Possible behaviours for the Mitchell ordering II

James Cummings Institute of Mathematics Hebrew University Givat Ram 91904 Jerusalem ISRAEL

              cummings@math.huji.ac.il

February 25, 1998

Abstract We analyse the Mitchell ordering in a model where κ is P_{2}κ-

hypermeasurable and 2^{2^{κ}} > (2^{κ})^{+}.

1 Introduction and preliminaries

The Mitchell ordering on normal measures was introduced by Mitchell [7], in his work on inner models for cardinals carrying many measures.

Definition 1: Let κ be measurable, let U_{0} and U_{1} be normal measures on a cardinal 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 ⊲.

We make the remark that there are certain limitations on the partial ordering ⊲; there can be at most 2^{2^{κ}} normal measures on κ, and the ordering can have height at most (2^{κ})^{+}.

If κ is supercompact then it is shown in [9] that these maximal values are achieved; in fact a much weaker hypothesis on κ will suffice [10].

Theorem 1 (Solovay): Let j : V ⟶ M be an elementary embedding where κ = crit(j) and V_{κ+2} ⊆ M, ^{κ}M ⊆ M. Then for every X ∈ V_{κ+2} there is a normal measure U on κ such that X ∈ Ult(V, U).

Proof: An easy adaptation of the proof in [9].

♦

A P_{2}κ-hypermeasurable cardinal is one which satisfies the hypothesis of this theorem.

We spell out the consequences for the Mitchell ordering.

1. Observe that |V_{κ+2} ∩ Ult(V, U)| = 2^{κ}, since any element of V_{κ+2} in Ult(V, U) is represented by a function from κ to V_{κ}. Since 2^{2^{κ}} > 2^{κ}, there must be 2^{2^{κ}} normal measures U.

2. Any 2^{κ} measures may be coded by an element of V_{κ+2}. So any family of 2^{κ} measures has a upper bound in the Mitchell ordering. In particular the height of the ordering has its maximal value, namely (2^{κ})^{+}.

Using inner model theory (of which more anon) we may get models where GCH holds, and the Mitchell ordering at κ is linear of ordertype κ^{++}. On the other hand if κ is a P_{2}κ-hypermeasurable cardinal) and 2^{2^{κ}} > (2^{κ})^{+} then the ordering is necessarily non-linear; in this paper we explore the ordering in a particular model where these circumstances prevail, and prove a result (theorem 13) which goes some way towards characterising it.

We have taken a fairly digressive approach to the proof. Some arguments needed here are so similar to those of [3] that we have just sketched them here.

2 Inner model theory

We sketch what we need from the theory of inner models for large cardinals below a strong cardinal. The reader is referred to [8] and [5] for more details. The theory is due to Mitchell, Jensen, Dodd and Koepke.

Definition 2: E⃗ is a (coherent, non-overlapping) extender sequence if E⃗ is a function with dom(E⃗) ⊆ On × On such that

1. There is some function o : On ⟶ On such that o(α) ≥ α + 1 and

dom(E⃗) = { (κ, λ) | κ < λ < o(κ) }.

2. If (κ, λ) ∈ dom(E⃗) then E⃗(κ, λ) is a (κ, λ)-extender.

3. (coherence) Let ≺ be the lexicographic order on On × On, so that

(κ_{0}, λ_{0}) ≺ (κ_{1}, λ_{1}) ⟺ (κ_{0} < κ_{1}) ∨ (κ_{0} = κ_{1} ∧ λ_{0} < λ_{1}).

If (κ, λ) ∈ dom(E⃗) and E = E⃗(κ, λ) then

• j_{E}(o)(κ) = λ, j_{E}(o) ↾ κ = o ↾ κ.

• If (α, β) ≺ (κ, λ) then E⃗(α, β) = j_{E}(E⃗)(α, β).

4. (non-overlap) If κ_{0} < κ_{1} and o(κ_{1}) > κ_{1} + 1 then o(κ_{0}) < κ_{1}.

The ideology of inner model theory is roughly this; if κ is a large cardinal in V, then we can build E⃗ reflecting this, and L[E⃗] is a well-behaved “Llike” model in which κ is still large. As a more concrete example, let κ be P_{2}κ-hypermeasurable. Then it may be shown that

1. There is an extender sequence E⃗ such that o(κ) > κ^{++}.

2. In L[E⃗], the embedding associated with the extender E⃗(κ, κ^{++}) witnesses that κ is P_{2}κ-hypermeasurable.

3. L[E⃗] is a model of GCH, has a Δ_{3}^{1} wellorder of the reals, and has a reasonable fine-structure theory.

The key to the good properties of L[E⃗] is the Comparison Lemma.

Lemma 1 (Comparison): If (M, F⃗) and (N, G⃗) are such that

M ⊨ F⃗ is an extender sequence
N ⊨ G⃗ is an extender sequence

and M and N are sufficiently iterable, then we may compare the models M and N. That is, we may iterate M by F⃗ and N by G⃗ so as to get (M^{∗}, F⃗^{∗}) and (N^{∗}, G⃗^{∗}), such that one of the sequences F^{∗}, G^{∗} is an initial segment of the other as far as sets in M^{∗} ∩ N^{∗} are concerned.

For the benefit of the reader, we collect some information about L[E⃗], with (very sketchy) proofs. The theory of L[E⃗] was worked out by Mitchell in [8], using the language of “hypermeasures”; here we just translate into the language of extenders. Assume till the end of this section that V = L[E⃗], where E⃗ is an arbitrary extender sequence whose domain is a set.

Lemma 2 (Uniqueness): E⃗(κ, λ) is the only (κ, λ)-extender which has the “coherence” properties demanded in clause 3 of definition 2.

Proof:[Sketch] Let E⃗ be a minimal counterexample, in the sense that no proper initial segment of E⃗ falsifies the lemma. Let (κ, λ) be the first place on E⃗ where the lemma is false.

Let F be an extender differing from E⃗(κ, λ), but having the same coherence property. Compare the ultrapowers Ult(V, F) and Ult(V, E⃗(κ, λ)), and argue as in [7] that they iterate to a common model N and that the diagram

[figure: N; Ult(V, E⃗(κ, λ)); k; Ult(V, F); l; V; j_{E⃗(κ,λ)}; j_{F}]

commutes, where k and l are the maps from the comparison iteration.

Since F has the coherence property, the least point of disagreement between the extender sequences of the two ultrapowers is greater than λ, so that k and l have critical points greater than λ. Using this, we may argue that in fact F = E⃗(κ, λ).

♦

Lemma 3 (Condensation): For θ some large regular cardinal, let X ≺ H_{θ} be a structure with V_{κ+1} ⊆ X and E⃗ ∈ X. Let X collapse to M and let E⃗ collapse to F⃗. Then for all η < oF⃗ (κ) we have F⃗ (κ, η) = E⃗(κ, η).

Proof: V and M agree to rank κ + 1, and below critical point κ their sequences agree. Use this to argue (by induction on η) that F⃗ (κ, η) is an extender with the same coherence properties as E⃗(κ, η), so that by uniqueness the two are equal.

♦

Lemma 4: GCH holds.

Proof: By induction on the cardinal λ. Suppose we have established GCH for cardinals less than λ, and notice that |H_{λ}| = λ. Now there are two cases.

1. For all α < λ we have o(α) < λ^{+}.

Then any subset of λ can be built into an iterable M ⊇ H_{λ} of cardinality λ, whose extender sequence agrees with E⃗ at critical points less than λ. If two such models M and N are compared then the iterations on each side fix the powerset of λ, hence one of Pλ ∩ M and Pλ ∩ N is a subset of the other. So the powerset of λ is the union of a chain of sets each of cardinality λ, so 2^{λ} = λ^{+}.

2. For some α < λ we have o(α) ≥ λ^{+}.

We claim that if A ⊆ λ, then A ∈ Ult(V, E⃗(α, β)) for some β < λ^{+}.

To see this, let us build A into X ≺ θ (θ large and regular) such that H_{λ} ⊆ X, E⃗ ∈ X, |X| = λ. Collapsing we get an iterable M = L_{μ}[F⃗] where F⃗ agrees with E⃗ at critical points below α, and (by condensation) F⃗ is an initial segment of E⃗ at critical point α. Let β = oF⃗ (α), then λ < β < λ^{+}.

Now compare M and Ult(V, E⃗(α, β)). Let the iteration maps be

i : M ⟶ M^{∗}
j : Ult(V, E⃗(α, β)) ⟶ N

Both have critical points greater than λ. It may be argued that M^{∗} is an initial segment of N, for otherwise we could iterate M^{∗} to get indiscernibles for N, and hence for V. M can be recovered from M^{∗} so M ∈ N, then by agreement M ∈ Ult(V, E⃗(α, β)).

This proves the claim. Now we are done, because in Ult(V, E⃗(α, β)) there are at worst 2^{α}.λ = λ subsets of λ.

♦

The claim in case 2 of the last proof is also the key to the hypermeasurability of κ in L[E⃗].

Lemma 5: Let o(κ) > κ^{++}. Let E = E⃗(κ, κ^{++}). Let j_{E} : V ⟶ M be the ultrapower by E. Then V_{κ+2} ⊆ M.

Proof: By GCH, it is enough to show Pκ^{+} ⊆ M. If A ⊆ κ^{+}, the claim from the last lemma shows that for some β < κ^{++} we have A ∈ Ult(V, E⃗(κ, β)). By coherence E⃗(κ, β) = j(E⃗)(κ, β), and we also have that Ult(V, E⃗(κ, β)) and Ult(M, E⃗(κ, β)) agree to a high rank. So A ∈ Ult(M, E⃗(κ, β)) ⊆ M.

♦

3 Core models

We look briefly at the theory of (non-overlapping) core models, which was developed in tandem with the inner model theory of the last chapter by the same authors (Mitchell, Jensen, Dodd, Koepke). The reader is referred to [5] for details. Our goal is to show that if V = L[E⃗] then the construction of the core model and maximal sequence as in [5] will recover V and E⃗. Combined with some facts about core models, this will let us do an analysis of elementary embeddings (in particular ultrapower maps) definable in certain generic extensions of L[E⃗].

The reader who is prepared to take our use of core models on trust should skip ahead to the results at the end of the section, which are all that will be used subsequently.

Claim 1: If V = L[E⃗], then there is no inner model of a strong cardinal.

Proof: [Sketch] If there is such an inner model, then there is an inner model of the form L[F⃗], where F⃗ is a sequence with similar properties to those in definition 2, but such that for some κ we have oF⃗ (κ) = ∞. That is to say, κ is the largest critical point on F⃗ and F⃗ has (κ, λ)-extenders for λ arbitrarily large.

It can be shown that the comparison of this model with V will terminate. This must lead to a situation in which an iterate M of V is a proper initial segment of an iterate N of L[F⃗], because the extender sequence E⃗ does not have unboundedly large extenders on any critical point. But were this situation to arise, we could iterate N to generate indiscernibles for M, so that V would be able to compute a set of indiscernibles for V. Contradiction.

♦

Theorem 2: Let V = L[E⃗]. If we perform the construction of the canonical maximal sequence F⃗ and core model K[F⃗] as in [5] then F⃗ = E⃗ and K = V.

Proof:

It is enough to show that F⃗ = E⃗, for then

V ⊇ K = K[F⃗] ⊇ L[F⃗] = V.

We will proceed by induction. Suppose that (κ, ν) ∈ dom(E⃗) and we have established that

E⃗ ↾ (κ, ν) = F⃗ ↾ (κ, ν).

Claim 2: Pκ ⊆ K[F⃗ ↾ (κ, ν)].

To see this let X ⊆ κ. Then X ∈ Ult(V, E⃗(κ, ν)), and this shows that X is in the lower part of an iterable premouse over E⃗ ↾ (κ, ν) = F⃗ ↾ (κ, ν). Hence X ∈ K[F⃗ ↾ (κ, ν)] as claimed.

The model K[F⃗ ↾ (κ, ν)] agrees with V to rank κ + 1, their extender sequences are the same, so by the Uniqueness Lemma we are forced to choose F⃗ (κ, ν) = E⃗(κ, ν).

To complete an inductive proof that F⃗ = E⃗ we need to show that the construction of F⃗ cannot go wrong in either of the following ways; by putting an extender at (κ, oE⃗(κ)), or by putting an extender on a critical point between κ and the next λ with oE⃗(λ) > λ + 1.

Claim 3: If ν = o(κ) then there is no (κ, ν)-extender in L[E⃗] which coheres E ↾ (κ, ν).

The proof of the claim is very similar to that of the Uniqueness Lemma. If it fails, let E⃗ be a minimal counterexample. Let E be the first (κ, ν) extender providing a counterexample, and compare the models V and Ult(V, E). We get a diagram

[figure: N; V; Ult(V, E)]

which commutes exactly as in the Uniqueness Lemma. But this cannot be because V and Ult(V, E) agree to such an extent that the comparison maps have critical point larger than κ, while the map from V to Ult(V, E) has critical point κ.

It follows from this that F⃗ cannot put an extender at (κ, oE⃗(κ)). The point is that V agrees with K[F⃗ ↾ (κ, oE⃗(κ))] to rank κ + 1, the extender sequences agree, so that such an extender would be a counterexample to the previous claim.

The last ingredient in the proof that F⃗ = E⃗ is the following claim.

Claim 4: If ν = oE⃗(κ) then there does not exist an extender E ∈ L[E⃗] which coheres E⃗ ↾ (κ, ν) and has critical point between κ and the next λ with oE⃗(λ) > λ + 1.

The proof is very similar to that of the last claim. Assume that E⃗ is a minimal counterexample, with E the first extender witnessing this. Coiterate as in the previous lemma, to get a commutative triangle exactly as before. Again there is a problem with critical points.

The proof is now routine, an induction on dom(E⃗) shows that we have dom(F⃗) = dom(E⃗) and F⃗ = E⃗. Hence V = L[E⃗] = K[F⃗] and we are done.

♦

We now recall some results about the non-overlapping core model, which are true under the assumption there is no inner model of a strong cardinal.

Fact 1: Let F⃗ be the canonical maximal sequence, and K[F⃗] the associated core model.

1. If π : K[F⃗] ⟶ W is an elementary embedding into a transitive class W then W is a normal iteration of K[F⃗] by F⃗.

2. If G is set-generic, then F⃗ and K[F⃗] as defined in V [G] coincide with F⃗ and K[F⃗] as defined in V.

Putting everything together, we can finally get the result that we need to analyse measures in generic extensions of L[E⃗].

Theorem 3: Let V = L[E⃗], where E⃗ is an extender sequence. Let G be setgeneric over V and let U ∈ V [G] be some normal measure on some cardinal κ. If j_{U} : V [G] ⟶ Ult(V [G], U) is the ultrapower map associated with U, then j_{U} ↾ V : V ⟶ j_{U}(V) is a normal iteration of V by E⃗.

Proof: The core of V [G] equals the core of V equals V.

♦

4 The model

Theorem 4: Let GCH hold, and let j : V ⟶ M be such that V_{κ+3} ⊆ M, where κ = crit(j). Suppose ^{1} that j is generated by a (κ, κ^{+++})-extender.

Let ℙ be the Reverse Easton iteration of length κ+1 in which we force with Add(α, α^{+})×Add(α^{+}, α^{+++}) (as defined in V [G_{α}]) at each strong inaccessible α ≤ κ. Let G be ℙ-generic over V.

Then in the generic extension κ is P_{2}κ-hypermeasurable, and we have 2^{κ} = κ^{+} and 2^{κ^{+}} = κ^{+++}.

Proof: We will use a number of forcing tricks without much explanation; the reader is referred to [2] for the details, and to [1] for general information about Reverse Easton forcing. Several similar arguments may also be found in [3] and [4].

1If such a j exists, then there is such a j which arises from such an extender.

We follow the standard strategy of building in V [G] an embedding which extends j, and which witnesses that κ has the desired property in V [G].

We shall adopt the convention that Add(α, β) adds β generic functions from α to α. Often we think of the generic object as a single function from α × β to α.

We let G_{α} denote the generic object for the forcing up to stage α, and g_{α} the generic for the forcing at stage α. We break up g_{α} as g_{α}^{0} × g_{α}^{1} where g_{α}^{0} is generic for Add(α, α^{+}) and g_{α}^{1} is generic for Add(α^{+}, α^{+++}). So G = G_{κ} ∗ g_{κ} = G_{κ} ∗ g_{κ}^{0} ∗ g_{κ}^{1}.

As an aid to constructing generic objects, we factor j through the ultrapower by a normal measure in the standard way. This gives a diagram

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

where as usual U = { X ⊆ κ | κ ∈ j(X) }, i is the ultrapower map from V to M_{0} = Ult(V, U), and k : [F]_{U} ⟼ j(F)(κ).

Let λ = (κ^{+++})_{M_{0}}. We note for later use that

M = { j(F)(a) | F ∈ V, dom(F) = [κ]^{<ω}, a ∈ [κ^{+++}]^{<ω} }
= { k(G)(b) | G ∈ M_{0}, dom(G) = [λ]^{<ω}, a ∈ [κ^{+++}]^{<ω} }

As usual, in V [G_{κ}] the models M[G_{κ}] and M_{0}[G_{κ}] are closed under κsequences. As a corollary we may use g_{κ} as the generic at stage κ in the iteration j(P). In i(P) we may use

g_{κ}^{∗} = g_{κ}^{0} × (g_{κ}^{1} ↾ κ^{+} × λ).

In M_{0}[G_{κ}] we define a certain term poset. The terms are names in the forcing Add(κ, κ^{+}) × Add(κ^{+}, λ) for elements of the forcing

R_{κ+1,i(κ)} ∗ Add(i(κ), i(κ^{+}))).

By the standard chain condition and closure arguments we may find X ∈ V [G_{κ}] generic for this term forcing. k : M_{0} ⟶ M extends easily to a map k : M_{0}[G_{κ}] ⟶ M[G_{κ}], and we may transfer X along k to get a generic X^{+} for the term forcing whose elements are names in Add(κ, κ^{+})×Add(κ^{+}, κ^{+++}) for elements of R_{κ+1,j(κ)} ∗ Add(j(κ), j(κ^{+}))).

Now we interpret this term generic using g_{κ}. This leaves us with

H ∗ a ∈ V [G_{κ}][g_{κ}]

which is generic over M[G_{κ}][g_{κ}] for R_{κ+1,j(κ)} ∗ Add(j(κ), j(κ^{+}))). It is now easy to build a map

j : V [G_{κ}] ⟶ M[G_{κ}][g_{κ}][H].

Now we alter a to get a new generic a^{∗} such that j“g_{κ}^{0} ⊆ a^{∗}. This works because M[G_{κ}][g_{κ}] is closed under κ-sequences inside V [G_{κ}][g_{κ}], and we only need alter any one condition at κ many places.

This enables us to build

j : V [G_{κ}][g_{κ}^{0}] ⟶ M[G_{κ}][g_{κ}][H][a^{∗}].

Now we may finish by transferring g_{κ}^{1} along j, which is legitimate because g_{κ}^{1} is generic over V [G_{κ}][g_{κ}^{0}] for κ^{+}-dense forcing. This gives an embedding

j : V [G] ⟶ M[j(G)]

defined in V [G].

As the cardinal arithmetic claim is trivial, we need only verify that this embedding witnesses P_{2}κ-hypermeasurability. As 2^{κ} = κ^{+} it suffices to check that

Pκ^{+} ∩ V [G] ⊆ M[j(G)].

But Pκ^{+} ∩ V [G] ⊆ M[G] by a standard chain condition argument, and M[G] ⊆ M[j(G)] by construction.

♦

5 Building measures

In this section we show how to construct measures in the generic extension. The methods used here are very similar to those of [3].

Theorem 5: Let i : V ⟶ N be an elementary embedding such that

1. κ = crit(i).

2. i(κ^{+}) < κ^{++}.

3. N = { i(f)(a) | f ∈ V, dom(f) = [κ]^{<ω}, a ∈ [i(κ)]^{<ω} }.

4. ^{κ}N ⊆ N.

Let ℙ be the forcing iteration defined in the last section, and let G be ℙgeneric.

Then in V [G] there are κ^{+++} ultrafilters U with the property that the ultrapower map j_{U} : V [G] ⟶ Ult(V [G], U) is an extension of i.

Proof:

Let G_{α}, g_{α}, g_{α}^{i} be as in the proof of theorem 4. The iterations ℙ and i(ℙ) agree up to κ, and V [G_{κ}] ⊨ ^{κ}N[G_{κ}] ⊆ N[G_{κ}]. At κ, the iteration i(ℙ) requires a generic over N[G_{κ}] for

Q = Add(κ, κ^{+}) × Add(κ^{+}, η)

where η = (κ^{+++})_{N}, and by closure we can see Q as an initial segment of the forcing which ℙ does at κ.

Claim 5: In V [G] there are exactly κ^{+++} generics h for Q over N[G_{κ}] with the property that

V [G] ⊨ ^{κ}N[G_{κ}][h] ⊆ N[G_{κ}][h].

Proof: Q has cardinality κ^{+} and 2^{κ^{+}} = κ^{+++} in V [G], so there are at most κ^{+++} such generics. We know that (in ordinal arithmetic) κ^{+++} = η.κ^{+++}, so we can break up g_{κ}^{1} into blocks of length η; let h_{α}^{1} be the α’th block. It is easily checked that h = g_{κ}^{0} ∗ h_{α}^{1} is generic (by Easton’s lemma) and has the desired property (as all κ-sequences from V [G] are actually in V [g_{κ}][g_{κ}^{0}]).

The claim is proved

♦

Now fix a generic h as above. Using the closure of N[G_{κ}][h] in V [G], we may prove by the standard methods the following claim.

Claim 6: In V [G] there are exactly κ^{+++} generics H for the forcing ℝ_{κ+1,i(κ)} over the model N[G_{κ}][h].

The key point here is that i(κ) has cardinality κ^{+}, so that we may enumerate the dense sets we must meet in order type κ^{+}; the closure hypothesis then enables us to meet them, and we have so much latitude in the construction that we may do it in 2^{κ^{+}} = κ^{+++} different ways.

Fix such an H. It is easy to extend i to an embedding

i : V [G_{κ}] ⟶ N[G_{κ}][h][H].

Notice also that V [G] ⊨ ^{κ}N[G_{κ}][h][H] ⊆ N[G_{κ}][h][H].

Claim 7: In V [G] there are exactly κ^{+++} generics a for the notion of forcing Add(i(κ), i(κ^{+})) over the model N[G_{κ}][h][H], with the following properties.

1. i“g_{κ}^{0} ⊆ a.

2. Let i : V [G_{κ}][g_{κ}^{0}] ⟶ N[G_{κ}][h][H][a] be the natural extension of i. Then for each β < i(κ) there is g : κ ⟶ κ in V [G_{κ}][g_{κ}^{0}] such that i(g)(κ) = β.

Proof: Just as in the construction of H above, we may use closure to build many generics a, using the fact that i(κ^{+}) < κ^{++}. We shall now describe how to alter such a generic a so as to get the properties demanded.

Fix an enumeration of i(κ) in order type β, say ⟨f(β) : β < κ^{+}⟩. Now define a^{∗} by altering a on the set (κ + 1) × i“κ^{+}, while leaving it unchanged elsewhere. For all β < κ^{+} let

a^{∗}(α, i(β)) = g_{κ}^{0}(α, β)

if α < κ, and

a^{∗}(κ, i(β)) = f(β).

a^{∗} gives rise to a generic filter; the key points are that each condition need only be altered at κ many places, and that the forcing is so highly closed that small perturbations do not kill genericity. We identify a^{∗} with the associated filter.

It is clear from the construction that i“g_{κ}^{0} ⊆ a^{∗}, so that as usual we may build

i : V [G_{κ}][g_{κ}^{0}] ⟶ N[G_{κ}][h][H][a^{∗}].

To see that a^{∗} has the other property demanded of it, define g_{β} ∈ V [G_{κ}][g_{κ}^{0}] by g_{β}(α) = g_{κ}^{0}(α, β). Now i(g_{β})(κ) = a^{∗}(κ, i(β)) = f(β).

It is easy to check that we may do this construction in κ^{+++} many ways. The claim is proved.

♦

Fix a as in the previous claim. It is now routine to transfer g_{κ}^{1} along i, to get at last a map

i : V [G] ⟶ N[G_{κ}][h][H][a][i(g_{κ}^{1})].

We will be done once we have the following claim.

Claim 8: The map i : V [G] ⟶ N[G_{κ}][h][H][a][i(g_{κ}^{1})] arises as the ultrapower map associated with an ultrafilter U. Moreover, different ultrafilters arise from different choices for (h, H, a).

Proof: It is easy to see that

N[G_{κ}][h][H][a][i(g_{κ}^{1})] = { i(f)(a) | f ∈ V [G], a ∈ [i(κ)]^{<ω} }.

Each β < i(κ) has form i(g)(κ) for some g ∈ V [G], so actually

N[G_{κ}][h][H][a][i(g_{κ}^{1})] = { i(f)(κ) | f ∈ V [G], dom(f) = κ }.

It follows that i arises from an ultrafilter U. For the other claim, simply notice that we can rebuild (h, H, a) from U in V [G] because

G_{κ} ∗ h ∗ H ∗ a = j_{U}(G_{κ} ∗ g_{κ}^{0}).

♦

The theorem is proved.

♦

For our later convenience, we introduce some notation for the ultrafilters that arise in this way. Let V and G be as in the last result.

Definition 3: In V [G], suppose that

1. i : V ⟶ N is an embedding with crit(i) = κ.

2. h is a generic over V [G_{κ}] for Q, and V [G] ⊨ ^{κ}N[G_{κ}][h] ⊆ N[G_{κ}][h].

3. H is generic over N[G_{κ}][h] for ℝ.

4. a is generic over N[G_{κ}][h][H] for Add(i(κ, κ^{++})) and i“g_{κ}^{0} ⊆ a.

5. The unique map

i : V [G] ⟶ N[G_{κ}][h][H][a][i(g_{κ}^{1})]

extending i : V ⟶ N and having i(G) = G_{κ} ∗ h ∗ H ∗ a ∗ i(g_{κ}^{1}) is the ultrapower by a normal measure.

Then U(i, h, H ∗ a) is the normal measure alluded to in the last clause.

6 Closure

In the construction of theorem 5 for generating ultrafilters, it was crucial that the target model of the embedding i be closed under κ-sequences, where κ = crit(i). In this section we prove some results about the closure of the target model of an elementary embedding. We then prove a theorem which says essentially that we need not worry about embeddings with bad closure, when we come to classifying the measures on κ in our final model.

Theorem 6: Let V = L[E⃗], where E⃗ is an extender sequence. Suppose κ is such that (κ, κ^{++}) ∈ dom(E⃗). Then there are many η < κ^{++} such that the ultrapower of V by E⃗(κ, η) is not closed under ω-sequences.

Proof: Let E = E⃗(κ, κ^{++}). Since V_{κ+2} ⊆ Ult(V, E) and GCH holds, we have that for every β < κ^{++}

E ↾ [β]^{<ω} ∈ Ult(V, E). Fix some large θ. We build a chain ⟨X_{α} : α < κ^{++}⟩ of elementary sub-

structures X_{α} ≺ H_{θ} with the properties 1. E⃗, κ, κ^{++} ∈ X_{0}.

2. V_{κ+1} ⊆ X_{0}.

3. X_{α} ⊆ X_{α+1} for all α, X_{λ} = ⋃_{α<λ} X_{α} for limit λ.

4. |X_{α}| = κ^{+} for all α.

5. X_{α} ∩ κ^{++} ∈ κ^{++}.

6. X_{α} ∩ κ^{++} < X_{α+1} ∩ κ^{++}, for all α.

Now let η_{α} = X_{α} ∩ κ^{++}. If X_{α} is collapsed to a transitive structure M_{α}, then κ^{++} collapses to η_{α} = (κ^{++})_{M_{α}}. What is more the Condensation Lemma implies that E collapses to E_{α} = E⃗(κ, η_{α}).

By elementarity,

M_{α} ⊨ ∀β < η_{α} E_{α} ↾ [β]^{<ω} ∈ Ult(M_{α}, E_{α}). As V_{κ+1} ⊆ M_{α} this implies that in fact

∀β < η_{α} E_{α} ↾ [β]^{<ω} ∈ Ult(V, E_{α}). Now let α be any ordinal in κ^{++} of cofinality ω. By construction the

sequence η⃗ is continuous, so η_{α} has cofinality ω. We claim that Ult(V, E_{α}) is not closed under ω-sequences. For were it so, we would have E_{α} ∈ Ult(V, E_{α}); but this cannot be, as an extender can never be in its own ultrapower [6].

♦

Next we prove a result providing target models which are closed under κ-sequences. We need a preliminary lemma.

Lemma 6: Let E be a (κ, η) extender, let M = Ult(V, E). Then ^{κ}M ⊆ M if and only if ^{κ}η ⊆ M.

Proof: One direction is trivial. For the other, suppose that ^{κ}η ⊆ M. Let ⟨j_{E}(F_{α})(a_{α}) : α < κ⟩ be some κ-sequence of elements of M, where for each α < κ we have dom(F_{α}) = [κ]^{<ω} and a_{α} ∈ [η]^{<ω}. Let F (α, x) = F_{α}(x), and observe that j_{E}(F) ∈ M and by hypothesis ⟨a_{α} : α < κ⟩ ∈ M. Now if α < κ

j_{E}(F)(α, a_{α}) = j_{E}(F_{α})(a_{α}),

so that the κ-sequence ⟨j_{E}(F_{α})(a_{α}) : α < κ⟩ is a member of M.

♦

We notice some useful corollaries.

Corollary 1: If V = L[E⃗] where E⃗ is an extender sequence, α < β < o(κ), then V and Ult(V, E⃗(κ, β)) agree as to whether the ultrapower by E⃗(κ, α) is closed under κ-sequences.

Proof: Ult(V, E⃗(κ, α)) and Ult(Ult(V, E⃗(κ, β)), E⃗(κ, α)) agree to rank α+1, so one model contains ^{κ}α exactly when the other model does.

♦

Corollary 2: If i is a normal iteration of V, E is a (κ, η)-extender which is the first extender applied in i, and the target model of i is closed under κ-sequences, then Ult(V, E) is closed under κ-sequences.

Proof: If not, then ^{κ}η is not contained in Ult(V, E). But Ult(V, E) and the target model of i agree to rank η + 1, so this is absurd.

♦

Theorem 7: Let V = L[E⃗], where E⃗ is an extender sequence. Suppose κ is such that (κ, κ^{++}) ∈ dom(E⃗). Then there are cofinally many η < κ^{++} such that the ultrapower of V by E⃗(κ, η) is closed under κ-sequences.

Proof: Let X_{α}, M_{α}, E_{α} be as in the proof of the last theorem, with the additional proviso that ^{κ}X_{α} ⊆ X_{α} for successor ordinals α. Let α < κ^{+} be a point of cofinality κ^{+}. It is easy to see that ^{κ}η_{α} ⊆ M_{α}. By elementarity

M_{α} ⊨ ^{κ}η_{α} ⊆ Ult(M_{α}, E_{α}),

so by the same argument as was used in the last theorem ^{κ}η_{α} ⊆ Ult(V, E_{α}). So ^{κ}Ult(V, E_{α}) ⊆ Ult(V, E_{α}).

♦

The next result will be used to show that we need not be concerned with these non-closed ultrapowers when it comes to classifying the measures in our final model.

Theorem 8: Let κ be some measurable cardinal, let ℙ be the forcing iteration of section 4.

Let j : V ⟶ M be an embedding into an inner model M, and suppose κ = crit(j). Suppose that G is ℙ-generic, and in V [G] there is a measure U on κ such that j_{U}^{V [G]} ↾ V = j.

Then V ⊨ ^{κ}M ⊆ M.

Proof: If f ∈ V, f : κ ⟶ κ, then f = j(f) ↾ κ so that f ∈ M.

Now fix g ∈ V, g : κ ⟶ ON. g ∈ V [G], so by the standard closure facts about ultrapowers g ∈ Ult(V [G], U). Ult(V [G], U) is M[H], where H is generic for j(ℙ). By the usual arguments g ∈ M[h], where h is the initial segment of H generic for a κ^{+}-c.c. initial segment of j(ℙ) (consisting of the part which adds subsets to α for α ≤ κ).

Now using the chain condition in the standard way, we may find in M a function G : κ × κ ⟶ ON such that

∀α ∃β g(α) = G(α, β).

As g ∈ V and M ⊆ V we may find in V a function f : κ ⟶ κ such that g(α) = G(α, f(α)) for all α. By the remark at the beginning of the proof f ∈ M, and hence g ∈ M.

♦

7 Classification and analysis of measures

With the machinery of the preceding sections in place, we are finally ready to classify the normal measures on κ and the Mitchell ordering between them in a certain model where κ is P_{2}κ-hypermeasurable and 2^{2^{κ}} > (2^{κ})^{+}.

Theorem 9: Let V = L[E⃗], where for some κ we have (κ, κ^{++}) ∈ dom(E⃗). Let ℙ be the forcing iteration of length κ + 1 defined in section 4, and let G be ℙ-generic over V.

Then in the model V [G]

1. κ is P_{2}κ-hypermeasurable.

2. 2^{κ} = κ^{+}, 2^{κ^{+}} = κ^{+++}.

3. Every normal measure on κ has form U(i, h, H ∗ a), where i : V ⟶ N is a finite normal iteration of V by E⃗, V ⊨ ^{κ}N ⊆ N, and i(κ) < κ^{++}.

Proof: This is mostly a summary of results already proved. The third claim is the only one with any novelty.

Let U ∈ V [G] be a measure on κ in V [G], giving rise to an embedding j_{U} : V [G] ⟶ Ult(V [G], U). Let i = j ↾ V, and let

j_{U}(G) = G_{κ} ∗ h ∗ H ∗ a ∗ j_{U}(g_{κ}^{1}).

As V = L[E⃗], the results from the section on core model theory show that i is an iteration of V by E⃗.

Claim 9: i is a finite iteration.

Proof: If not, then it is easy to see that the sequence x⃗ consisting of the first ω critical points is not a member of the target model N. By closure of the forcing, x⃗ cannot be in N[G_{κ}][h]. But i is defined in V [G] and V [G] ⊨ ^{κ}N[G_{κ}][h] ⊆ N[G_{κ}][h], so this is absurd and the claim is proved.

♦

It now follows from the section on closure of models that V ⊨ ^{κ}N ⊆ N. j_{U} is an ultrapower map so i(κ) = j_{U}(κ) < κ^{++}. j_{U} is an extension of i and j_{U}(G) = G_{κ} ∗ h ∗ H ∗ a, so it must be the case that U = U(i, h, H ∗ a).

♦

Notice that whenever i is an iteration as above and (h, H, a) are appropriate generics then U(i, h, H ∗ a) is a normal measure in V [G]. So we have a complete classification of the measures in V [G].

It remains to analyse the ordering ⊲ on these measures. Unfortunately the analysis here is not as complete as was achieved in [3]

We start with a general analysis of subsets of κ^{+} in V [G]. Observe that

• An ultrafilter on κ in V [G] may be coded as a subset of κ^{+}, since 2^{κ} = κ^{+} in V [G].

• κ is P_{2}κ-hypermeasurable in V [G], so we should expect that each subset of κ^{+} should appear in the ultrapower of V [G] by some normal measure on κ.

If x ⊆ κ^{+} with x ∈ V [G], then by the usual chain condition arguments x ∈ V [G_{κ}][g_{κ}^{0}][h], where h is Add(κ^{+}, 1)-generic and for some δ < κ^{+++} we have h ∈ V [G_{κ}][g_{κ}^{0}][g_{κ}^{1} ↾ δ × κ]. What is more x = τ̇^{G_{κ}∗g_{κ}^{0}∗h}, where τ̇ can be construed as a subset of κ^{+}.

By the analysis we did in section 2, τ̇ is in Ult(V, E⃗(κ, η)) for all sufficiently large η < κ^{++}. From the results in section 6, we may as well take it that τ̇ ∈ N = Ult(V, E⃗(κ, η)) where V ⊨ ^{κ}N ⊆ N. That is x ∈ N[G_{κ}][g_{κ}^{0}][h].

We may now use the methods of of section 5 to expand this model to one of the form Ult(V [G], U) for some measure U.

This analysis leaves some questions open:

1. Given ζ < κ^{++}, is there some x which only appears in extensions of Ult(V, E⃗(κ, η)) as above for η > ζ?

2. if x appears in some such extension, what can we say about the set of extensions in which it appears?

Some information about these questions will come from a more specific analysis of measures.

In what follows, let i and j be two finite normal iterations of V by E, with target models closed under κ-sequences, and with i(κ) and j(κ) both less than κ^{++}. Let j_{01} : V ⟶ N_{1} be the first step in the iteration j.

Let

U_{0} = U(i, h_{0}, H_{0} ∗ a_{0})
U_{1} = U(j, h_{1}, H_{1} ∗ a_{1})

for appropriate generics (h_{0}, H_{0}, a_{0}) and (h_{1}, H_{1}, a_{1}).

Lemma 7: V [G] ⊨ U_{0} ⊲ U_{1} iff U_{0} ∈ N_{1}[G_{κ}][h_{1}].

Proof: N_{1} and j(V) agree to a high rank, G_{κ} ∗ h_{1} is generic over j(V) for small forcing, so it is generic over N_{1} also and the extensions j(V)[G_{κ}][h_{1}] and N_{1}[G_{κ}][h_{1}] agree to a high rank. The usual closure arguments show that j(V)[G_{κ}][h_{1}] and Ult(V [G], U_{1}) agree to a high rank, so that the models Ult(V [G], U_{1}) and N_{1}[G_{κ}][h_{1}] agree to a high rank, in particular they contain the same measures on κ.

♦

Theorem 10: If U_{0} ⊲ U_{1} then

1. h_{0}, H_{0} ∈ N_{1}[G_{κ}][h_{1}].

2. i ↾ N_{1} is an iteration of N_{1} by j_{01}(E⃗).

Proof: Let N^{∗} = N_{1}[G_{κ}][h_{1}]. We know that U_{0} ∈ N^{∗} and V [G] ⊨ ^{κ}N^{∗} ⊆ N^{∗}, so that j_{U_{0}}^{V [G]} ↾ N^{∗} = j_{U_{0}}^{N^{∗}}.

For the first claim observe that j_{U_{0}}(G_{κ}) = G_{κ} ∗ h_{0} ∗ H_{0}, so that N^{∗} can reconstruct h_{0} and H_{0}.

For the second claim, we know that N_{1} is the core of N^{∗} and that N^{∗} can define j_{U_{0}} ↾ N^{∗}, so j_{U_{0}} ↾ N_{1} must be an iteration of N_{1}. But N_{1} ⊆ V and j_{U_{0}} ↾ V is i, so i ↾ N_{1} is an iteration of N_{1}.

♦

The following result is proved in [3] in the context of measures, and the proof works equally well for iterations of extenders in the context of L[E⃗].

Theorem 11: Let j_{01} : V ⟶ N_{1} be the ultrapower map associated with the extender E⃗(κ, β). Let i be a normal iteration with critical point κ, in which i_{0n}(E⃗)(κ_{n}, λ_{n}) is applied to M_{n} = i_{0n}(V) at stage n. Then i ↾ N_{1} is an internal iteration of N_{1} if and only if for all n either κ_{n} < i_{0n}(κ), or κ_{n} = i_{0n}(κ) and λ_{n} < i_{0n}(β).

This answers one of the questions we posed about subsets of κ^{+} earlier on. If an ultrafilter U arises from an iteration which starts off with E(κ, α), and U ⊲ V, then the iteration associated with V must start with E(κ, β) for β > α.

We can also say something about the measures on κ which occur in models of the form N_{1}[G_{κ}][h_{1}]

Theorem 12: Suppose N = Ult(V, E⃗(κ, η)) for some η < κ^{++}, ^{κ}N ⊆ N, and h is generic over N[G_{κ}] for Add(κ, κ^{+}) × Add(κ^{+}, κ^{+++}) as computed in N[G_{κ}]. If i is a finite internal iteration of N which has a target closed under κ-sequences, and i(κ) < (κ^{++})_{N}, then there are κ^{+} measures U ∈ N[G_{κ}][h] such that j_{u} ↾ N = i.

Proof: The analysis we have done for measures in the model V works equally well for the model N. There are κ^{+} measures for each i because |κ_{N}^{+++}| = κ^{+} in V.

♦

We can finally derive some consequences for the structure of the Mitchell ordering in V [G].

Definition 4: Let U ∈ V [G] be a normal measure on κ. Then the level of U is α, where E⃗(κ, α) is the first extender applied in the iteration of V induced by U. The block of U is h, where h is the generic object at κ in the forcing iteration j_{U}(G_{κ}).

Notice that if U has level α and block h, the ⊲-ancestors of U are precisely the measures in Ult(V, E⃗(κ, α))[G_{κ}][h].

The following theorem is just a summary of what has been proved.

Theorem 13: In the model V [G] 1. If U ⊲ V, the level of U is less than the level of V.

2. If U ⊲ V, U ⊲ W for all W with the same level and block as V.

3. There are precisely κ^{+} measures with a prescribed block and level.

4. There are κ^{++} possible levels.

5. For each level, there are κ^{+++} possible blocks.

6. The ultrafilters of a prescribed level and block have at most κ^{+} ancestors.

7. For each U, there is a final segment of levels containing a block which has U as ancestor.

References

[1] J. E. Baumgartner, Iterated forcing. In Surveys in set theory, pp 1–55, Cambridge, 1983. Cambridge University Press.

[2] J. Cummings, A model in which GCH holds at successors but fails at limits. Transactions of the AMS, 329:1–39, 1992.

[3] J. Cummings, Possible behaviours for the Mitchell ordering I. To appear in Annals of Pure and Applied Logic.

[4] J. Cummings, Strong ultrapowers and long core models. Journal of Symbolic Logic 58, 1993.

[5] P. Koepke, An introduction to extenders and core models for extender sequences. In Logic Colloquium ’87, pp 137–182, Amsterdam, 1989. North-Holland.

[6] D. Martin and J. Steel, Iteration trees. To appear.

[7] W. Mitchell, Sets constructible from sequences of ultrafilters. Journal of Symbolic Logic, 39:57–66, 1974.

[8] W. Mitchell, Hypermeasurable cardinals. In Logic Colloquium ’78, pp 303–316, Amsterdam, 1979. North-Holland.

[9] Solovay, Reinhardt and Kanamori, Strong axioms of infinity and elementary embeddings. Annals of Mathematical Logic, 13:73–116, 1978.

[10] Solovay, Berkeley course notes.