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 κ.

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.

Definition 1: Let κ be measurable, let U_{0} and U_{1} be normal measures on κ. Then U_{0} ⊲ U_{1} if and only if U_{0} ∈ Ult(V, U_{1}), the ultrapower of V by U_{1}.

The following facts are standard.

• ⊲ is transitive.

• ⊲ is well-founded.

• ⊲ is strict.

• An ultrafilter has at most 2^{κ} ancestors in the ordering ⊲.

Definition 2: o(κ) is the height of the well-founded relation ⊲.

Notice that we must have o(κ) ≤ (2^{κ})^{+}. 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(κ) = κ^{++}, 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 κ.

• If κ is the critical point of j : V ⟶ M such that V_{κ+2} ⊆ M, then we may show that every element of V_{κ+2} is in Ult(V, U) for some U on κ. In particular any 2^{κ} measures on κ will have an upper bound in the ordering ⊲. What is more, for any particular U there will only be 2^{κ} elements of V_{κ+2} in Ult(V, U), so that there must be 2^{2^{κ}} measures on κ. If it happens that 2^{2^{κ}} > (2^{κ})^{+} 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 κ is measurable, and all measures on κ may be divided into “blocks” in the following way:

1. For each α < o(κ) and β ∈ (α, o(κ)) ∪ ∞ there is a block M(α, β).

2. All the measures in M(α, β) have height α in the Mitchell ordering.

3. M(α, β) has cardinality κ^{+} if β ∈ (α, o(κ)), and cardinality κ^{++} if β = ∞.

4. For U ∈ M(α, β) and V ∈ M(γ, δ), U ⊲ V iff β ≤ γ (with the convention that ∞ is bigger than any ordinal).

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 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 κ can be regarded as having conditions which are functions p : α ⟶ κ for α < κ (rather than the more standard functions from α < κ to {0, 1}). In this form we can consider the forcing as adding a generic function from κ to κ.

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 a class of M or that N ⊆ M (notice that the former implies the latter, as N = ⋃_{α} k(V_{α}^{M})).

If a model M believes that U (with U ∈ M) is a measure on κ, we will denote the natural embedding from M into Ult(M, U) by j_{U}^{M}.

Lemma 1: Let j : V ⟶ M be an elementary embedding with j a class of V, κ = crit(j), such that every element of M is j(F)(κ) for some function F ∈ V. Then j is the ultrapower by the normal measure U = { X | κ ∈ j(X) }.

Proof: Factor j through the ultrapower of V by U,

[figure: M_{0}; k; V; j_{0}; M; j]

by defining k : [f] ⟼ j(f)(κ). k is a surjection, and M_{0} is the transitive collapse of the range of k, so M_{0} = M and j_{0} = j.

♦

Lemma 1 will prove useful in identifying certain embeddings as ultrapowers.

Lemma 2: Let M and N be inner models of ZFC such that • M ⊆ N.

• N ⊨ ^{κ}M ⊆ M.

• M ⊨ U is a normal measure on κ.

Then U is a normal measure in N and j_{U}^{N} ↾ M = j_{U}^{M}.

Proof: It follows immediately from the closure of M that U is a normal measure in N. Let x ∈ M. j_{U}^{N}(x) is the transitive collapse of the structure (F, E_{U}) where

F = { f : κ ⟶ x | f ∈ N },

and

fE_{U}g ⟺ { α | f(α) ∈ g(α) } ∈ U.

By the closure of M inside N we have

F = { f : κ ⟶ x | f ∈ M },

which is the set of functions whose collapse is j_{U}^{M}(x), so by the absoluteness of the collapsing construction j_{U}^{N}(x) = j_{U}^{M}(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 κ, W a measure on λ ≤ κ and suppose that W ∈ Ult(V, U). Let M_{U} be the ultrapower of V by U, M_{W} the ultrapower of V by W. Then

Ult(M_{U}, W) = Ult(M_{W} , j_{W}^{V} (U))

and the following diagram commutes.

[figure: V; j_{W}^{V}; M_{U}; j_{W}^{M_{U}}; j_{U}^{V}; M_{W}; Ult(M_{U}, W); j_{j^{V}_{W} (U)}^{M_{W}}]

Proof: Let x ∈ V.

j_{j_{W}^{V} (U)}^{M_{W}}(j_{W}^{V} (x)) = j_{W}^{V} (j_{U}^{V} (x)),

by elementarity. W ∈ M_{U} and (as λ ≤ κ) ^{λ}M_{U} ⊆ M_{U}, so that

j_{W}^{V} ↾ M_{U} = j_{W}^{M_{U}} .

In particular

j_{W}^{V} (j_{U}^{V} (x)) = j_{W}^{M_{U}} (j_{U}^{V} (x)).

From this we can deduce that the two ultrapowers are equal (let x = V_{α}), 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 ∈ M be a forcing notion, let G be ℙ-generic over M and let H be k(ℙ)-generic over N. Suppose that

p ∈ G ⟹ k(p) ∈ H.

Then

1. There is a unique extension of k to a map k^{∗} : M[G] ⟶ N[H] such that k^{∗} : G ⟼ H.

2. If Λ is a set of ordinals such that

N = { k(F)(a) | F ∈ M, a ∈ [Λ]^{<ω} },

then

N[H] = { k^{∗}(F)(a) | F ∈ M[G], a ∈ [Λ]^{<ω} }.

Proof: For the first claim, it is clear that if k^{∗} exists it must be given by

k^{∗} : τ̇^{G} ⟼ k(τ̇)^{H},

where τ̇^{G} denotes the interpretation of the term τ by the generic G.

We check that this is well-defined. Let τ̇^{G} = σ̇^{G}, then there is p ∈ G such that p ⊩_{ℙ}^{M} τ̇ = σ̇. By elementarity k(p) ⊩_{k(ℙ)}^{N} k(τ̇) = k(σ̇). By assumption k(p) ∈ H, so that k(τ̇)^{H} = k(σ̇)^{H}. The proof that k^{∗} is elementary is entirely similar.

For the second claim, let τ̇^{H} ∈ N[H]. Then τ̇ = k(F)(a) for some F ∈ M and a ∈ [Λ]^{<ω}, and we may take it that for all x in the domain F (x) is a ℙ-term. In M[G] we may define a function F_{1} : x ⟼ F (x)^{G}, and then

k^{∗}(F_{1})(a) = k(F)(a)^{k^{∗}(G)} = τ̇^{H}.

♦

Lemma 4 will be used to take elementary embeddings (usually finitely iterated ultrapowers) and extend them onto certain generic extensions of V. The second claim will play a key rôle 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 κ + 1, in which one Cohen subset of α is added at each strong inaccessible α ≤ κ. Let G_{1} and G_{2} be ℙ-generics over V, with the property that V [G_{1}] = V [G_{2}]. Then for any model V^{∗} agreeing with V to rank κ + 1, G_{1} and G_{2} are ℙ-generic over V^{∗} and V^{∗}[G_{1}] = V^{∗}[G_{2}].

Proof: By the agreement ℙ ∈ V^{∗} and (since |ℙ| = κ) both models compute the same maximal antichains, so G_{1} and G_{2} are generic over V^{∗} for ℙ. G_{1} is the interpretation under G_{2} of some term τ̇, and by the agreement again we may take it that τ̇ ∈ V^{∗}. So G_{1} ∈ V^{∗}[G_{2}] and vice versa, so that V^{∗}[G_{1}] = V^{∗}[G_{2}].

♦

Next we give the lemma that will be used to generate measures.

Lemma 6: Let GCH hold, and let j : V ⟶ M be an embedding which is a class in V, such that κ = crit(j) and ^{κ}M ⊆ M. Suppose also that the ordinal j(κ^{+}) has cardinality κ^{+} in V. Let ℙ be as in lemma 5, and observe that ℙ can be factored as ℙ_{κ} followed by Add(κ, 1) as computed by V^{ℙ_{κ}}.

Let G = G_{κ} ∗ g be ℙ-generic, and suppose that there is G_{1} = G_{κ} ∗ g_{1} with V [G] = V [G_{1}].

Then in V [G] there are κ^{++} many H such that G_{1} ∗ H is j(ℙ)-generic over M and j extends to ȷ^{∗} : V [G] ⟶ M[G_{1}][H].

Proof: By lemma 5 M[G] = M[G_{1}]. In M[G_{1}] the factor iteration j(ℙ)/G_{1} is highly-closed and has j(κ^{+}) many antichains. As ℙ has the κ^{+}-chain condition and M[G] = M[G_{1}] we have V [G] ⊨ ^{κ}M[G_{1}] ⊆ M[G_{1}]. Hence in V [G] the forcing j(ℙ)/G_{1} is κ^{+}-closed, and the set of its maximal antichains which lie in M[G_{1}] has cardinality κ^{+}.

We wish to build generics which are compatible with G. Working in M[G_{1}], define a function q with domain the M-inaccessibles η such that κ < η ≤ j(κ), by q(η) = ∅ for η < j(κ) and q(j(κ)) = g. q is a condition in j(ℙ)/G_{1}. We build in V [G] a binary tree of height κ^{+} such that

• The top node is q.

• Any path is a descending sequence in j(ℙ)/G_{1}, meeting each antichain in M[G_{1}].

• Every element has incompatible immediate successors.

The construction proceeds for the requisite κ^{+} steps, because j(ℙ)/G_{1} is κ^{+}-closed in V [G]. This construction will give us κ^{++} distinct generic filters H, each with the property that j“G ⊆ G_{1} ∗ H. We can use these to build extensions ȷ^{∗} of j such that ȷ^{∗}(G) = G_{1} ∗ H.

♦

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 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. Definition 3: U⃗ is a coherent sequence of measures if and only if

• U⃗ is a function, with dom(U⃗) ⊆ On × On.

• For some function oU⃗ : On ⟶ On, dom(U⃗) = { (κ, η) | 0 ≤ η < oU⃗(κ) }.

• If (κ, η) ∈ dom(U⃗) then U⃗(κ, η) is a normal measure on κ.

• If (κ, η) ∈ dom(U⃗), and j : V ⟶ M is the ultrapower of V by the

measure U⃗(κ, η) then

– For all α ≤ κ, (α, β) ∈ dom(j(U⃗)) if and only if α ≤ κ or α = κ

and β < η.

– If α ≤ κ and (α, β) ∈ dom(j(U⃗)) then

j(U⃗)(α, β) = U⃗(α, β).

Definition 4: Let M be an inner model of ZFC, let

M ⊨ U⃗ is a coherent sequence of measures. A normal iteration of M by U⃗, of length η is a pair

(⟨M_{α} : α < η⟩, ⟨j_{αβ} : α ≤ β < η⟩) where

• M_{0} = M.

• M_{α} is an inner model of ZFC for each α < η.

• For α ≤ β < η, j_{αβ} : M_{α} ⟶ M_{β} is an elementary embedding.

• j_{αα} = id, and for α ≤ β ≤ γ, j_{αγ} = j_{βγ} ∘ j_{αβ}.

• M_{α+1} = Ult(M_{α}, j_{0α}(U⃗)(κ_{α}, η_{α})), and j_{αα+1} : M_{α} ⟶ M_{α+1} is the associated ultrapower map, if α + 1 < η.

• If λ < η, λ is limit, then M_{λ} and j_{αλ} are had by taking a direct limit in the natural way.

• The sequence ⟨κ_{α} : α + 1 < η⟩ is strictly increasing.

The following structural fact is easy, by induction on α < η.

Lemma 7: If (M⃗, j⃗ ) is a normal iteration of M by U⃗ in length η then for every α < η

M_{α} = { j_{0α}(F)(a) | F ∈ M, a ∈ [Λ]^{<ω} },

where Λ = { κ_{ν} | ν < α }.

We will denote by K Mitchell’s core model K[U⃗_{max}], which exists under the assumption that there is no inner model in which ∃κ o(κ) = κ^{++}. We will use the following facts about K (see section 2 of [5]).

Lemma 8 (Mitchell): Suppose that ¬∃κ o(κ) = κ^{++} in any inner model of ZFC. Then

• K is a uniformly definable inner model of ZFC+GCH.

• K ⊨ V = K.

• K ⊨ U⃗_{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 U⃗_{max}.

It is worth making the following easy observations about K and U⃗_{max}.

Lemma 9: If K, U⃗_{max} are as above then

• All measures in K appear on the sequence U⃗_{max}.

• If α < β < oU⃗_{max}(κ) then U⃗_{max}(κ, α) ≠ U⃗_{max}(κ, β).

• K ⊨ U⃗_{max}(κ, α) ⊲ U⃗_{max}(κ, β) iff α < β.

We will be particularly interested in finite normal iterations of K, in the case when there is a largest measurable on U⃗_{max}. Lemma 10: Suppose that κ is the largest ordinal with oU⃗_{max}(κ) > 0. Let n + 1 < ω, let (M⃗, j⃗ ) be a normal iteration of K by U⃗_{max} of length n + 1, with j_{01} the ultrapower of K by U⃗_{max}(κ, η) for some η. Then

1. M_{n} ⊆ K, and K ⊨ ^{κ}M_{n} ⊆ M_{n}.

2. For each i < n, κ_{i} < j_{0n}(κ).

3. In M_{0}, the ordinal j_{0n}(κ^{+}) has cardinality κ^{+} Proof:

1. The critical points are increasing and each model is closed inside the previous one.

2. κ_{i} ≤ j_{0i}(κ), as κ is the largest measurable on U⃗_{max}. If κ_{i} < j_{0i}(κ) then we are done as j_{0n}(κ) = j_{in}(j_{0i}(κ)) ≥ j_{0i}(κ); if κ_{i} = j_{0i}(κ) then this is the critical point of j_{in} so κ_{i} < j_{in}(j_{0i}(κ)) = j_{0n}(κ).

3. The ordinals less than j_{0n}(κ^{+}) all have the form

j_{0n}(F)(κ_{0}, …, κ_{n−1}), where F : [κ]^{n} ⟶ κ^{+}. By GCH there are κ^{+} such functions F.

♦

The next result puts some limits on the possible closure of the models in a normal iteration of infinite length.

Lemma 11: If (M⃗, j⃗ ) is a normal iteration of M by U⃗, of length η ≥ ω, then the sequence of ordinals κ⃗ = ⟨κ_{n} : n < ω⟩ is not a member of M_{α} for ω ≤ α < η.

Proof: The model M_{α} agrees with M_{ω} to rank κ_{ω} + 1, so it is enough to show that κ⃗ ∉ M_{ω}. M_{ω} was constructed as a direct limit, so if κ⃗ ∈ M_{ω} then κ⃗ = j_{nω}(λ⃗) for some λ⃗ ∈ M_{n}; in particular κ_{n} = j_{nω}(λ_{n}). But crit(j_{nω}) = κ_{n} as we are in a normal iteration, so that κ_{n} ∉ rge(j_{nω}).

♦

This completes the preliminaries. We make the remark that in what follows we assume that the ground model is of form K[U⃗_{max}], but could have taken it in the form L[U⃗] because for suitable U⃗ we have L[U⃗] ⊨ V = K[U⃗_{max}].

3 Classifying measures

In this section we will take the core model K[U⃗_{max}] discussed in the last section, in the case when there is a largest measurable on U⃗_{max}, and force over it with an iteration ℙ as in lemma 6. We will then classify completely the measures on κ 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 κ maximal with oU⃗_{max}(κ) > 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 α ≤ κ, as computed in K. As in lemma 6 we may factor ℙ as ℙ_{κ} ∗ Add(κ, 1), and correspondingly we may factor G as G_{κ} ∗ g.

Lemma 12: Let U be a measure on κ 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_{κ} ∗ g_{1} ∗ H, where g_{1} is Add(κ, 1)-generic over K^{∗}[G_{κ}] and H is j(ℙ)/G_{κ} ∗ g_{1}-generic over K^{∗}[G_{κ}][g_{1}].

2. If G_{1} = G_{κ} ∗ g_{1} then K[G_{1}] = K[G].

3. N = K^{∗}[i(G)].

4. j“G ⊆ i(G).

5. j : K ⟶ K^{∗} is a finite normal iteration of K by U⃗_{max}, with the first step being an ultrapower map with critical point κ.

Proof:

By elementarity N = K^{∗}[i(G)], where K^{∗} is K[U⃗_{max}] as computed in the sense of N. i(G) is generic over K^{∗} for i(ℙ), which equals j(ℙ) since ℙ ∈ K.

j : K ⟶ K^{∗} must be a normal iteration with first step an ultrapower by a measure on κ, because K is still K[U⃗_{max}] in K[G]. In particular K and K^{∗} agree to rank κ + 1.

i(G) = G_{κ} ∗ g_{1} ∗ H, where g_{1} is generic for Add(κ, 1) as computed in K^{∗}[G_{κ}] and H is generic for j(ℙ)/G_{κ} ∗ g_{1}. K[G_{κ}] and K^{∗}[G_{κ}] agree to rank κ + 1, so g_{1} is actually K[G_{κ}] generic for Add(κ, 1). Also K[G_{1}] and K^{∗}[G_{1}] agree to rank κ + 1.

As N is an ultrapower, K[G] ⊨ ^{κ}N ⊆ N. As H is generic for highly closed forcing, K[G] ⊨ ^{κ}K^{∗}[G_{1}] ⊆ K^{∗}[G_{1}]. In particular g ∈ K^{∗}[G_{1}], so that by the last paragraph g ∈ K[G_{1}]. Hence K[G] = K[G_{1}].

If j is not a finite iteration, then lemma 11 implies that there is an ωsequence of ordinals κ⃗ ∈ K[G] such that κ⃗ ∉ K^{∗}. But ℙ is ω_{1}-closed, and so κ⃗ ∉ K^{∗}[G], in contradiction to what we just proved about the closure of K^{∗}[G].

♦

Definition 5: U ∈ K[G] is an n-step extension of U⃗_{max}(κ, η) if, when we define j as in the last lemma, j has length n + 1 and the first step in j is the application of U⃗_{max}(κ, η) to K.

Notice that this is reasonable terminology, as when U is an n-step extension of U⃗_{max}(κ, η) we certainly have U⃗_{max}(κ, η) ⊆ U. The one-step extensions are the easiest ones to understand.

Lemma 13: Let η < oU⃗_{max}(κ), and let j_{η} : K ⟶ M_{η} be the ultrapower of K by U⃗_{max}(κ, η). Then in K[G] the set of H_{1} = g_{1} ∗ H such that (setting G_{1} = G_{κ} ∗ g_{1})

• G_{1} is ℙ-generic over K.

• K[G] = K[G_{1}].

• H is j_{η}(P)/G_{1}-generic over M_{η}[G_{1}].

• j_{η}“G ⊆ G_{κ} ∗ H_{1}. has cardinality κ^{++}, and each one gives rise to a distinct one-step extension U_{H_{1}} of U⃗_{max}(κ, η). Proof: There are κ^{+} generics g_{1} such that K[G] = K[G_{κ}][g_{1}]. Fix one such, and observe that by lemma 5 M_{η}[G] = M_{η}[G_{1}]. By lemma 6 we may build κ^{++} many appropriate generics H, and by cardinality considerations there can be at most κ^{++} many.

Let H be one such, and consider the unique map

ȷ_{η}^{∗} : K[G] ⟶ M_{η}[G_{1}][H] such that ȷ_{η}^{∗} extends j_{η} and ȷ_{η}^{∗}(G) = G_{1} ∗ H. By lemma 4,

M_{η}[G_{1}][H] = { ȷ_{η}^{∗}(F)(κ) | F ∈ K[G] }, so lemma 1 tells us that ȷ_{η}^{∗} is the ultrapower of K[G] by the measure

U_{H} = { X ⊆ κ | κ ∈ ȷ_{η}^{∗}(X) }. Distinct generics H_{1} give distinct one-step extensions, because given U_{H_{1}}

we may recover H_{1} by computing j_{U_{H_{1}}}^{K[G]}(G) = G_{κ} ∗ H_{1}.

♦

This last lemma gives a complete description of the one-step extensions of measures U⃗_{max}(κ, η). 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 defined from κ in a certain way.

Lemma 14: Let j : K ⟶ K^{∗} be a normal iteration of K by U⃗_{max} of length n + 1, with j_{0i}(U⃗_{max})(κ_{i}, η_{i}) being applied at stage i in the iteration, and κ_{0} = κ. Then in K[G] there are κ^{++} many H_{1} = g_{1} ∗ H such that (setting G_{1} = G_{κ} ∗ g_{1})

• G_{1} is ℙ-generic over K.

• K[G] = K[G_{1}].

• H is j(P)/G_{1}-generic over K^{∗}[G_{1}].

• j“G ⊆ G_{κ} ∗ H_{1}.

• If

ȷ^{∗} : K[G] ⟶ K^{∗}[G_{κ}][H_{1}]

is the unique extension of j with ȷ^{∗}(G) = G_{κ} ∗ H_{1}, then

K^{∗}[G_{κ}][H_{1}] = { ȷ^{∗}(F)(κ) | F ∈ K[G] }.

Proof: As before there are κ^{+} appropriate g_{1}, and we will fix one. Then we know that K^{∗}[G] = K^{∗}[G_{1}].

We will define a “master condition” for j(P)/G_{1}, much as in lemma 6. As there the condition q will have value ∅ at M-inaccessible η with κ < η < j(κ), but q(j(κ)) will be slightly bigger than in lemma 6. Define q(j(κ)) by

• dom(q(j(κ))) = κ + n.

• q(j(κ)) ↾ κ = g.

• q(j(κ) + i) = κ_{i}, for i < n.

Just as in lemma 6 we may build κ^{++} many H with q as a member, and argue that H is generic and that j“G ⊆ G_{1} ∗ H. It will suffice to show that for every i < n the ordinal κ_{i} has the form ȷ^{∗}(F)(κ), as lemma 7 then shows that every element of K^{∗}[G_{κ}][H_{1}] may be written in this form. Now fix i < n, and define a function F in K[G] by

F (α) = g(α + i).

We have

ȷ^{∗}(F)(κ) = ȷ^{∗}(g)(κ + i) = H(j(κ))(κ + i) = q(κ)(κ + i) = κ_{i},

so the lemma is proved.

♦

This result classifies the n-step extensions of measures on κ 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 κ in K[G]. Suppose further that U is a 1-step extension of U_{0} = U⃗_{max}(κ, α), using some generic H_{U}^{1} = g_{U} ∗H_{U}, and that V is a 1-step extension of V_{0} = U⃗_{max}(κ, β) using some generic H_{V}^{1} = g_{V} ∗ H_{V}. Set G_{U} = G_{κ} ∗ g_{U}, G_{V} = G_{κ} ∗ g_{V}.

Then K[G] ⊨ U ⊲ V if and only if

• α < β.

• H_{U}^{1} ∈ Ult(K, V_{0})[G].

Proof: Let M = Ult(K, U_{0}), let N = Ult(K, V_{0}).

• First suppose that K[G] ⊨ U ⊲ V. This means that

U ∈ Ult(K[G], V) = N[G_{V}][H_{V}].

As K[G] = K[G_{V}] we know that N[G] = N[G_{V}]. H_{V} is generic for highly closed forcing, so this will imply that U ∈ N[G]. Since K[G] ⊨ ^{κ}N[G] ⊆ N[G], K[G] and N[G] agree to rank κ + 1, so that there is agreement between j_{U}^{K[G]} and j_{U}^{N[G]} to that rank. In particular

G_{U} ∗ H_{U} = j_{U}^{K[G]}(G) = j_{U}^{N[G]}(G),

so that H_{U}^{1} ∈ N[G].

To show that α < β, observe that N ⊆ K ⊆ K[G]. Also

j_{U}^{K[G]} ↾ N[G] = j_{U}^{N[G]},

so that the restriction of j_{U}^{K[G]} to N is an embedding definable 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 j_{U}^{K[G]} ↾ K = j_{U_{0}}^{K}, so that j_{U}^{K[G]} ↾ N = j_{U_{0}}^{K} ↾ N. It is easy to see that the first step in the iteration of N induced by this restriction is to take the ultrapower by

U_{0} = { X ⊆ κ | X ∈ N, κ ∈ j_{U_{0}}^{K}(X) },

so that U_{0} ∈ N. Hence U_{0} ⊲ V_{0}, and α < β.

• For the other direction, suppose that H_{U}^{1} ∈ N[G] and α < β, that is K ⊨ U_{0} ⊲ V_{0} and so U_{0} ∈ N.

We will show that N[G] can reconstruct U from H_{U}^{1}. K[G] and N[G] (which equals N[G_{V}]) agree to rank κ + 1, and j_{U_{0}}^{K} ↾ N = j_{U_{0}}^{N}, what is more N contains all canonical P -names for subsets of κ. So if τ̇ is such a name then N[G] can compute

j_{U}^{K[G]}(τ̇^{G}) = j_{U_{0}}^{K}(τ̇)^{G_{U} ∗H_{U}} = j_{U_{0}}^{N}(τ̇)^{G_{U} ∗H_{U}} ,

and hence N[G] can compute U, so

U ∈ N[G_{V}] ⊆ N[G_{V}][H_{V}] = 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 j_{U_{0}}^{V}(P)/G are constructed by models of the form Ult(K, V_{0})[G] as V_{0} runs through the measures on κ with U_{0} ⊲ V_{0}. The next lemma will provide us with this information.

Lemma 16: Let α < β < γ < oU⃗_{max}(κ). Let us define U = U⃗_{max}(κ, α), V = U⃗_{max}(κ, β), and finally W = U⃗_{max}(κ, γ). Then the Ult(K, U)[G]- generics on j_{U}^{K}(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 j_{U}“G ⊆ G ∗ H.

Proof: Let M_{U} = Ult(K, U) and define M_{V}, M_{W} similarly. K and M_{W} agree to rank κ + 1, so that M_{V} and N = Ult(M_{W} , V) agree to rank j_{V} (κ) + 1. As P is relatively small, M_{V} [G] and N[G] also agree to this level, which is much greater than j_{U}(κ). So M_{V} [G] and N[G] construct the same generics H for the forcing j_{U}(P)/G.

But now by the same arguments as in lemma 6, M_{W} [G] believes that it can construct κ^{++} many generics, but that the inner model N[G] can only build κ^{+} many. This proves the lemma.

♦

We use this to get a picture of the ordering on one-step extensions in the case when oU⃗_{max}(κ) = 3. This is fairly representative of the general case.

Lemma 17: Let oU⃗_{max}(κ) = 3, with U = U⃗_{max}(κ, 0), V = U⃗_{max}(κ, 1), W = U⃗_{max}(κ, 2). Let M_{U}, M_{V}, M_{W} 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_{0}: extensions of U via generics in M_{V} [G]. |C_{0}| = κ^{+}

• C_{1}: extensions of U via generics in M_{W} [G] \ M_{V} [G]. |C_{1}| = κ^{+}.

• C_{2}: extensions of U via generics in K[G] \ M_{W} [G]. |C_{2}| = κ^{++}.

• C_{3}: extensions of V via generics in M_{W} [G]. |C_{1}| = κ^{+}.

• C_{4}: extensions of V via generics in K[G] \ M_{W} [G]. |C_{4}| = κ^{++}.

• C_{5}: extensions of W. |C_{5}| = κ^{++}.

A measure from C_{i} is below a measure from C_{j} in the Mitchell ordering if and only if

• i = 0 and j ∈ {3, 4, 5} OR

• i = 1 and j = 5 OR

• i = 3 and j = 5.

The proof is immediate. We give a picture which may make the shape of the partial ordering clearer.

[figure: C_{5}; C_{3}; C_{4}; C_{0}; C_{1}; C_{2}]

If instead of o(κ) = 3 we take o(κ) = ω, we get an infinite 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 finite 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.

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 first step in the iteration associated with V.

Theorem 1: Let U be an m + 1-step extension of U_{0}, via a generic object H_{U}^{1} = g_{U} ∗ H_{U} and an iteration (M⃗, j⃗) of length m + 1, with the ultrafilter j_{0i}(U⃗_{max})(κ_{i}, λ_{i}) being applied to M_{i} at stage i. Let V be an n + 1-step extension of V_{0}, via a generic H_{V}^{1} = g_{V} ∗H_{V} and an iteration (N⃗, k⃗) of length n + 1, with the ultrafilter k_{0i}(U⃗_{max})(μ_{i}, ν_{i}) being applied to N_{i} at stage i.

Then K[G] ⊨ U ⊲ V if and only if

• H_{U} ∈ Ult(K, V_{0})[G].

• j_{0m} ↾ Ult(K, V_{0}) is a finite normal iteration of Ult(K, V_{0}) by k_{01}(U⃗_{max}).

Proof: Notice that N_{1} = Ult(K, V_{0}). As before we let G_{U} = G_{κ} ∗ g_{U} and G_{V} = G_{κ} ∗ g_{V}.

• Suppose that K[G] ⊨ U ⊲ V. Then

U ∈ Ult(K[G], V) = N_{n}[G_{V}][H_{V}],

so as in lemma 15 U ∈ N_{n}[G]. N_{1} and N_{n} agree to rank κ_{1} + 1, so by an easy chain condition argument the models N_{1}[G] and N_{n}[G] also agree to this rank, hence U ∈ N_{1}[G].

As in lemma 15 N_{1}[G] can reconstruct H_{U}^{1}, so that H_{U}^{1} ∈ N_{1}[G].

For the second part just observe that j_{U}^{K[G]} ↾ N_{1}[G] = j_{U}^{N_{1}[G]}, so that j_{U}^{K[G]} ↾ N_{1} must give rise to a normal iteration of N_{1} by its version of U⃗_{max}, which is k_{01}(U⃗_{max}). But N_{1} ⊆ K and j_{U}^{K[G]} ↾ K = j_{0m}, so this amounts to saying that j_{0m} ↾ N_{1} is a normal iteration of N_{1} by k_{01}(U⃗_{max}).

This iteration must be finite, as usual, because otherwise the first ω critical points will give a sequence which is in N_{1}[G] but not in Ult(N_{1}[G], U).

• Suppose that H_{U}^{1} ∈ N_{1}[G], and that j_{0m} ↾ N_{1} can be written as an iteration (N⃗^{∗}, ȷ^{∗}) of length s+1, so that N_{s}^{∗} = j_{0m}(N_{1}) and j_{0m} ↾ N_{1} = ȷ_{0s}^{∗}.

We will show that N_{1}[G] can compute U; the proof is precisely parallel to that in lemma 15. K[G] and N_{1}[G] agree to rank κ + 1, K and N_{1} agree on the set of canonical names for subsets of κ. If τ̇ is such a name then (since ȷ_{0s}^{∗} is a class in N_{1})) N_{1}[G] can compute

j_{U}^{K[G]}(τ̇^{G}) = j_{0m}(τ̇)^{G_{U} ∗H_{U}} = ȷ_{0s}^{∗}(τ̇)^{G_{U} ∗H_{U}} .

Just as in lemma 15 this gives U ∈ N_{1}[G], and by the same arguments as we used in the first part of the proof this implies that U ∈ N_{n}[G], hence that K[G] ⊨ U ⊲ V.

♦

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 finite iteration to a one-step ultrapower model.

Lemma 18: Let M be a model of ZFC, and assume

M ⊨ U⃗ is a coherent sequence.

Let κ be the largest critical point on U⃗. Let j be a finite normal iteration of M, in which a measure

U_{m} = j_{0m}(U⃗)(κ_{m}, μ_{m})

is applied to M_{m} to get j_{mm+1} : M_{m} ⟶ M_{m+1} for each m < n. Let κ_{0} = κ, μ_{0} = α. Let N = Ult(M, U⃗(κ, β)) for some β, and suppose that i = j ↾ N : N ⟶ j(N) is a finite normal iteration of N.

Then

1. For each m < n, U_{m} ∈ N_{m}.

2. i has length n, and step m in the iteration i is the application of U_{m} to N_{m}.

3. The diagram

[figure: M_{0}; k_{0}; M_{1}; k_{1}; M_{n}; k_{n}; j_{01}; N_{0}; N_{1}; N_{n}; i_{01}]

commutes, where k_{i} : M_{i} ⟶ N_{i} is the ultrapower map arising from the measure j_{0i}(U⃗(κ, β)).

Proof: M_{0} can recover U_{0} by computing

U_{0} = { X ∈ Pκ ∩ M_{0} | κ ∈ j(X) }.

We can then build a commutative triangle

[figure: M_{0}; j_{01}; M_{n}; j; M_{1}; j_{1n}]

Since Pκ ∩ M_{0} = Pκ ∩ N_{0} and i = j ↾ N we have

U_{0} = { X ∈ Pκ ∩ N_{0} | κ ∈ i(X) },

and a commutative triangle

[figure: N_{0}; i_{01}; N_{n}; i; N_{1}; i_{1n}]

So U_{0} ∈ N_{0}. We make the easy observation that α < β, because N_{0} is the ultrapower of M_{0} by the measure U⃗(κ, β) on the coherent sequence U⃗.

Applying lemma 3 the square

[figure: M_{0}; k_{0}; M_{1}; k_{1}; j_{01}; N_{0}; N_{1}; i_{01}]

commutes.

We now attempt to argue that i_{1n} and j_{1n} resemble each other. Let λ_{1} = j_{01}(κ).

Claim 1: In the situation described above

1. λ_{1} = i_{01}(κ).

2. V_{λ_{1}+1}^{M_{1}} = V_{λ_{1}+1}^{N_{1}}.

3. j_{1n} ↾ V_{λ_{1}+1}^{M_{1}} = i_{1n} ↾ V_{λ_{1}+1}^{N_{1}}.

Proof: M_{0} and N_{0} agree to rank κ + 1, so by standard arguments

i_{01} ↾ V_{κ+1}^{N_{0}} = j_{01} ↾ V_{κ+1}^{M_{0}}

and

V_{λ_{1}+1}^{M_{1}} = i_{01}(V_{κ+1}^{M_{0}}) = j_{01}(V_{κ+1}^{N_{0}}) = V_{λ_{1}+1}^{N_{1}}.

The key point is that both models compute the same set of functions from κ to V_{κ+1}.

If x ∈ V_{λ_{1}+1}^{M_{1}} then x = j_{01}(F)(κ) = i_{01}(F)(κ) for some such function, and so

j_{1n}(x) = j(F)(κ) = i(F)(κ) = i_{1n}(x)

by the normality of the iterations.

♦

Since κ is the largest measurable on U⃗, λ_{1} is the largest on j_{01}(U⃗) and hence κ_{1} ≤ λ_{1}. We know that κ_{1} = crit(j_{1n}), so also κ_{1} = crit(i_{1n}). What is more

U_{1} = { X ∈ Pκ_{1} ∩ M_{1} | κ_{1} ∈ j_{1n}(X) }
= { X ∈ Pκ_{1} ∩ N_{1} | κ_{1} ∈ i_{1n}(X) }.

Hence U_{1} is in N_{1} and i_{12} is the ultrapower of N_{1} by U_{1}.

At this point we observe that since N_{1} is the ultrapower of M_{1} by the measure j_{01}(U⃗(κ, β)), there is a certain agreement between the measure sequences in these models: namely these sequences agree below λ_{1}, and at λ_{1} the model N_{1} has the same measures as M_{1} up to the point j_{01}(β).

As a consequence we see that either κ_{1} < λ_{1} or κ_{1} = λ_{1} and μ_{1} < j_{01}(β). By lemma 3, we see that the diagram

[figure: M_{1}; k_{1}; M_{2}; k_{2}; j_{12}; N_{1}; N_{2}; i_{12}]

commutes.

To finish the proof we just repeat these arguments, showing step by step that the diagrams commute and the models N_{n} construct the measures U_{n}.

♦

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 sufficient for j ↾ N to be an iteration of N that for all m < n either κ_{m} < j_{0m}(κ) or κ_{m} = j_{0m}(κ) and μ_{m} < j_{0m}(β).

We observe that as a consequence, if j_{0n} induces an internal iteration of Ult(K, U⃗_{max}(κ, β)), then it induces such an iteration of Ult(K, U⃗_{max}(κ, γ)) for any γ > β.

We can finally undertake the general analysis of the ordering between n-step extensions in K[G].

Definition 6: Let α < o(κ), and let β ∈ (α, o(κ)) ∪ {∞}.

For β ∈ (α, o(κ)) let M(α, β) be the set of extensions U of U⃗_{max}(κ, α) such that β is the least γ with the following two properties:

1. The constructing generic H_{U} is in Ult(K, U⃗_{max}(κ, γ))[G].

2. j_{U}^{K[G]} induces an internal iteration of Ult(K, U⃗_{max}(κ, γ)).

For β = ∞ let M(α, β) be the set of those U such that no γ 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 κ in K[G] is in a unique M(α, β). M(α, β) has cardinality κ^{+} if β ∈ (α, o(κ)) and cardinality κ^{++} if β = ∞.

If U ∈ M(α, β) and V ∈ M(γ, δ), then U ⊲ V if and only if β ≤ γ.

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 ultrafilters. 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 ultrafilters. Journal of Symbolic Logic 51 (1985) 936–952.