1 Introduction - CMU Math
Possible behaviours for the Mitchell ordering James Cummings Math and CS Department Dartmouth College Hanover NH 03755 January 23, 1998 Abstract
We use a mixture of forcing and inner models techniques to get some results on the possible behaviours of the Mitchell ordering at a measurable .
1 Introduction The Mitchell ordering on normal measures was invented by Mitchell [3] as a tool in his study of inner models for large cardinals.
De nition 1: Let be measurable, let U and U be normal measures on . Then U U if and only if U 2 Ult(V; U ), the ultrapower of V by U . 0
0
1
0
1
1
The following facts are standard.
is transitive. is well-founded. is strict. An ultra lter has at most 2 ancestors in the ordering . 1
1
De nition 2: o() is the height of the well-founded relation . Notice that we must have o() (2) . Much is known about the possible behaviours of . For example Mitchell has shown [3] that in a highly structured inner model we can have GCH holding and o() = , with being a linear ordering. Baldwin has shown [6] that from suitable hypotheses we can have models in which is a given prewellordering of cardinality less than . If is the critical point of j : V ! M such that V M , then we may show that every element of V is in Ult(V; U ) for some U on . In particular any 2 measures on will have an upper bound in the ordering . What is more, for any particular U there will only be 2 elements of V in Ult(V; U ), so that there must be 2 measures on . If it happens that 2 > (2) then cannot be linear, and it is not clear what the structure of will be. This question is addressed in [1]. In this paper we will produce a model in which is measurable, and all measures on may be divided into \blocks" in the following way: 1. For each < o() and 2 ( ; o()) [ 1 there is a block M ( ; ). 2. All the measures in M ( ; ) have height in the Mitchell ordering. 3. M ( ; ) has cardinality if 2 ( ; o()), and cardinality if = 1. 4. For U 2 M ( ; ) and V 2 M ( ; ), U V i (with the convention that 1 is bigger than any ordinal). +
++
+2
+2
+2
2
2
+
+
++
2 Preliminaries In this paper we will use large cardinals and forcing to produce some models where the Mitchell ordering is rather complex. In the interests of clarity and self-containedness we have collected various key facts in this section, facts 2
which we will use repeatedly in the sequel. None of them are due to us; in many cases we are unsure to whom they should be attributed. We start with a remark about Cohen forcing. The forcing for adding a single Cohen subset to a regular cardinal can be regarded as having conditions which are functions p : ! for < (rather than the more standard functions from < to f0; 1g). In this form we can consider the forcing as adding a generic function from to . We will be interested in elementary embeddings k : M ! N between inner models of ZFC. In general it will not be the case that k is aSclass of M or that N M (notice that the former implies the latter, as N = k(V M )). If a model M believes that U (with U 2 M ) is a measure on , we will denote the natural embedding from M into Ult(M; U ) by jUM .
Lemma 1: Let j : V ! M be an elementary embedding with j a class of V , = crit(j ), such that every element of M is j (F )() for some function F 2 V . Then j is the ultrapower by the normal measure U = f X j 2 j (X ) g. Proof: Factor j through the ultrapower of V by U ,
j0
V
3
M
0
Q
Q
Q Q
Q k Q
Q
Q Q
j
QQ s -
M
by de ning k : [f ] 7 ! j (f )(). k is a surjection, and M is the transitive collapse of the range of k, so M = M and j = j . 0
0
0
Lemma 1 will prove useful in identifying certain embeddings as ultrapowers.
Lemma 2: Let M and N be inner models of ZFC such that 3
M N. N M M . M U is a normal measure on .
Then U is a normal measure in N and jUN M = jUM . Proof: It follows immediately from the closure of M that U is a normal measure in N . Let x 2 M . jUN (x) is the transitive collapse of the structure (F; EU ) where F = f f : ! x j f 2 N g; and fEU g () f j f ( ) 2 g( ) g 2 U: By the closure of M inside N we have F = f f : ! x j f 2 M g; which is the set of functions whose collapse is jUM (x), so by the absoluteness of the collapsing construction jUN (x) = jUM (x). Lemma 2 will be useful in understanding restrictions of ultrapower maps, as for example in the proof of the following lemma. Lemma 3: Let U be a measure on , W a measure on and suppose that W 2 Ult(V; U ). Let MU be the ultrapower of V by U , MW the ultrapower of V by W . Then Ult(MU ; W ) = Ult(MW ; jWV (U )) and the following diagram commutes. jUV
V
-
MU MU jW
V jW ?
MW
jjMVW(U )
-
?
Ult(MU ; W )
W
4
Proof: Let x 2 V . jjMWV WU (jWV (x)) = jWV (jUV (x)); (
)
by elementarity. W 2 MU and (as ) MU MU , so that jWV MU = jWMU : In particular jWV (jUV (x)) = jWMU (jUV (x)): From this we can deduce that the two ultrapowers are equal (let x = V ), and that the diagram commutes. We will use lemma 3 to analyse restrictions of iterated ultrapowers.
Lemma 4: Let k : M ! N be an elementary embedding between inner models of ZFC. Let P 2 M be a forcing notion, let G be P-generic over M
and let H be k(P)-generic over N . Suppose that p 2 G =) k(p) 2 H: Then 1. There is a unique extension of k to a map k : M [G] ! N [H ] such that k : G 7 ! H . 2. If is a set of ordinals such that
N = f k(F )(a) j F 2 M; a 2 []
N [H ] = f k(F )(a) j F 2 M [G]; a 2 []
5
Proof: For the rst claim, it is clear that if k exists it must be given by k : _ G 7 ! k( _ )H ; where _ G denotes the interpretation of the term by the generic G. We check that this is well-de ned. Let _ G = _ G, then there is p 2 G such that p MP _ = _ . By elementarity k(p) Nk P k( _ ) = k(_ ). By assumption k(p) 2 H , so that k( _ )H = k(_ )H . The proof that k is elementary is entirely similar. For the second claim, let _ H 2 N [H ]. Then _ = k(F )(a) for some F 2 M and a 2 []
1
(
1
)
Lemma 4 will be used to take elementary embeddings (usually nitely iterated ultrapowers) and extend them onto certain generic extensions of V . The second claim will play a key r^ole in understanding the nature of the extended embedding. The next lemmas goes into more detail about the extensions that we will make. We start with a technical result about equivalence between generics.
Lemma 5: Let P be the forcing notion given by a Reverse Easton iteration
of length + 1, in which one Cohen subset of is added at each strong inaccessible . Let G and G be P-generics over V , with the property that V [G ] = V [G ]. Then for any model V agreeing with V to rank + 1, G and G are P-generic over V and V [G ] = V [G ]. 1
1
1
2
2
2
1
2
Proof: By the agreement P 2 V and (since jPj = ) both models compute
the same maximal antichains, so G and G are generic over V for P. G is the interpretation under G of some term _ , and by the agreement again we may take it that _ 2 V . So G 2 V [G ] and vice versa, so that V [G ] = V [G ]. 1
2
1
2
1
1
2
2
Next we give the lemma that will be used to generate measures. 6
Lemma 6: Let GCH hold, and let j : V ! M be an embedding which is a class in V , such that = crit(j ) and M M . Suppose also that the
ordinal j ( ) has cardinality in V . Let P be as in lemma 5, and observe that P can be factored as P followed by Add(; 1) as computed by V P. Let G = G g be P-generic, and suppose that there is G = G g with V [G] = V [G ]. Then in V [G] there are many H such that G H is j (P)-generic over M and j extends to | : V [G] ! M [G ][H ]. +
+
1
1
++
1
1
1
Proof: By lemma 5 M [G] = M [G ]. In M [G ] the factor iteration j (P)=G 1
1
1
is highly-closed and has j ( ) many antichains. As P has the -chain condition and M [G] = M [G ] we have V [G] M [G ] M [G ]. Hence in V [G] the forcing j (P)=G is -closed, and the set of its maximal antichains which lie in M [G ] has cardinality . We wish to build generics which are compatible with G. Working in M [G ], de ne a function q with domain the M -inaccessibles such that < j (), by q() = ; for < j () and q(j ()) = g. q is a condition in j (P)=G . We build in V [G] a binary tree of height such that The top node is q. Any path is a descending sequence in j (P)=G , meeting each antichain in M [G ]. Every element has incompatible immediate successors. The construction proceeds for the requisite steps, because j (P)=G is -closed in V [G]. This construction will give us distinct generic lters H , each with the property that j \G G H . We can use these to build extensions | of j such that |(G) = G H . +
1
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1
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1
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This last construction was a \master condition" argument a la Silver; notice that any extension of q in j (P)=G would have done equally well as the top node of the tree. We will make heavy use of Mitchell's theory of core models for sequences of measures; nowadays this should be seen as a special case of the core model 7
theory for non-overlapping extenders (due to Mitchell, Dodd, Jensen and Koepke) in which every extender happens to be equivalent to a measure. The reader is referred to Mitchell's paper [4] for proofs.
De nition 3: U~ is a coherent sequence of measures if and only if U~ is a function, with dom(U~ ) On On. For some function oU~ : On ! On, dom(U~ ) = f (; ) j 0 < oU~ () g:
If (; ) 2 dom(U~ ) then U~ (; ) is a normal measure on . If (; ) 2 dom(U~ ), and j : V ! M is the ultrapower of V by the measure U~ (; ) then
{ For all , ( ; ) 2 dom(j (U~ )) if and only if or = and < . { If and ( ; ) 2 dom(j (U~ )) then j (U~ )( ; ) = U~ ( ; ):
De nition 4: Let M be an inner model of ZFC, let
M U~ is a coherent sequence of measures: A normal iteration of M by U~ , of length is a pair (hM : < i; hj : < i) where M = M. M is an inner model of ZFC for each < . For < , j : M ! M is an elementary embedding. j = id, and for , j = j j . 0
8
M = Ult(M ; j (U~ )( ; )), and j : M ! M is the +1
0
+1
+1
associated ultrapower map, if + 1 < . If < , is limit, then M and j are had by taking a direct limit in the natural way. The sequence h : + 1 < i is strictly increasing. The following structural fact is easy, by induction on < .
~ | ) is a normal iteration of M by U~ in length then for Lemma 7: If (M;~
every < M = f j (F )(a) j F 2 M; a 2 []
We will denote by K Mitchell's core model K [U~ max ], which exists under the assumption that there is no inner model in which 9 o() = . We will use the following facts about K (see section 2 of [5]). ++
Lemma 8 (Mitchell): Suppose that :9 o() = in any inner model ++
of ZFC. Then K is a uniformly de nable inner model of ZFC+GCH. K V = K. K U~ max is a coherent sequence of measures. K is invariant under set forcing. If i : K ! M is an elementary embedding into an inner model M then i arises from a normal iteration of K by U~ max .
It is worth making the following easy observations about K and U~ max . Lemma 9: If K , U~ max are as above then All measures in K appear on the sequence U~ max . 9
If < < oU~ max () then U~ max (; ) 6= U~ max (; ). K U~ max (; ) U~ max (; ) i < . We will be particularly interested in nite normal iterations of K , in the case when there is a largest measurable on U~ max .
Lemma 10: Suppose that is the largest ordinal with oU~ max () > 0. Let
~ | ) be a normal iteration of K by U~ max of length n + 1, n + 1 < !, let (M;~ with j the ultrapower of K by U~ max (; ) for some . Then 1. Mn K , and K Mn Mn. 2. For each i < n, i < j n (). 3. In M , the ordinal j n( ) has cardinality 01
0
0
0
+
+
Proof: 1. The critical points are increasing and each model is closed inside the previous one. 2. i j i (), as is the largest measurable on U~ max . If i < j i () then we are done as j n () = jin(j i ()) j i (); if i = j i () then this is the critical point of jin so i < jin(j i ()) = j n (). 3. The ordinals less than j n ( ) all have the form 0
0
0
0
0
0
0
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0
+
j n(F )( ; : : : ; n ); 0
0
1
where F : []n ! . By GCH there are such functions F . +
+
The next result puts some limits on the possible closure of the models in a normal iteration of in nite length.
10
~ | ) is a normal iteration of M by U~ , of length !, Lemma 11: If (M;~ then the sequence of ordinals ~ = hn : n < !i is not a member of M for ! < . Proof: The model M agrees with M! to rank ! + 1, so it is enough to show that ~ 2= M! . M! was constructed as a direct limit, so if ~ 2 M! then ~ = jn! (~) for some ~ 2 Mn; in particular n = jn! (n). But crit(jn! ) = n as we are in a normal iteration, so that n 2= rge(jn! ).
This completes the preliminaries. We make the remark that in what follows we assume that the ground model is of form K [U~ max ], but could have taken it in the form L[U~ ] because for suitable U~ we have L[U~ ] V = K [U~ max ].
3 Classifying measures
In this section we will take the core model K [U~ max ] discussed in the last section, in the case when there is a largest measurable on U~ max , and force over it with an iteration P as in lemma 6. We will then classify completely the measures on in K [G], and will describe the Mitchell ordering on these measures. For~the rest of this section let V = K , and suppose that there is maximal with oUmax () > 0. Fix G which is P-generic over K , where P is the Reverse Easton iteration in which a Cohen subset is added to each inaccessible , as computed in K . As in lemma 6 we may factor P as P Add(; 1), and correspondingly we may factor G as G g.
Lemma 12: Let U be a measure on in the model K [G]. Let i : K [G] ! N
be the ultrapower of K [G] by U . Let j : K ! K = i(K ) be the restriction of i to K . Then 1. i(G) = G g H , where g is Add(; 1)-generic over K [G] and H is j (P)=G g -generic over K [G][g ]. 1
1
1
1
11
2. If G = G g then K [G ] = K [G]. 3. N = K [i(G)]. 4. j \G i(G). 1
1
1
5. j : K ! K is a nite normal iteration of K by U~ max , with the rst step being an ultrapower map with critical point .
Proof:
By elementarity N = K [i(G)], where K is K [U~ max ] as computed in the sense of N . i(G) is generic over K for i(P), which equals j (P) since P 2 K . j : K ! K must be a normal iteration with rst step an ultrapower by a measure on , because K is still K [U~ max ] in K [G]. In particular K and K agree to rank + 1. i(G) = G g H , where g is generic for Add(; 1) as computed in K [G] and H is generic for j (P)=G g . K [G] and K [G] agree to rank + 1, so g is actually K [G] generic for Add(; 1). Also K [G ] and K [G ] agree to rank + 1. As N is an ultrapower, K [G] N N . As H is generic for highly closed forcing, K [G] K [G ] K [G ]. In particular g 2 K [G ], so that by the last paragraph g 2 K [G ]. Hence K [G] = K [G ]. If j is not a nite iteration, then lemma 11 implies that there is an !sequence of ordinals ~ 2 K [G] such that ~ 2= K . But P is ! -closed, and so ~ 2= K [G], in contradiction to what we just proved about the closure of K [G]. 1
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De nition 5: U 2 K [G] is an n-step extension of U~ max (; ) if, when we de ne j as in the last lemma, j has length n + 1 and the rst step in j is the application of U~ max (; ) to K .
Notice that this is reasonable terminology, as when U is an n-step extension of U~ max (; ) we certainly have U~ max (; ) U . The one-step extensions are the easiest ones to understand. 12
Lemma 13: Let < oU~ max (), and let j : K ! M be the ultrapower of K by U~ max (; ). Then in K [G] the set of H = g H such that (setting G = G g ) G is P-generic over K . K [G] = K [G ]. H is j (P )=G -generic over M [G ]. j \G G H . 1
1
1
1
1
1
1
1
1
has cardinality , and each one gives rise to a distinct one-step extension UH of U~ max (; ). ++
1
Proof: There are generics g such that K [G] = K [G][g ]. Fix one such, +
1
1
and observe that by lemma 5 M [G] = M [G ]. By lemma 6 we may build many appropriate generics H , and by cardinality considerations there can be at most many. Let H be one such, and consider the unique map | : K [G] ! M [G ][H ] such that | extends j and | (G) = G H . By lemma 4, M [G ][H ] = f | (F )() j F 2 K [G] g; so lemma 1 tells us that | is the ultrapower of K [G] by the measure UH = f X j 2 | (X ) g: Distinct generics H give distinct one-step extensions, because given UH we may recover H by computing jUKHG (G) = G H . 1
++
++
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[
]
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This last lemma gives a complete description of the one-step extensions of measures U~ max (; ). We need to do a bit more work to produce n-step extensions; the point will be to guarantee that each critical point we use can be de ned from in a certain way. 13
Lemma 14: Let j : K ! K be a normal iteration of K by U~ max of length
n + 1, with j i (U~ max )(i; i) being applied at stage i in the iteration, and = . Then in K [G] there are many H = g H such that (setting G = G g ) G is P-generic over K . K [G] = K [G ]. H is j (P )=G -generic over K [G ]. j \G G H . If 0
++
0
1
1
1
1
1
1
1
1
1
| : K [G] ! K [G][H ] 1
is the unique extension of j with |(G) = G H , then 1
K [G][H ] = f |(F )() j F 2 K [G] g: 1
Proof: As before there are appropriate g , and we will x one. Then we that K [G] = K [G1].
+
1
know We will de ne a \master condition" for j (P )=G , much as in lemma 6. As there the condition q will have value ; at M -inaccessible with < < j (), but q(j ()) will be slightly bigger than in lemma 6. De ne q(j ()) by dom(q(j ())) = + n. q(j ()) = g. q(j () + i) = i, for i < n. Just as in lemma 6 we may build many H with q as a member, and argue that H is generic and that j \G G H . It will suce to show that for every i < n the ordinal i has the form |(F )(), as lemma 7 then shows that every element of K [G][H ] may be written in this form. Now x i < n, and de ne a function F in K [G] by F ( ) = g ( + i): 1
++
1
1
14
We have |(F )() = |(g)( + i) = H (j ())( + i) = q()( + i) = i ; so the lemma is proved.
This result classi es the n-step extensions of measures on in K . It remains to determine when the relation holds between two such extension measures. As one might expect, the situation is simplest when considering one-step extensions.
Lemma 15: Let U , V be two measures on in K [G]. Suppose further that U is a 1-step extension of U = U~ max (; ), using some generic HU = gU HU , 1
0
and that V is a 1-step extension of V = U~ max (; ) using some generic HV = gV HV . Set GU = G gU , GV = G gV . Then K [G] U V if and only if < . HU 2 Ult(K; V )[G]. 0
1
1
0
Proof: Let M = Ult(K; U ), let N = Ult(K; V ). First suppose that K [G] U V . This means that U 2 Ult(K [G]; V ) = N [GV ][HV ]: 0
0
As K [G] = K [GV ] we know that N [G] = N [GV ]. HV is generic for highly closed forcing, so this will imply that U 2 N [G]. Since K [G] N [G] N [G], K [G] and N [G] agree to rank + 1, so that there is agreement between jUK G and jUN G to that rank. In particular [
]
[
]
GU HU = jUK G (G) = jUN G (G); [
]
[
]
so that HU 2 N [G]. To show that < , observe that N K K [G]. Also 1
jUK G N [G] = jUN G ; [
]
[
]
15
so that the restriction of jUK G to N is an embedding de nable in N [G], from N to some well-founded model. It must therefore be a normal iteration of N , since N is the core model of N [G]. But jUK G K = jUK , so that jUK G N = jUK N . It is easy to see that the rst step in the iteration of N induced by this restriction is to take the ultrapower by [
]
[
[
]
0
]
0
U = f X j X 2 N; 2 jUK (X ) g; 0
0
so that U 2 N . Hence U V , and < . For the other direction, suppose that HU 2 N [G] and < , that is K U V and so U 2 N . We will show that N [G] can reconstruct U from HU . K [G] and N [G] (which equals N [GV ]) agree to rank + 1, and jUK N = jUN , what is more N contains all canonical P -names for subsets of . So if _ is such a name then N [G] can compute 0
0
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jUK G ( _ G ) = jUK ( _ )GU HU = jUN ( _ )GU HU ; [
]
0
0
and hence N [G] can compute U , so
U 2 N [GV ] N [GV ][HV ] = Ult(K [G]; V ): Hence K [G] U V and we are done.
At this point we are almost ready to describe the ordering of onestep extensions. What we still need is some idea of how many generics on jUV (P )=G are constructed by models of the form Ult(K; V )[G] as V runs through the measures on with U V . The next lemma will provide us with this information. Lemma 16: Let < < < oU~ max (). Let us de ne U = U~ max (; ), V = U~ max (; ), and nally W = U~ max (; ). Then the Ult(K; U )[G]generics on jUK (P )=G constructed in the model Ult(K; V )[G] form a proper subset of those constructed in the model Ult(K; W )[G], and the same is true if we restrict to those generics H such that jU \G G H . 0
0
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0
16
0
Proof: Let MU = Ult(K; U ) and de ne MV , MW similarly. K and MW agree
to rank + 1, so that MV and N = Ult(MW ; V ) agree to rank jV () + 1. As P is relatively small, MV [G] and N [G] also agree to this level, which is much greater than jU (). So MV [G] and N [G] construct the same generics H for the forcing jU (P )=G. But now by the same arguments as in lemma 6, MW [G] believes that it can construct many generics, but that the inner model N [G] can only build many. This proves the lemma. ++
+
We use this to get a picture of the ordering on one-step extensions in the ~ max U case when o () = 3. This is fairly representative of the general case.
Lemma 17: Let oU~ max () = 3, with U = U~ max (; 0), V = U~ max (; 1),
W = U~ max (; 2). Let MU , MV , MW denote the ultrapowers of K by these measures. Work in K [G]. Then we may divide the one-step extensions of these measures into classes C : extensions of U via generics in MV [G]. jC j = C : extensions of U via generics in MW [G] n MV [G]. jC j = . C : extensions of U via generics in K [G] n MW [G]. jC j = . C : extensions of V via generics in MW [G]. jC j = . C : extensions of V via generics in K [G] n MW [G]. jC j = . C : extensions of W . jC j = . A measure from Ci is below a measure from Cj in the Mitchell ordering if and only if i = 0 and j 2 f3; 4; 5g OR i = 1 and j = 5 OR i = 3 and j = 5. 0
0
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++
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The proof is immediate. We give a picture which may make the shape of the partial ordering clearer.
C
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6AKA
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A A A A A A
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A A A A
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If instead of o() = 3 we take o() = !, we get an in nite partial ordering P with an interesting universal property; if Q is the four-element poset
then P does not embed Q, and P embeds every nite poset which does not embed Q. This was pointed out to me by Andrew Jergens [2]. Baldwin speculated that the methods of [6] might extend to all wellfounded posets which embed neither Q nor the poset R given by
We observe that P does embed R. 18
Now we consider the general case of the Mitchell ordering between n-step extensions. This problem is not quite as hard as one might expect, largely because the question whether U V is controlled by the rst step in the iteration associated with V .
Theorem 1: Let U be an m + 1-step extension of U , via a generic object ~ |) of length m + 1, with the ultra lter HU = gU HU and an iteration (M;~ 0
1
j i (U~ max )(i; i) being applied to Mi at stage i. Let V be an n + 1-step ~ ~k) of length extension of V , via a generic HV = gV HV and an iteration (N; n + 1, with the ultra lter k i(U~ max )(i; i) being applied to Ni at stage i. Then K [G] U V if and only if HU 2 Ult(K; V )[G]. j m Ult(K; V ) is a nite normal iteration of Ult(K; V ) by k (U~ max ). 0
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01
Proof: Notice that N = Ult(K; V ). As before we let GU = G gU and GV = G gV . 1
0
Suppose that K [G] U V . Then U 2 Ult(K [G]; V ) = Nn[GV ][HV ]; so as in lemma 15 U 2 Nn[G]. N and Nn agree to rank + 1, so by an easy chain condition argument the models N [G] and Nn[G] also agree to this rank, hence U 2 N [G]. As in lemma 15 N [G] can reconstruct HU , so that HU 2 N [G]. For the second part just observe that jUK G N [G] = jUN G , so that jUK G N must give rise to a normal iteration of N by its version of U~ max , which is k (U~ max ). But N K and jUK G K = j m , so this amounts to saying that j m N is a normal iteration of N by k (U~ max ). This iteration must be nite, as usual, because otherwise the rst ! critical points will give a sequence which is in N [G] but not in Ult(N [G]; U ). 1
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Suppose that HU 2 N [G], and that j m N can be written as an iter1
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0
1
ation (N~ ;~|) of length s +1, so that Ns = j m (N ) and j m N = |s. We will show that N [G] can compute U ; the proof is precisely parallel to that in lemma 15. K [G] and N [G] agree to rank + 1, K and N agree on the set of canonical names for subsets of . If _ is such a name then (since |s is a class in N )) N [G] can compute 0
1
0
1
0
1
1
1
0
1
1
jUK G ( _ G ) = j m ( _ )GU HU = |s( _ )GU HU : [
]
0
0
Just as in lemma 15 this gives U 2 N [G], and by the same arguments as we used in the rst part of the proof this implies that U 2 Nn [G], hence that K [G] U V . 1
Our next task is to explore the circumstances under which an iterated ultrapower of K restricted to a one-step ultrapower N gives rise to a map which is an iterated ultrapower of N . The following lemma resolves the question about the restriction of a nite iteration to a one-step ultrapower model.
Lemma 18: Let M be a model of ZFC, and assume
M U~ is a coherent sequence. Let be the largest critical point on U~ . Let j be a nite normal iteration of M , in which a measure Um = j m (U~ )(m; m) is applied to Mm to get jmm : Mm ! Mm for each m < n. Let = , = . Let N = Ult(M; U~ (; )) for some , and suppose that i = j N : N ! j (N ) is a nite normal iteration of N . Then 1. For each m < n, Um 2 Nm . 0
+1
0
+1
0
20
2. i has length n, and step m in the iteration i is the application of Um to Nm . 3. The diagram
M
j01
0
-
M
1
k0
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
k1
?
i01
N
0
-
kn
?
N
1
Mn
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
?
Nn
commutes, where ki : Mi ! Ni is the ultrapower map arising from the measure j i (U~ (; )). 0
Proof: M can recover U by computing U = f X 2 P \ M j 2 j (X ) g: 0
0
0
0
We can then build a commutative triangle
M
j
0
@
@ @
-
Mn
@
j01 @
@
j1n @ @ R @
M
1
Since P \ M = P \ N and i = j N we have U = f X 2 P \ N j 2 i(X ) g; and a commutative triangle 0
0
0
0
21
i
N
0
@
@ @
-
Nn
@
i01 @
@
i1n @ @ R @
N
1
So U 2 N . We make the easy observation that < , because N is the ultrapower of M by the measure U~ (; ) on the coherent sequence U~ . Applying lemma 3 the square 0
0
0
0
M
j01
0
-
M
k0
k1
?
N
1
i01
0
-
?
N
1
commutes. We now attempt to argue that i n and j n resemble each other. Let = j (). 1
1
1
01
Claim 1: In the situation described above 1. = i (). 2. VM = VN . 1
01
1 1 +1
1 1 +1
3. j n VM = i n VN . 1 1 +1
1
1 1 +1
1
Proof: M and N agree to rank + 1, so by standard arguments 0
0
i VN = j VM 01
0
+1
01
0
+1
22
and
VM = i (VM ) = j (VN ) = VN : The key point is that both models compute the same set of functions from to V . If x 2 VM then x = j (F )() = i (F )() for some such function, and so j n(x) = j (F )() = i(F )() = i n(x) by the normality of the iterations. 1 1 +1
0
01
1 1 +1
0
01
+1
+1
+1
1 1 +1
01
01
1
1
Since is the largest measurable on U~ , is the largest on j (U~ ) and hence . We know that = crit(j n), so also = crit(i n). What is more U = f X 2 P \ M j 2 j n (X ) g = f X 2 P \ N j 2 i n (X ) g: Hence U is in N and i is the ultrapower of N by U . At this point we observe that since N is the ultrapower of M by the measure j (U~ (; )), there is a certain agreement between the measure sequences in these models: namely these sequences agree below , and at the model N has the same measures as M up to the point j ( ). As a consequence we see that either < or = and < j ( ). By lemma 3, we see that the diagram 1
1
1
1
1
1
1
01
1
1
1
1
1
1
1
1
1
1
1
12
1
1
1
1
01
1
1
1
1
M
1
j12
-
M
k1 ?
N
1
k2 i12
-
commutes. 23
?
N
01
1
2
2
1
1
1
1
01
To nish the proof we just repeat these arguments, showing step by step that the diagrams commute and the models Nn construct the measures Un.
The following corollary can be derived by a close inspection of the proof of the preceding lemma.
Corollary 1: Given an iteration j of M and a model N as described above, it is necessary and sucient for j N to be an iteration of N that for all m < n either m < j m () or m = j m () and m < j m ( ). 0
0
0
We observe that as a consequence, if j n induces an internal iteration of Ult(K; U~ max (; )), then it induces such an iteration of Ult(K; U~ max (; )) for any > . We can nally undertake the general analysis of the ordering between n-step extensions in K [G]. 0
De nition 6: Let < o(), and let 2 ( ; o()) [ f1g. For 2 ( ; o()) let M ( ; ) be the set of extensions U of U~ max (; ) such that is the least with the following two properties: 1. The constructing generic HU is in Ult(K; U~ max (; ))[G]. 2. jUK G induces an internal iteration of Ult(K; U~ max (; )). [
]
For = 1 let M ( ; ) be the set of those U such that no as described above exists. The description of the ordering is given by the following result, whose proof follows immediately from the work above.
Theorem 2: Every measure on in K [G] is in a unique M ( ; ). M ( ; ) has cardinality if 2 ( ; o()) and cardinality if = 1. If U 2 M ( ; ) and V 2 M ( ; ), then U V if and only if . +
++
24
References [1] J. Cummings, Possible behaviours for the Mitchell ordering II. In preparation. [2] A. Jergens, Private communication. [3] W. Mitchell, Sets constructible from sequences of ultra lters. Journal of Symbolic Logic 39 (1974) 57{66. [4] W. Mitchell, The core model for sequences of measures. I. Math. Proc. Camb. Phil. Soc. 95 (1984) 229{260. [5] W. Mitchell, Indiscernibles, skies and ideals. In Contemporary Mathematics 31 (1984) 161{182. [6] S. Baldwin, The -ordering on normal ultra lters. Journal of Symbolic Logic 51 (1985) 936{952.
25
We use a mixture of forcing and inner models techniques to get some results on the possible behaviours of the Mitchell ordering at a measurable .
1 Introduction The Mitchell ordering on normal measures was invented by Mitchell [3] as a tool in his study of inner models for large cardinals.
De nition 1: Let be measurable, let U and U be normal measures on . Then U U if and only if U 2 Ult(V; U ), the ultrapower of V by U . 0
0
1
0
1
1
The following facts are standard.
is transitive. is well-founded. is strict. An ultra lter has at most 2 ancestors in the ordering . 1
1
De nition 2: o() is the height of the well-founded relation . Notice that we must have o() (2) . Much is known about the possible behaviours of . For example Mitchell has shown [3] that in a highly structured inner model we can have GCH holding and o() = , with being a linear ordering. Baldwin has shown [6] that from suitable hypotheses we can have models in which is a given prewellordering of cardinality less than . If is the critical point of j : V ! M such that V M , then we may show that every element of V is in Ult(V; U ) for some U on . In particular any 2 measures on will have an upper bound in the ordering . What is more, for any particular U there will only be 2 elements of V in Ult(V; U ), so that there must be 2 measures on . If it happens that 2 > (2) then cannot be linear, and it is not clear what the structure of will be. This question is addressed in [1]. In this paper we will produce a model in which is measurable, and all measures on may be divided into \blocks" in the following way: 1. For each < o() and 2 ( ; o()) [ 1 there is a block M ( ; ). 2. All the measures in M ( ; ) have height in the Mitchell ordering. 3. M ( ; ) has cardinality if 2 ( ; o()), and cardinality if = 1. 4. For U 2 M ( ; ) and V 2 M ( ; ), U V i (with the convention that 1 is bigger than any ordinal). +
++
+2
+2
+2
2
2
+
+
++
2 Preliminaries In this paper we will use large cardinals and forcing to produce some models where the Mitchell ordering is rather complex. In the interests of clarity and self-containedness we have collected various key facts in this section, facts 2
which we will use repeatedly in the sequel. None of them are due to us; in many cases we are unsure to whom they should be attributed. We start with a remark about Cohen forcing. The forcing for adding a single Cohen subset to a regular cardinal can be regarded as having conditions which are functions p : ! for < (rather than the more standard functions from < to f0; 1g). In this form we can consider the forcing as adding a generic function from to . We will be interested in elementary embeddings k : M ! N between inner models of ZFC. In general it will not be the case that k is aSclass of M or that N M (notice that the former implies the latter, as N = k(V M )). If a model M believes that U (with U 2 M ) is a measure on , we will denote the natural embedding from M into Ult(M; U ) by jUM .
Lemma 1: Let j : V ! M be an elementary embedding with j a class of V , = crit(j ), such that every element of M is j (F )() for some function F 2 V . Then j is the ultrapower by the normal measure U = f X j 2 j (X ) g. Proof: Factor j through the ultrapower of V by U ,
j0
V
3
M
0
Q
Q
Q Q
Q k Q
Q
Q Q
j
QQ s -
M
by de ning k : [f ] 7 ! j (f )(). k is a surjection, and M is the transitive collapse of the range of k, so M = M and j = j . 0
0
0
Lemma 1 will prove useful in identifying certain embeddings as ultrapowers.
Lemma 2: Let M and N be inner models of ZFC such that 3
M N. N M M . M U is a normal measure on .
Then U is a normal measure in N and jUN M = jUM . Proof: It follows immediately from the closure of M that U is a normal measure in N . Let x 2 M . jUN (x) is the transitive collapse of the structure (F; EU ) where F = f f : ! x j f 2 N g; and fEU g () f j f ( ) 2 g( ) g 2 U: By the closure of M inside N we have F = f f : ! x j f 2 M g; which is the set of functions whose collapse is jUM (x), so by the absoluteness of the collapsing construction jUN (x) = jUM (x). Lemma 2 will be useful in understanding restrictions of ultrapower maps, as for example in the proof of the following lemma. Lemma 3: Let U be a measure on , W a measure on and suppose that W 2 Ult(V; U ). Let MU be the ultrapower of V by U , MW the ultrapower of V by W . Then Ult(MU ; W ) = Ult(MW ; jWV (U )) and the following diagram commutes. jUV
V
-
MU MU jW
V jW ?
MW
jjMVW(U )
-
?
Ult(MU ; W )
W
4
Proof: Let x 2 V . jjMWV WU (jWV (x)) = jWV (jUV (x)); (
)
by elementarity. W 2 MU and (as ) MU MU , so that jWV MU = jWMU : In particular jWV (jUV (x)) = jWMU (jUV (x)): From this we can deduce that the two ultrapowers are equal (let x = V ), and that the diagram commutes. We will use lemma 3 to analyse restrictions of iterated ultrapowers.
Lemma 4: Let k : M ! N be an elementary embedding between inner models of ZFC. Let P 2 M be a forcing notion, let G be P-generic over M
and let H be k(P)-generic over N . Suppose that p 2 G =) k(p) 2 H: Then 1. There is a unique extension of k to a map k : M [G] ! N [H ] such that k : G 7 ! H . 2. If is a set of ordinals such that
N = f k(F )(a) j F 2 M; a 2 []
N [H ] = f k(F )(a) j F 2 M [G]; a 2 []
5
Proof: For the rst claim, it is clear that if k exists it must be given by k : _ G 7 ! k( _ )H ; where _ G denotes the interpretation of the term by the generic G. We check that this is well-de ned. Let _ G = _ G, then there is p 2 G such that p MP _ = _ . By elementarity k(p) Nk P k( _ ) = k(_ ). By assumption k(p) 2 H , so that k( _ )H = k(_ )H . The proof that k is elementary is entirely similar. For the second claim, let _ H 2 N [H ]. Then _ = k(F )(a) for some F 2 M and a 2 []
1
(
1
)
Lemma 4 will be used to take elementary embeddings (usually nitely iterated ultrapowers) and extend them onto certain generic extensions of V . The second claim will play a key r^ole in understanding the nature of the extended embedding. The next lemmas goes into more detail about the extensions that we will make. We start with a technical result about equivalence between generics.
Lemma 5: Let P be the forcing notion given by a Reverse Easton iteration
of length + 1, in which one Cohen subset of is added at each strong inaccessible . Let G and G be P-generics over V , with the property that V [G ] = V [G ]. Then for any model V agreeing with V to rank + 1, G and G are P-generic over V and V [G ] = V [G ]. 1
1
1
2
2
2
1
2
Proof: By the agreement P 2 V and (since jPj = ) both models compute
the same maximal antichains, so G and G are generic over V for P. G is the interpretation under G of some term _ , and by the agreement again we may take it that _ 2 V . So G 2 V [G ] and vice versa, so that V [G ] = V [G ]. 1
2
1
2
1
1
2
2
Next we give the lemma that will be used to generate measures. 6
Lemma 6: Let GCH hold, and let j : V ! M be an embedding which is a class in V , such that = crit(j ) and M M . Suppose also that the
ordinal j ( ) has cardinality in V . Let P be as in lemma 5, and observe that P can be factored as P followed by Add(; 1) as computed by V P. Let G = G g be P-generic, and suppose that there is G = G g with V [G] = V [G ]. Then in V [G] there are many H such that G H is j (P)-generic over M and j extends to | : V [G] ! M [G ][H ]. +
+
1
1
++
1
1
1
Proof: By lemma 5 M [G] = M [G ]. In M [G ] the factor iteration j (P)=G 1
1
1
is highly-closed and has j ( ) many antichains. As P has the -chain condition and M [G] = M [G ] we have V [G] M [G ] M [G ]. Hence in V [G] the forcing j (P)=G is -closed, and the set of its maximal antichains which lie in M [G ] has cardinality . We wish to build generics which are compatible with G. Working in M [G ], de ne a function q with domain the M -inaccessibles such that < j (), by q() = ; for < j () and q(j ()) = g. q is a condition in j (P)=G . We build in V [G] a binary tree of height such that The top node is q. Any path is a descending sequence in j (P)=G , meeting each antichain in M [G ]. Every element has incompatible immediate successors. The construction proceeds for the requisite steps, because j (P)=G is -closed in V [G]. This construction will give us distinct generic lters H , each with the property that j \G G H . We can use these to build extensions | of j such that |(G) = G H . +
1
1
1
+
1
+
1
+
1
+
1
1
1
+
+
++
1
1
1
This last construction was a \master condition" argument a la Silver; notice that any extension of q in j (P)=G would have done equally well as the top node of the tree. We will make heavy use of Mitchell's theory of core models for sequences of measures; nowadays this should be seen as a special case of the core model 7
theory for non-overlapping extenders (due to Mitchell, Dodd, Jensen and Koepke) in which every extender happens to be equivalent to a measure. The reader is referred to Mitchell's paper [4] for proofs.
De nition 3: U~ is a coherent sequence of measures if and only if U~ is a function, with dom(U~ ) On On. For some function oU~ : On ! On, dom(U~ ) = f (; ) j 0 < oU~ () g:
If (; ) 2 dom(U~ ) then U~ (; ) is a normal measure on . If (; ) 2 dom(U~ ), and j : V ! M is the ultrapower of V by the measure U~ (; ) then
{ For all , ( ; ) 2 dom(j (U~ )) if and only if or = and < . { If and ( ; ) 2 dom(j (U~ )) then j (U~ )( ; ) = U~ ( ; ):
De nition 4: Let M be an inner model of ZFC, let
M U~ is a coherent sequence of measures: A normal iteration of M by U~ , of length is a pair (hM : < i; hj : < i) where M = M. M is an inner model of ZFC for each < . For < , j : M ! M is an elementary embedding. j = id, and for , j = j j . 0
8
M = Ult(M ; j (U~ )( ; )), and j : M ! M is the +1
0
+1
+1
associated ultrapower map, if + 1 < . If < , is limit, then M and j are had by taking a direct limit in the natural way. The sequence h : + 1 < i is strictly increasing. The following structural fact is easy, by induction on < .
~ | ) is a normal iteration of M by U~ in length then for Lemma 7: If (M;~
every < M = f j (F )(a) j F 2 M; a 2 []
We will denote by K Mitchell's core model K [U~ max ], which exists under the assumption that there is no inner model in which 9 o() = . We will use the following facts about K (see section 2 of [5]). ++
Lemma 8 (Mitchell): Suppose that :9 o() = in any inner model ++
of ZFC. Then K is a uniformly de nable inner model of ZFC+GCH. K V = K. K U~ max is a coherent sequence of measures. K is invariant under set forcing. If i : K ! M is an elementary embedding into an inner model M then i arises from a normal iteration of K by U~ max .
It is worth making the following easy observations about K and U~ max . Lemma 9: If K , U~ max are as above then All measures in K appear on the sequence U~ max . 9
If < < oU~ max () then U~ max (; ) 6= U~ max (; ). K U~ max (; ) U~ max (; ) i < . We will be particularly interested in nite normal iterations of K , in the case when there is a largest measurable on U~ max .
Lemma 10: Suppose that is the largest ordinal with oU~ max () > 0. Let
~ | ) be a normal iteration of K by U~ max of length n + 1, n + 1 < !, let (M;~ with j the ultrapower of K by U~ max (; ) for some . Then 1. Mn K , and K Mn Mn. 2. For each i < n, i < j n (). 3. In M , the ordinal j n( ) has cardinality 01
0
0
0
+
+
Proof: 1. The critical points are increasing and each model is closed inside the previous one. 2. i j i (), as is the largest measurable on U~ max . If i < j i () then we are done as j n () = jin(j i ()) j i (); if i = j i () then this is the critical point of jin so i < jin(j i ()) = j n (). 3. The ordinals less than j n ( ) all have the form 0
0
0
0
0
0
0
0
0
+
j n(F )( ; : : : ; n ); 0
0
1
where F : []n ! . By GCH there are such functions F . +
+
The next result puts some limits on the possible closure of the models in a normal iteration of in nite length.
10
~ | ) is a normal iteration of M by U~ , of length !, Lemma 11: If (M;~ then the sequence of ordinals ~ = hn : n < !i is not a member of M for ! < . Proof: The model M agrees with M! to rank ! + 1, so it is enough to show that ~ 2= M! . M! was constructed as a direct limit, so if ~ 2 M! then ~ = jn! (~) for some ~ 2 Mn; in particular n = jn! (n). But crit(jn! ) = n as we are in a normal iteration, so that n 2= rge(jn! ).
This completes the preliminaries. We make the remark that in what follows we assume that the ground model is of form K [U~ max ], but could have taken it in the form L[U~ ] because for suitable U~ we have L[U~ ] V = K [U~ max ].
3 Classifying measures
In this section we will take the core model K [U~ max ] discussed in the last section, in the case when there is a largest measurable on U~ max , and force over it with an iteration P as in lemma 6. We will then classify completely the measures on in K [G], and will describe the Mitchell ordering on these measures. For~the rest of this section let V = K , and suppose that there is maximal with oUmax () > 0. Fix G which is P-generic over K , where P is the Reverse Easton iteration in which a Cohen subset is added to each inaccessible , as computed in K . As in lemma 6 we may factor P as P Add(; 1), and correspondingly we may factor G as G g.
Lemma 12: Let U be a measure on in the model K [G]. Let i : K [G] ! N
be the ultrapower of K [G] by U . Let j : K ! K = i(K ) be the restriction of i to K . Then 1. i(G) = G g H , where g is Add(; 1)-generic over K [G] and H is j (P)=G g -generic over K [G][g ]. 1
1
1
1
11
2. If G = G g then K [G ] = K [G]. 3. N = K [i(G)]. 4. j \G i(G). 1
1
1
5. j : K ! K is a nite normal iteration of K by U~ max , with the rst step being an ultrapower map with critical point .
Proof:
By elementarity N = K [i(G)], where K is K [U~ max ] as computed in the sense of N . i(G) is generic over K for i(P), which equals j (P) since P 2 K . j : K ! K must be a normal iteration with rst step an ultrapower by a measure on , because K is still K [U~ max ] in K [G]. In particular K and K agree to rank + 1. i(G) = G g H , where g is generic for Add(; 1) as computed in K [G] and H is generic for j (P)=G g . K [G] and K [G] agree to rank + 1, so g is actually K [G] generic for Add(; 1). Also K [G ] and K [G ] agree to rank + 1. As N is an ultrapower, K [G] N N . As H is generic for highly closed forcing, K [G] K [G ] K [G ]. In particular g 2 K [G ], so that by the last paragraph g 2 K [G ]. Hence K [G] = K [G ]. If j is not a nite iteration, then lemma 11 implies that there is an !sequence of ordinals ~ 2 K [G] such that ~ 2= K . But P is ! -closed, and so ~ 2= K [G], in contradiction to what we just proved about the closure of K [G]. 1
1
1
1
1
1
1
1
1
1
1
1
De nition 5: U 2 K [G] is an n-step extension of U~ max (; ) if, when we de ne j as in the last lemma, j has length n + 1 and the rst step in j is the application of U~ max (; ) to K .
Notice that this is reasonable terminology, as when U is an n-step extension of U~ max (; ) we certainly have U~ max (; ) U . The one-step extensions are the easiest ones to understand. 12
Lemma 13: Let < oU~ max (), and let j : K ! M be the ultrapower of K by U~ max (; ). Then in K [G] the set of H = g H such that (setting G = G g ) G is P-generic over K . K [G] = K [G ]. H is j (P )=G -generic over M [G ]. j \G G H . 1
1
1
1
1
1
1
1
1
has cardinality , and each one gives rise to a distinct one-step extension UH of U~ max (; ). ++
1
Proof: There are generics g such that K [G] = K [G][g ]. Fix one such, +
1
1
and observe that by lemma 5 M [G] = M [G ]. By lemma 6 we may build many appropriate generics H , and by cardinality considerations there can be at most many. Let H be one such, and consider the unique map | : K [G] ! M [G ][H ] such that | extends j and | (G) = G H . By lemma 4, M [G ][H ] = f | (F )() j F 2 K [G] g; so lemma 1 tells us that | is the ultrapower of K [G] by the measure UH = f X j 2 | (X ) g: Distinct generics H give distinct one-step extensions, because given UH we may recover H by computing jUKHG (G) = G H . 1
++
++
1
1
1
1
1
[
]
1
1
1
This last lemma gives a complete description of the one-step extensions of measures U~ max (; ). We need to do a bit more work to produce n-step extensions; the point will be to guarantee that each critical point we use can be de ned from in a certain way. 13
Lemma 14: Let j : K ! K be a normal iteration of K by U~ max of length
n + 1, with j i (U~ max )(i; i) being applied at stage i in the iteration, and = . Then in K [G] there are many H = g H such that (setting G = G g ) G is P-generic over K . K [G] = K [G ]. H is j (P )=G -generic over K [G ]. j \G G H . If 0
++
0
1
1
1
1
1
1
1
1
1
| : K [G] ! K [G][H ] 1
is the unique extension of j with |(G) = G H , then 1
K [G][H ] = f |(F )() j F 2 K [G] g: 1
Proof: As before there are appropriate g , and we will x one. Then we that K [G] = K [G1].
+
1
know We will de ne a \master condition" for j (P )=G , much as in lemma 6. As there the condition q will have value ; at M -inaccessible with < < j (), but q(j ()) will be slightly bigger than in lemma 6. De ne q(j ()) by dom(q(j ())) = + n. q(j ()) = g. q(j () + i) = i, for i < n. Just as in lemma 6 we may build many H with q as a member, and argue that H is generic and that j \G G H . It will suce to show that for every i < n the ordinal i has the form |(F )(), as lemma 7 then shows that every element of K [G][H ] may be written in this form. Now x i < n, and de ne a function F in K [G] by F ( ) = g ( + i): 1
++
1
1
14
We have |(F )() = |(g)( + i) = H (j ())( + i) = q()( + i) = i ; so the lemma is proved.
This result classi es the n-step extensions of measures on in K . It remains to determine when the relation holds between two such extension measures. As one might expect, the situation is simplest when considering one-step extensions.
Lemma 15: Let U , V be two measures on in K [G]. Suppose further that U is a 1-step extension of U = U~ max (; ), using some generic HU = gU HU , 1
0
and that V is a 1-step extension of V = U~ max (; ) using some generic HV = gV HV . Set GU = G gU , GV = G gV . Then K [G] U V if and only if < . HU 2 Ult(K; V )[G]. 0
1
1
0
Proof: Let M = Ult(K; U ), let N = Ult(K; V ). First suppose that K [G] U V . This means that U 2 Ult(K [G]; V ) = N [GV ][HV ]: 0
0
As K [G] = K [GV ] we know that N [G] = N [GV ]. HV is generic for highly closed forcing, so this will imply that U 2 N [G]. Since K [G] N [G] N [G], K [G] and N [G] agree to rank + 1, so that there is agreement between jUK G and jUN G to that rank. In particular [
]
[
]
GU HU = jUK G (G) = jUN G (G); [
]
[
]
so that HU 2 N [G]. To show that < , observe that N K K [G]. Also 1
jUK G N [G] = jUN G ; [
]
[
]
15
so that the restriction of jUK G to N is an embedding de nable in N [G], from N to some well-founded model. It must therefore be a normal iteration of N , since N is the core model of N [G]. But jUK G K = jUK , so that jUK G N = jUK N . It is easy to see that the rst step in the iteration of N induced by this restriction is to take the ultrapower by [
]
[
[
]
0
]
0
U = f X j X 2 N; 2 jUK (X ) g; 0
0
so that U 2 N . Hence U V , and < . For the other direction, suppose that HU 2 N [G] and < , that is K U V and so U 2 N . We will show that N [G] can reconstruct U from HU . K [G] and N [G] (which equals N [GV ]) agree to rank + 1, and jUK N = jUN , what is more N contains all canonical P -names for subsets of . So if _ is such a name then N [G] can compute 0
0
0
1
0
0
0
1
0
0
jUK G ( _ G ) = jUK ( _ )GU HU = jUN ( _ )GU HU ; [
]
0
0
and hence N [G] can compute U , so
U 2 N [GV ] N [GV ][HV ] = Ult(K [G]; V ): Hence K [G] U V and we are done.
At this point we are almost ready to describe the ordering of onestep extensions. What we still need is some idea of how many generics on jUV (P )=G are constructed by models of the form Ult(K; V )[G] as V runs through the measures on with U V . The next lemma will provide us with this information. Lemma 16: Let < < < oU~ max (). Let us de ne U = U~ max (; ), V = U~ max (; ), and nally W = U~ max (; ). Then the Ult(K; U )[G]generics on jUK (P )=G constructed in the model Ult(K; V )[G] form a proper subset of those constructed in the model Ult(K; W )[G], and the same is true if we restrict to those generics H such that jU \G G H . 0
0
0
0
16
0
Proof: Let MU = Ult(K; U ) and de ne MV , MW similarly. K and MW agree
to rank + 1, so that MV and N = Ult(MW ; V ) agree to rank jV () + 1. As P is relatively small, MV [G] and N [G] also agree to this level, which is much greater than jU (). So MV [G] and N [G] construct the same generics H for the forcing jU (P )=G. But now by the same arguments as in lemma 6, MW [G] believes that it can construct many generics, but that the inner model N [G] can only build many. This proves the lemma. ++
+
We use this to get a picture of the ordering on one-step extensions in the ~ max U case when o () = 3. This is fairly representative of the general case.
Lemma 17: Let oU~ max () = 3, with U = U~ max (; 0), V = U~ max (; 1),
W = U~ max (; 2). Let MU , MV , MW denote the ultrapowers of K by these measures. Work in K [G]. Then we may divide the one-step extensions of these measures into classes C : extensions of U via generics in MV [G]. jC j = C : extensions of U via generics in MW [G] n MV [G]. jC j = . C : extensions of U via generics in K [G] n MW [G]. jC j = . C : extensions of V via generics in MW [G]. jC j = . C : extensions of V via generics in K [G] n MW [G]. jC j = . C : extensions of W . jC j = . A measure from Ci is below a measure from Cj in the Mitchell ordering if and only if i = 0 and j 2 f3; 4; 5g OR i = 1 and j = 5 OR i = 3 and j = 5. 0
0
+
1
1
2
2
3
1
4
5
++
17
++
+
4
5
+
++
The proof is immediate. We give a picture which may make the shape of the partial ordering clearer.
C
5
6AKA
C
3
6
C
0
A A A A A A
C
4
A A A A
A
C
1
C
2
If instead of o() = 3 we take o() = !, we get an in nite partial ordering P with an interesting universal property; if Q is the four-element poset
then P does not embed Q, and P embeds every nite poset which does not embed Q. This was pointed out to me by Andrew Jergens [2]. Baldwin speculated that the methods of [6] might extend to all wellfounded posets which embed neither Q nor the poset R given by
We observe that P does embed R. 18
Now we consider the general case of the Mitchell ordering between n-step extensions. This problem is not quite as hard as one might expect, largely because the question whether U V is controlled by the rst step in the iteration associated with V .
Theorem 1: Let U be an m + 1-step extension of U , via a generic object ~ |) of length m + 1, with the ultra lter HU = gU HU and an iteration (M;~ 0
1
j i (U~ max )(i; i) being applied to Mi at stage i. Let V be an n + 1-step ~ ~k) of length extension of V , via a generic HV = gV HV and an iteration (N; n + 1, with the ultra lter k i(U~ max )(i; i) being applied to Ni at stage i. Then K [G] U V if and only if HU 2 Ult(K; V )[G]. j m Ult(K; V ) is a nite normal iteration of Ult(K; V ) by k (U~ max ). 0
1
0
0
0
0
0
0
01
Proof: Notice that N = Ult(K; V ). As before we let GU = G gU and GV = G gV . 1
0
Suppose that K [G] U V . Then U 2 Ult(K [G]; V ) = Nn[GV ][HV ]; so as in lemma 15 U 2 Nn[G]. N and Nn agree to rank + 1, so by an easy chain condition argument the models N [G] and Nn[G] also agree to this rank, hence U 2 N [G]. As in lemma 15 N [G] can reconstruct HU , so that HU 2 N [G]. For the second part just observe that jUK G N [G] = jUN G , so that jUK G N must give rise to a normal iteration of N by its version of U~ max , which is k (U~ max ). But N K and jUK G K = j m , so this amounts to saying that j m N is a normal iteration of N by k (U~ max ). This iteration must be nite, as usual, because otherwise the rst ! critical points will give a sequence which is in N [G] but not in Ult(N [G]; U ). 1
1
1
1
1
1
[
[
]
1
01
1
0
1
]
1
[
]
1
0
1
1
19
]
1
01
1
1
1[
Suppose that HU 2 N [G], and that j m N can be written as an iter1
1
0
1
ation (N~ ;~|) of length s +1, so that Ns = j m (N ) and j m N = |s. We will show that N [G] can compute U ; the proof is precisely parallel to that in lemma 15. K [G] and N [G] agree to rank + 1, K and N agree on the set of canonical names for subsets of . If _ is such a name then (since |s is a class in N )) N [G] can compute 0
1
0
1
0
1
1
1
0
1
1
jUK G ( _ G ) = j m ( _ )GU HU = |s( _ )GU HU : [
]
0
0
Just as in lemma 15 this gives U 2 N [G], and by the same arguments as we used in the rst part of the proof this implies that U 2 Nn [G], hence that K [G] U V . 1
Our next task is to explore the circumstances under which an iterated ultrapower of K restricted to a one-step ultrapower N gives rise to a map which is an iterated ultrapower of N . The following lemma resolves the question about the restriction of a nite iteration to a one-step ultrapower model.
Lemma 18: Let M be a model of ZFC, and assume
M U~ is a coherent sequence. Let be the largest critical point on U~ . Let j be a nite normal iteration of M , in which a measure Um = j m (U~ )(m; m) is applied to Mm to get jmm : Mm ! Mm for each m < n. Let = , = . Let N = Ult(M; U~ (; )) for some , and suppose that i = j N : N ! j (N ) is a nite normal iteration of N . Then 1. For each m < n, Um 2 Nm . 0
+1
0
+1
0
20
2. i has length n, and step m in the iteration i is the application of Um to Nm . 3. The diagram
M
j01
0
-
M
1
k0
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
k1
?
i01
N
0
-
kn
?
N
1
Mn
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
?
Nn
commutes, where ki : Mi ! Ni is the ultrapower map arising from the measure j i (U~ (; )). 0
Proof: M can recover U by computing U = f X 2 P \ M j 2 j (X ) g: 0
0
0
0
We can then build a commutative triangle
M
j
0
@
@ @
-
Mn
@
j01 @
@
j1n @ @ R @
M
1
Since P \ M = P \ N and i = j N we have U = f X 2 P \ N j 2 i(X ) g; and a commutative triangle 0
0
0
0
21
i
N
0
@
@ @
-
Nn
@
i01 @
@
i1n @ @ R @
N
1
So U 2 N . We make the easy observation that < , because N is the ultrapower of M by the measure U~ (; ) on the coherent sequence U~ . Applying lemma 3 the square 0
0
0
0
M
j01
0
-
M
k0
k1
?
N
1
i01
0
-
?
N
1
commutes. We now attempt to argue that i n and j n resemble each other. Let = j (). 1
1
1
01
Claim 1: In the situation described above 1. = i (). 2. VM = VN . 1
01
1 1 +1
1 1 +1
3. j n VM = i n VN . 1 1 +1
1
1 1 +1
1
Proof: M and N agree to rank + 1, so by standard arguments 0
0
i VN = j VM 01
0
+1
01
0
+1
22
and
VM = i (VM ) = j (VN ) = VN : The key point is that both models compute the same set of functions from to V . If x 2 VM then x = j (F )() = i (F )() for some such function, and so j n(x) = j (F )() = i(F )() = i n(x) by the normality of the iterations. 1 1 +1
0
01
1 1 +1
0
01
+1
+1
+1
1 1 +1
01
01
1
1
Since is the largest measurable on U~ , is the largest on j (U~ ) and hence . We know that = crit(j n), so also = crit(i n). What is more U = f X 2 P \ M j 2 j n (X ) g = f X 2 P \ N j 2 i n (X ) g: Hence U is in N and i is the ultrapower of N by U . At this point we observe that since N is the ultrapower of M by the measure j (U~ (; )), there is a certain agreement between the measure sequences in these models: namely these sequences agree below , and at the model N has the same measures as M up to the point j ( ). As a consequence we see that either < or = and < j ( ). By lemma 3, we see that the diagram 1
1
1
1
1
1
1
01
1
1
1
1
1
1
1
1
1
1
1
12
1
1
1
1
01
1
1
1
1
M
1
j12
-
M
k1 ?
N
1
k2 i12
-
commutes. 23
?
N
01
1
2
2
1
1
1
1
01
To nish the proof we just repeat these arguments, showing step by step that the diagrams commute and the models Nn construct the measures Un.
The following corollary can be derived by a close inspection of the proof of the preceding lemma.
Corollary 1: Given an iteration j of M and a model N as described above, it is necessary and sucient for j N to be an iteration of N that for all m < n either m < j m () or m = j m () and m < j m ( ). 0
0
0
We observe that as a consequence, if j n induces an internal iteration of Ult(K; U~ max (; )), then it induces such an iteration of Ult(K; U~ max (; )) for any > . We can nally undertake the general analysis of the ordering between n-step extensions in K [G]. 0
De nition 6: Let < o(), and let 2 ( ; o()) [ f1g. For 2 ( ; o()) let M ( ; ) be the set of extensions U of U~ max (; ) such that is the least with the following two properties: 1. The constructing generic HU is in Ult(K; U~ max (; ))[G]. 2. jUK G induces an internal iteration of Ult(K; U~ max (; )). [
]
For = 1 let M ( ; ) be the set of those U such that no as described above exists. The description of the ordering is given by the following result, whose proof follows immediately from the work above.
Theorem 2: Every measure on in K [G] is in a unique M ( ; ). M ( ; ) has cardinality if 2 ( ; o()) and cardinality if = 1. If U 2 M ( ; ) and V 2 M ( ; ), then U V if and only if . +
++
24
References [1] J. Cummings, Possible behaviours for the Mitchell ordering II. In preparation. [2] A. Jergens, Private communication. [3] W. Mitchell, Sets constructible from sequences of ultra lters. Journal of Symbolic Logic 39 (1974) 57{66. [4] W. Mitchell, The core model for sequences of measures. I. Math. Proc. Camb. Phil. Soc. 95 (1984) 229{260. [5] W. Mitchell, Indiscernibles, skies and ideals. In Contemporary Mathematics 31 (1984) 161{182. [6] S. Baldwin, The -ordering on normal ultra lters. Journal of Symbolic Logic 51 (1985) 936{952.
25
