + |T|, and for M, N K, M F N means that M is an F-elementarysubmodel of N.

1991 Mathematics Subject Classification. 03C25, 03C75, 03C20.Key words and phrases. Model theory, infinitary logics, classification theory, categoriticy, Los

Theorem, measurable cardinal, limit ultrapower.Research supported by the United States-Israel Binational Science Foundation. Publication

number 472.

1


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2 SAHARON SHELAH

The principal relevant results from [2] are: K


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CATEGORICITY OF THEORIES IN L . . . 3

Definition 1.2. Suppose that M K


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4 SAHARON SHELAH

N

N1 N2

U

tttttt

g1 g2

a1 N1

h10

a2 N2 a3 N3tttt

ttt

h11

tttt

tt

h20 h21

M

T

k

Q

id id id

U

Just notice now that N, g1h10, g2h

21 witness that (a1, N1)EM(a3, N3), since:

g1h10(a1) = g1(h

11(a2)) = g2h

20(a2) = g2h

21(a3).

2), 3) Left to reader.

Definition 1.4. Suppose that M, N K


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(5) T is -stable if N K |S(N)| .(6) We say N is -universal over M when: M F N, N K and if M F

N

K then there is a F-embedding of N

into N over M.(7) We say N is (, )-saturated over M if there is a F- increasing continuous

sequence Mi : i < such that: M0 = M, N =i


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6 SAHARON SHELAH

one has M F N, Mt F N and for s = t I, t < s iff M1t

N

MM2s .

(2) Assume(a) LS(T), M K is nice,(b) for = 1, 2 we have Op is defined by (I, D, G), f,

IM for < with eq(f,) G, i.e., such that eq(f,)/D M

ID|G,

(c) for = 1, 2 we have M0 = M, M1 = Op(M

0), M

2 = Op3(M

1),

a,1 = f,/D1 (M0)ID

|G = M1 andal,2 = f3,/D2 (M

2)I3D3

|G3 =

M2.Then there are , i ( = 0, i < or {1, 2}, i < ) such that() is a blueprint for E.M. models, |L| , L the vocabulary of so

L L,() for any linear order I we have EM(I, ) = EML(I, ) is the Lreduct

of EML(I, ), (an L)model) which is a model of T of cardinality

+ |I| and

I J EM(I, ) F EM(J, ),

() li are unary function symbols in L,() EM(, ) is M,() for any linear order I, and s < t in I we have: the type which

(i) 1(xs) : < 12(xt) : < 2 realizes overM inEM(I, )

is the same type as a1,1 : < 1a1,2 : < 2 realizes over

M in M12 ,(ii) 1(xt) : < 1

2(xs) : < 2 realizes overM inEM(I, )

the same type as a2,2 : < 1a2,1 : < 2 realizes over M

in M22 .

Remark: Note M0 nice M3 is automatic in the interesting case since M0 K


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if s, t I, s < t then (Fs M1) (Ft M2) can be extended to an F-embedding

of M3 into N, and hence by the assumption it follows that M

1

t

NM0 M

2

s .

(2) A similar proof.

Corollary 1.7. Assume T categorical in or just I(, T) < 2. Then+


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8 SAHARON SHELAH

() EM(J+ , )EM(J, )

EM(J, )

EM(J, ).

Now easily there is an automorphism f of EM(J+ , ) over EM(J, ) whichmaps as to as+ . The Op which witnesses () extends f to an automorphism ofOp(EM(J+ , ) which is the identity over EM(J

, ) continuous.

It may be helpful, though somewhat vague, to add the remark that

-asymmetry

enables one to define order and to build many complicated models; so 1.7 removesa potential obstacle to a categoricity theorem. Note that we could have put 3.11(2)here.

Definition 1.8. Let A be a set. We write M1M3M0

A (where A M3, M0 F

M1 F M3) to mean that there exist M2, M3 such that A |M2|, M3 F M

3 and

M1M3M0

M2. In this situation we say that A/M1 = tp(A, M1, M3) does not fork over

M0 in M3.

We will write M1M3M0

a to mean M1M3M0

{a}, we then say tp(a, M1, M3) does not

fork on M0.

We write A1M3M0

A2 if for some M3, M3 F M3 K


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(3) Notice that quite generally speaking, N1N3

N0N2 implies that N1 N2 = N0

(see above).

Definition 1.10. We define

(T) =(K) = { : cf() = and there exist a continuous F-chain

Mi : i + 1 K and a M+1 such thatfor all i < , a/M forks over Mi in M+1}.

I.e., for (T) there are Mi K : i + 1 and a M+1 such that

i < MM+1

Mia.

Example 1.11. Fix and . Let ( , E)


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10 SAHARON SHELAH

Again we will mean the same thing by saying that T has continuous non-forking in(, ).

Our next goal is to show that if T fails to possess these features for some < such that + LS(K), then T has many models in .

Let us recall in this context a further important result from [11, II, 3.10]:

Theorem 1.14. Assume T be a -categorical theory, or just K


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Proposition 1.16. (T categorical in )

(1) Any M K is saturated.

(2) Every N K


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12 SAHARON SHELAH

Proposition 1.21. (1) Suppose Ni : i is nice-increasing continuous-

inuous for = 1, 2, N1i F N2i K


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By every N K is saturated there are an operation Op and N K suchthat M0 F N F Op(M0) hence there are M

+0 , M

+1 , M

+2 in K


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14 SAHARON SHELAH

Op(M0), such that N0 = , hence N0 is saturated hence there is an automor-phism g0 of N0 such that g0 M12 = f2 (so g0 M0 = idM0). So there is N2,

2=1

M2 N2 F N0, N2+ < , N2 closed under g0, g10 . Now there is N3,

N0 M1 N3 F Op(M1), N3 K, hence N3 is saturated. So M1N3M0

N2 and

hence N2N3M0

M1 (by symmetry, i.e., 1.7). Hence for some N3, N

3 F N3 K


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Note: we do not require even Mi Mi+1.

To achieve this, let us define by induction on i < i, Ni , Mi

and fi . W.l.o.g.

0 = and i limit implies lg(i) limit. Let N0 = M0 = Sk(M0), the Skolemizationof M0, f = idM0 . If i is a limit ordinal, let N

i =

j


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16 SAHARON SHELAH

(4) for T, letting T[] = { T : ( )}, T[] = { T[] : lex }, T[] = { T[] : lex } (so T[] = T[] T[]) and

< g() we have MT[]MT

MM so we can replace MT by MT[] and

MT[]MT

MMT[] for < ;

(5) if lim(T) and / T, then MT =


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There are two different ways to carry on the construction (under Data 2.2.1).We will consider each in its turn.

Construction 2.3. Recall that it is possible to iterate the operation Op withrespect to the linear order (T,


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18 SAHARON SHELAH

Construction 2.4. In this approach, we employ the generalized Ehrenfeucht-Mostowski models EM(I, ) from chapter VII in [7] or [12]. For this we need

to specify the generators of the model and what the types are.Let M+0 be the model obtained from M0 by adding Skolem functions and indi-

vidual constants for each element of M0. We know that there is an operation Opsuch that, for i , Mi F Mi+1 F Op(Mi). As in [2, 1.7.4] this means thatthere are I, D and G such that Op(M) = Op(M , I , D , G) where I is a non-emptyset, D is an ultrafilter on I, and G is a suitable set of equivalence relations on I,i.e.,

(i) ife G and e is an equivalence relation on I coarser than e, then e G;(ii) G is closed under finite intersections;

(iii) if e G, then D/e = {A I/e :xA

x D} is a -complete ultrafilter on

I/e.

For each b Mi+1 \ Mi, let xb

t : t I/D be the image of b in Op(Mi). Wellalso write xbt : t I/D for the canonical image d(b) of b Mi in Op(Mi).

Mi+1 b xbt : t I/D Op(Mi)

Mi

ttt

tttt

U

We define a model M+, M+0 L, M+, as follows. M+ is generated by the

set {xb : b Mi+1 \ Mi, T,g() = i + 1}. Note that this set does generate a

model since M+0 is closed under Skolem functions. Since functions have finite arity,it is enough to specify, for each finite set of the xb, what quantifier-free type itrealizes. Since there is monotonicity, we shall obtain indiscernibility as in [7]. Thetype of a finite set xb : = 1, . . . , n depends on the set b1, . . . bn and the atomic

(i.e., quantifier-free) type of 1, . . . , n in the model T,,


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identifying elements of M with their images in the ultrapower. Now define

h (b) = d(H (b)) if b Mi,

h (xbt) : t I/D if b Mi+1 \ Mi,

where d(h(b)) is the canonical image of H(b) in the ultrapower. The type of

xb : = 1, . . . , n is defined to be the type of h(b) : = 1, . . . , n in N.

Remark 2.4.1. It is possible to split the construction into two steps. For i j + 1, there is an operation Opi,j , Mi Mj = Op

i,j(Mi), moving b to i,jabt : t I, b Mj, i,jabt Mi, with the obvious commutativity and continuity properties.Now the construction is done on a finite tree : = 1, . . . , n, m : ,m < .We omit the details of monotonicity.

Notation 2.4.2. Let MT = M be the Skolem closure. If S T is closed withrespect to initial segments, let MS = SkMT(x

b : S, b Mg()) and M

=

M{:g()}. Define h : Mg() M

by h(b) = x

b

(T) and N = h[M].Remark 2.4.3. The construction can be used to get many fairly saturated models.We list the principal properties below.

Fact 2.4.4. Suppose that S T is closed with respect to initial segments, S0 =S1 S2 and

S1 & S2 \ S1


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20 SAHARON SHELAH

Theorem 3.2. Assume the conclusion of 1.7 for every (e.g., + < )and +. Suppose that the tree T is as in Claim 2.2.2 and suppose further:

Mi K : i + 1 is nice-increasing continuous sequence of members ofK, such that M+1 = M and we apply 2 and

()1 there is no F-increasing continuous sequence Ni K : i suchthat:

(i) Mi F Ni,(ii) M+1 F N,

(iii) if i < j and Nj < , then NiNjMi

Mj.

Then TFAE for lim(T)def= { (Ord) :

i


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(1) T is -based if + < and LS(T); also it is (< )-based if = cf(),LS(T) < , < ;

(2) (T) = for every such that +

< and LS(T).

Proof. (1), (2) We use 3.2, 3.3 to contradict -categoricity. In the first phraseof (1) let = , = +, in the second let us repeat the proofs (i.e., prove theappropriate variants of 3.2, 3.3 be regular; so =cf() and + < .

Case 1: = .By [11, III, 5.1] = [5, IV, 2.1].

Case 2: is regular.We can find a stationary W I[], W { < : cf() = } (by [13, 1]).Hence, possibly replacing W by its intersection with some club of , there is W+,W W+ and a : W+ such that: a (so W+) implies W+,a = a a and otp(a) and

= sup a cf() = W.

Now let enumerate a in increasing order (for W+), and for any W W

let

TW = { : W+ but / W \ W} {

0 : W}.

Now if W1, W2 W, W1 \ W2 is stationary, then MTW1 cannot be F-embeddedinto MTW2 (again by [11, III, 5] = [5, IV 2]).

Case 3: singular.Choose , > = cf() > + and act as in case 2 using instead exceptadding to TW the set {i : i < } (to get 2 we need more, see in [5, IV,VI] onpairwise non isomorphic models).

Hypothesis 3.6. The conclusion of 3.5 (in addition to 3.1 of course).

Conclusion 3.7. Suppose LS(T), + < , M K

(1) Ifp S(M) then p is determined by {p N : N F M and N = LS(T)}(2) Assume further

()M{Nt:tI} (a) I is a directed partial order,

(b) Nt F M,(c) I |= t s implies Nt Ns

(hence Nt F Ns by clause (b)),(d)

tI

Nt = M.

Then() every p S(M) is determined by {p Nt : t I} which mean just

that if q S(M) and for every t I we have p Nt = q Nt thenp = q,

() for some t I, p does not fork over Nt, {p Nt : t I}.

Proof. (1) Follows from part (2): We can find N = Nt : t I such that()M{Nt:tI} holds, Nt LS(T) and on it use part (2). Why N exists? E.g.,

as the proof of part (2) which I = {}, N = M and use Nu : u I

forI = ([M]


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22 SAHARON SHELAH

() we can choose by induction on n < for every u [M]n, t[u] I and Nusuch that: u Nu , N

u F Nt[u], N

u LS(T) and

u v [|M|] LS(T). So we will

suppose that < . Suppose N M = M, N < and p S(N). Let =: N + + LS(T) so < hence + < . W.l.o.g. there is no N1, N FN1 M, N1 and p1, p p1 S(N1) such that p1 forks over N (by 3.3but not used). If there is i < such that N Mi, then p is realized in Mi+1.By the choice of the models M+i,j , it is easy to find N

such that N N M,


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N = def= N + + LS(T) and, for every i ,

MiM

Mi NN.

Now let Ni = N Mi and note that N = N. The sequence Ni : i iscontinuous increasing and there is an extension p of p in S(N) = S(N). Hencethere exists i < such that (i j < ) (p does not fork over Nj). If weare proving part (2), then Mi+1 is saturated but Mi = > = Ni+1 andhence there is a Mi+1 realizing p

Ni+1. But by the non forking relation abovewe get tp(a, N, M) does not fork over Ni+1, hence is p

, as required. If we areproving part (1), Mi+1 is universal over Mi hence we can find a saturated modelN F Mi+1 which contains Mi N

. Hence we can find N : < + which is

F-increasing continuous such that: Ni F N F Mi+1, N+1 is a -universal

extension of N and N0 = Mi N

, and let a N be such that tp(a, N , N

+1)

does not fork over Mi N and extend p (Mi N). By 3.5(1), for some there

is N, N N N

and N

NN Mi

N, so a realizes p. (Recall symmetry and

uniqueness of extensions).

(3) Similar proof for the second sentence, using 1.20 for the first sentence.

Remark: Using categoricity we can prove 3.8 also by 1.20(2) (and uniqueness).

Conclusion 3.9. Assume LS(T) < (LS(T), ), M K is not +-saturated; let Nu : u [|M|]


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24 SAHARON SHELAH

M1 omitting every p satisfying M1 = 1 > |PM

1 | implies the existenceof an L-model M2 omitting every p and satisfying M2 = 2 > |P

M2 | .

So by 3.9 we have Q(LS(T), 1, 2), T categorical in = 1 > LS(T) and2 < 1 implies T is categorical in 2 (the need for 2 < 1 is as only over modelsin K


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Lemma 3.12. In K


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26 SAHARON SHELAH

K), possible by 1.22 + 1.16(6). For limit take unions (the result are saturatedby the definition, and clause (g) holds by 3.5(2)). Lastly for = + 1, limit, if

there are no such M, N then N is isolated over M {a}.Now both M and M = M0 =

n


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tp(A,N,N+) does not fork over Nj . W.l.o.g., i = j is a successor ordinal and

tp(A {a}, M) does not fork over Mi1. So as N

N+M A, necessarily tp(A, N

i, N+

)

forks over Mi, hence (by clause (c) above), aN+Mi

A. But M and Mi are by the

construction saturated over Mi1, and hence there is an isomorphism f from Mionto M which is the identity over Mi1. So by using uniqueness of does not fork, it

maps tp(A{a}, Mi1, N+) to tp((A{a}, M , N +) and hence a

N+M

A (by 1.21(4)).

Thus we get aMM

M, contradiction to the choice of N+, A , M .

Alternatively repeat the proof of 3.13 using 3.11(2)s second sentence.

Theorem 3.16. Assume is a successor cardinal, i.e., = +0 . Then T iscategorical in every [(2LS(T))+ , ) (really for some 0 < (2LS(T))+, [0, )suffices).

Proof. As in [3]. By 3.10, for some 1 < (2LS(T))+ every M K[1,] is LS(T)+-

saturated. Let [1, ), and assume M K is not saturated, so for some (LS(T), ) the model M is -saturated not +-saturated. Let p, Nu : u [|M|]


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28 SAHARON SHELAH

Another way to say it: the stationarization of p over N+ is realized.But is every q S(N+) a stationarization of some p S(N), N F N+,

N

LS(T)? We can find N0 F N+

, N0 (T), such that: [N0 FN1 N+ & N1 LS(T) q N1 does not fork over N0], we can getit for N1 < , but does it hold for N1 = N

+? A central point is() Does K satisfy amalgamation?

Again it seems that P(n)-systems are called for. See more in [15].(4) If |T| < we can do better, as Op(EM(I, )) = EM(Op(I), ), will

discuss elsewhere.

References

[1] Bradd Hart and Saharon Shelah. Categoricity over P for first order T or categoricity for L1 can stop at k while holding for 0, ,k1. Israel Journal of Mathematics,70:219235, 1990. math.LO/9201240. [HaSh:323]

[2] Oren Kolman and Saharon Shelah. Categoricity of Theories in L,, when is a measur-

able cardinal. Part 1. Fundamenta Mathematicae, 151:209240, 1996. math.LO/9602216.[KlSh:362]

[3] Michael Makkai and Saharon Shelah. Categoricity of theories in L, with a compactcardinal. Annals of Pure and Applied Logic, 47:4197, 1990. [MaSh:285]

[4] Saharon Shelah. Categoricity in abstract elementary classes: going up inductively.math.LO/0011215. [Sh:600]

[5] Saharon Shelah. Nonstructure theory, volume accepted. Oxford University Press. [Sh:e][6] Saharon Shelah. Categoricity in 1 of sentences in L1,(Q). Israel Journal of Mathematics,

20:127148, 1975. [Sh:48][7] Saharon Shelah. Classification theory and the number of nonisomorphic models, volume 92

of Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Co.,Amsterdam-New York, xvi+544 pp, $62.25, 1978. [Sh:a]

[8] Saharon Shelah. Classification theory for nonelementary classes, I. The number of uncountablemodels of L1,. Part A. Israel Journal of Mathematics, 46:212240, 1983. [Sh:87a]

[9] Saharon Shelah. Classification theory for nonelementary classes, I. The number of uncountable

models of L1,. Part B. Israel Journal of Mathematics, 46:241273, 1983. [Sh:87b][10] Saharon Shelah. Classification of nonelementary classes. II. Abstract elementary classes. In

Classification theory (Chicago, IL, 1985), volume 1292 of Lecture Notes in Mathematics,pages 419497. Springer, Berlin, 1987. Proceedings of the USAIsrael Conference on Classi-fication Theory, Chicago, December 1985; ed. Baldwin, J.T. [Sh:88]

[11] Saharon Shelah. Universal classes. In Classification theory (Chicago, IL, 1985), volume 1292of Lecture Notes in Mathematics, pages 264418. Springer, Berlin, 1987. Proceedings of theUSAIsrael Conference on Classification Theory, Chicago, December 1985; ed. Baldwin, J.T.[Sh:300]

[12] Saharon Shelah. Classification theory and the number of nonisomorphic models, volume 92of Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Co.,Amsterdam, xxxiv+705 pp, 1990. [Sh:c]

[13] Saharon Shelah. Advances in Cardinal Arithmetic. In Finite and Infinite Combinatorics inSets and Logic, pages 355383. Kluwer Academic Publishers, 1993. N.W. Sauer et al (eds.).0708.1979. [Sh:420]

[14] Saharon Shelah. Categoricity for abstract classes with amalgamation. Annals of Pure andApplied Logic, 98:261294, 1999. math.LO/9809197. [Sh:394]

[15] Saharon Shelah. Categoricity of an abstract elementary class in two successive cardinals.Israel Journal of Mathematics, 126:29128, 2001. math.LO/9805146. [Sh:576]

[16] Saharon Shelah. Classification Theory for Abstract Elementary Classes, volume 18 ofStudiesin Logic: Mathematical logic and foundations. College Publications, 2009. [Sh:h]

Institute of Mathematics, The Hebrew University of Jerusalem, 91904 Jerusalem,

Israel, and, Department of Mathematics, Rutgers University, New Brunswick, NJ 08854,

USA

saharon shelah- categoricity of theories in l-kappa^*-omega when kappa^* is a measurable cardinal....

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saharon shelah- categoricity of theories in l-kappa^-omega when kappa^ is a measurable cardinal....