Coherent sequences versus Radin sequences

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

          cummings@math.huji.ac.il

February 25, 1998

Abstract

We attempt to make a connection between the sequences of measures used to define Radin forcing and the coherent sequences of extenders which are the basis of modern inner model theory. We show that in certain circumstances we can read off sequences of measures as defined by Radin from coherent sequences of extenders, and that we can define Radin forcing directly from a coherent extender sequence and a sequence of ordinals; this generalises Mitchell’s construction of Radin forcing from a coherent sequence of measures.

1 Introduction

This paper was inspired by the following questions:

1. What is the relationship between Radin’s version of Radin forcing, as described in [5], and Mitchell’s as described in [3] ?

2. Let V = L[E⃗] where E⃗ is a coherent non-overlapping sequence of extenders, let j : V ⟶ M be the embedding arising from E⃗(κ, β), and following Radin let a sequence u be defined by setting

u(0) = κ,

and then

u(α) = { X ⊆ V_{κ} | u ↾ α ∈ j(X) },

for as long as u ↾ α ∈ M.

If α > 0 then u(α) is a κ-complete ultrafilter on V_{κ}. What is the relationship between the measures u(α) and the extenders on the sequence E⃗?

We will give a partial answer to question 2 in the case that β = κ^{++}, and the form that the answer takes will give us the clue to a partial answer for question 1. We give a brief review of the facts about extenders and L[E⃗]-models that we shall use.

Let κ be a cardinal, λ an ordinal. E is said to be a (κ, λ)-extender if E = ⟨E_{a} : a ∈ [λ]^{<ω}⟩ where

1. E_{a} is a κ-complete ultrafilter on [κ]^{|a|}.

2. The ultrafilters E_{a} have a certain coherence property, enabling us to form a limit ultrapower

j_{E} : V ⟶ Ult(V, E).

3. Ult(V, E) is well-founded, so we may identify it with a transitive class M_{E}.

4. j_{E} has critical point κ.

5. λ < j_{E}(κ), and for a ∈ [λ]^{<ω} and X ⊆ [κ]^{|a|}

X ∈ E_{a} ⟺ a ∈ j_{E}(X).

The reader is referred to [1] for a full treatment of extenders. Our main interest here is in coherent non-overlapping extender sequences (hereafter referred to simply as “extender sequences”) and the associated inner models.

E⃗ is an 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. 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. If κ_{0} < κ_{1} and o(κ_{1}) > κ_{1} + 1 then o(κ_{0}) < κ_{1}.

The motivation for this definition is that if E⃗ is an extender sequence then L[E⃗] is a good inner model, having many properties in common with Kunen’s L[μ] and Mitchell’s L[U⃗], but capable of accommodating larger cardinals. In fact L[E⃗] models are the canonical inner models for cardinals below a strong cardinal, the theory of these models being due to Mitchell, Dodd and Jensen.

The key to the good properties of L[E⃗] is the Comparison Lemma, due to Mitchell, of which the following is a special case. Let E⃗ be a coherent sequence in L[E⃗] ^{1} and let F⃗ be coherent in L[F⃗]. Then we can find normal iterations of the two models, say i : L[E⃗] ⟶ M = L[E⃗_{M}] and j : L[F⃗] ⟶ N = L[F⃗_{N}], such that one of the sequences E⃗_{M} and F⃗_{N} is an initial segment of the other as far as sets in M ∩ N are concerned.

1With the definition we have adopted, if E⃗ is coherent then E⃗ ∩ L[E⃗] is coherent in L[E⃗].

For a proof of the Comparison Lemma in the case when E⃗ and F⃗ are sequences of measures, see [2]. Using the Comparison Lemma (in a version involving iterable set models) it may be shown that L[E⃗] enjoys the GCH, among other pleasant “L-like” properties.

2 Identifying the Radin measures

Let V = L[E⃗] where E⃗ is a coherent sequence. Let κ be a cardinal such that (κ, κ^{++}) ∈ dom(E⃗), and let E = E⃗(κ, κ^{++}). Mitchell showed in [4] that if j_{E} : V ⟶ M_{E} is the ultrapower by E then V_{κ+2} ⊆ M_{E}. This will mean that if we define a “Radin” sequence by the recursion

u(0) = κ,
u(α) = { X ⊆ V_{κ} | u ↾ α ∈ j_{E}(X) },

then the recursion runs for at least κ^{++} steps. This is significant because for some α < κ^{++} the measure u(α) will be a so-called weak repeat point, that is every set in u(α) has already been given measure one by some u(β) with β < α; if we let v = u ↾ α for such an α then the associated Radin forcing ℝ_{v} will preserve the measurability of κ.

We will characterise the measures u(α) in terms of the extenders on E⃗, for α belonging to a long initial segment of κ^{++}. We will use an idea from Mitchell’s paper [2], where in order to classify a normal measure V ∈ L[U⃗] he applies the Comparison Lemma to L[U⃗] and its ultrapower by V.

We could have used core model theory to do this, but that is not necessary here (and in any case the ideas in the core model proofs are very similar).

We start by identifying the measures u(1) and u(2), which will illustrate the ideas we need to do the general case.

2.1 The measure u(1)

Recall that u(1) is the ultrafilter on V_{κ} given by

X ∈ u(1) ⟺ ⟨κ⟩ ∈ j(X).

If M_{1} = Ult(V, u(1)), then it is a standard fact (easily checked) that M_{1} is the transitive collapse of the subclass of M given by

{ j(F)(κ) | dom(F) = V_{κ} },

and that if j_{1} : V ⟶ M_{1} is the ultrapower map then we have a commutative triangle

[figure: M_{1}; k; V; j_{1}; M; j]

with k being the inverse of the collapsing map.

We need to know the critical point of k. Since as usual Pκ ⊆ M_{1} we have κ^{+} = (κ^{+})_{M_{1}}, and since GCH holds in V we have by the usual arguments that (κ^{++})_{M_{1}} < j_{1}(κ) < κ^{++}. But V_{κ+2} ⊆ M, so that (κ^{++})_{M} = κ^{++}. The upshot of all this is that the critical point of k is (κ^{++})_{M_{1}}, an ordinal which we will denote by β_{1}.

Claim 1: For X ⊆ V_{κ}, X ∈ u(1) if and only if

⟨κ⟩ ∈ j_{E⃗(κ,β_{1})}(X).

Proof: By elementarity M_{1} = L[E⃗_{1}], where E⃗_{1} is a coherent sequence in M_{1}.

Let ϕ(α, F⃗) abbreviate the conjunction of the propositions:

1. F⃗ is a coherent sequence in L[F⃗].

2. In the sense of the model L[F⃗]:

(a) (α, α^{++}) ∈ dom(F⃗).

(b) If i : V ⟶ N is the ultrapower of V by F⃗ (α, α^{++}), i_{1} : V ⟶ N_{1} the ultrapower by { X | ⟨α⟩ ∈ i(X) }, and β = (α^{++})_{N_{1}}, then for

some X ⊆ V_{α} we have ⟨α⟩ ∈ i(X) and ⟨α⟩ ∉ j_{F⃗ (α,β)}(X).

The claim is proven if ϕ(κ, E⃗) is false, so assume for a contradiction that it is true.

We may assume without loss of generality that

• If G⃗ is a proper initial segment of E⃗, then ϕ(α, G⃗) is false for all α.

• ϕ(α, E⃗) is false for all α < κ.

In Mitchell’s terminology from [2], E⃗ is ϕ-minimal. Also, κ is definable from E⃗ as the least α such that ϕ(α, E⃗), a fact which we exploit later.

We co-iterate V = L[E⃗] and M_{1} = L[E⃗_{1}]; exactly as in [2] the ϕ-minimal choice of E⃗ guarantees that the co-iteration terminates with the same model on both sides, because it is impossible that one sequence be aligned with a proper initial segment of the other. Let N be the common iterate, with iteration maps i : V ⟶ N and l : M_{1} ⟶ N.

We claim that the diagram

[figure: N; M_{1}; l; V; i; j_{1}]

is commutative. Again we use an idea from [2]. Suppose it fails to commute, and let x be the minimal element (in the standard well-order of L[E⃗]) such that i(x) ≠ l(j_{1}(x)). The maps j_{1}, l and i are all definable from κ and E⃗, so since κ is definable from E⃗ we may conclude that x is definable from E⃗. If N = L[G⃗] and y denotes the result of working that definition in N from parameter G⃗, then clearly y = i(x) = l(j_{1}(x)) and we have a contradiction.

So we have a commutative diagram

[figure: N; M_{1}; l; k; V; i; j_{1}; M; j]

We inspect the first step of the comparison process; k(E⃗_{1}) = j(E⃗), and in M we have o^{j(E⃗)}(κ) = κ^{++}, because E⃗ is coherent and M was had by applying E⃗(κ, κ^{++}) to V. Therefore oE⃗_{1}(κ) = (κ^{++})_{M_{1}} = β_{1}. Since β_{1} is the critical point of k, the extenders E⃗(κ, λ) and E⃗_{1}(κ, λ) must coincide for κ < λ < β_{1}, so that (since we certainly have agreement on critical points less than κ) (κ, β_{1}) is the least point of disagreement between E⃗ and E⃗_{1}.

Thus, the first step in the comparison process is to apply E(κ, β_{1}) to V and do nothing to M_{1}. This handles the disagreement at κ, so that subsequent steps in the comparison will involve applying extenders with critical points greater than κ.

Suppose that X is a witness to the truth of ϕ(κ, E⃗), which is to say that ⟨κ⟩ ∈ j(X) and ⟨κ⟩ ∉ j_{E⃗(κ,β_{1})}(X). ⟨κ⟩ ∈ j_{1}(X) because crit(k) = β_{1} > κ, so ⟨κ⟩ ∈ i(X) since we just argued that crit(l) > κ. But the analysis of the iteration i shows that ⟨κ⟩ ∈ j_{E⃗(κ,β_{1})}(X), which is a contradiction. This finishes the proof of the claim.

♦

2.2 The measure u(2)

The analysis of u(2) is a little trickier, because while u(1) is essentially just a measure on ordinals, u(2) is essentially a measure on measures.

Recall that u(2) is the measure on V_{κ} defined by

X ∈ u(2) ⟺ ⟨κ, u(1)⟩ ∈ j(X).

Just as in the analysis of u(1), we may factor j through the ultrapower by u(2) to get a diagram

[figure: M_{2}; k; V; j_{2}; M; j]

where the critical point of k is now β_{2} = (κ^{++})_{M_{2}}, and where M_{2} = L[E⃗_{2}]. We need a little analysis of this situation. If we let

X = { j(F)(κ, u(1)) | F ∈ V }

then M_{2} is the transitive collapse of X and k is the inverse of the collapsing map. Since Pκ ∪ {u(1)} ⊆ X it is easy to see that u(1) ∈ M_{2}. We claim that β_{1} < β_{2}.

To see this observe that V and M_{2} agree to rank κ + 1, so that their ultrapowers by u(1) agree to rank j_{1}(κ) + 1. In particular β_{1} is the κ^{++} of Ult(M_{2}, u(1)), so that by the standard arguments applied inside the model M_{2} we have β_{1} < κ_{M_{2}}^{++} = β_{2}.

It is convenient to have a notation for reading off measures from coherent sequences.

Definition 1: If F⃗ is a coherent sequence and (γ, δ) ∈ dom(F⃗), μ_{1}(F⃗ , γ, δ) is the measure on V_{γ} given by

μ_{1}(F⃗ , γ, δ) = { X ⊆ V_{γ} | ⟨γ⟩ ∈ j_{F⃗ (γ,δ)}(X) }.

Claim 2: For X ⊆ V_{κ},

X ∈ u(2) ⟺ ⟨κ, β_{1}⟩ ∈ j_{E⃗(κ,β_{2})}(X^{∗}),

where X^{∗} is defined from X by

X^{∗} = { (γ, δ) | ⟨γ, μ_{1}(E⃗, γ, δ)⟩ ∈ X }.

Proof: As in the case of u(1), we really need to proceed by contradiction, but in this section we will ignore that aspect of the proof and simply show that if i : V ⟶ N and l : M_{2} ⟶ N are the co-iteration maps for the comparison of V and M_{2}, and i = l ∘ j_{2}, then the claim holds. The reader who is disturbed by this is referred to the next subsection, in which we give a complete treatment for the general case.

We are now assuming that the diagram

[figure: N; M_{2}; l; k; V; i; j_{2}; M; j]

commutes, where i and l are the co-iteration maps. As before we may argue that oE⃗_{2}(κ) = β_{2}, so that the initial step of the co-iteration is to apply E⃗(κ, β_{2}) to V while applying nothing to M_{2}. Notice that, by the nonoverlapping nature of the coherent sequences, all extenders subsequently applied during the comparison process have critical points greater than β_{2}. In particular crit(l) > β_{2}.

In the notation we just introduced, we showed in the last section that u(1) = μ_{1}(E⃗, κ, β_{1}). By coherence

E⃗(κ, β_{1}) = j(E⃗)(κ, β_{1}), and then as crit(k) = β_{2} > β_{1} we also have

j(E⃗)(κ, β_{1}) = E⃗_{2}(κ, β_{1}). The models Ult(V, E⃗(κ, β_{1})) and Ult(M_{2}, E⃗(κ, β_{1})) agree to a high rank, so that

M_{2} ⊨ u(1) = μ_{1}(E⃗_{2}, κ, β_{1}). Let X ∈ u(2). This means that ⟨κ, u(1)⟩ ∈ j(X), which is equivalent to

⟨κ, u(1)⟩ ∈ j_{2}(X). That is to say

M_{2} ⊨ ⟨κ, μ_{1}(j_{2}(E⃗), κ, β_{1})⟩ ∈ j_{2}(X). By the same kind of diagram-chasing as we did for u(1), we may show that

Ult(V, E⃗(κ, β_{2})) ⊨ ⟨κ, μ_{1}(j_{E⃗(κ,β_{2})}(E⃗), κ, β_{1})⟩ ∈ j_{E⃗(κ,β_{2})}(X). Equivalently,

⟨κ, β_{1}⟩ ∈ j_{E⃗(κ,β_{2})}({ ⟨γ, δ⟩ | ⟨γ, μ_{1}(E⃗, γ, δ)⟩ ∈ X }), which is to say

⟨κ, β_{1}⟩ ∈ j_{E⃗(κ,β_{2})}(X^{∗}).

♦

2.3 The measure u(α)

Now we prove a general statement relating the measures u(α) to the sequence E⃗.

Definition 2: α is nice if the critical point of the elementary embedding

k : Ult(V, u(α)) ⟶ M

given by k : [F] ⟼ j(F)(u ↾ α) is greater than α.

Observe that if α is nice then u ↾ α ∈ Ult(V, u(α)), and k(u ↾ α) = u ↾ α. All this follows from the standard analysis of Ult(V, u(α)) as the transitive collapse of { j(F)(u ↾ α) | F ∈ V }, and of k as the inverse of the collapsing map.

We show that a reasonably long initial segment of κ^{++} consists of nice α. We use some information about so-called “repeat points” which will be familiar to devotees of Radin forcing.

Definition 3: α is a repeat point of u if u(α) ⊆ ⋃_{β<α} u(β).

Lemma 1: There is no repeat point α less than κ^{+}, and there are κ^{++} many in [κ^{+}, κ^{++}).

Proof: Let α < κ^{+}, suppose that α is the order type of some ordering relation X ⊆ κ × κ. Define

A = { v ∈ V_{κ} | lh(v) = o.t.(X ∩ (v(0) × v(0))) }.

Then it is easy to see that A ∈ u(α), and A ∉ u(β) for β < α.

For the second part of the claim, observe that if α is not a repeat we may choose X_{α} to witness this. By GCH there are only κ^{+} choices for X_{α}, so there must be κ^{++} repeat points below κ^{++}.

♦

The following result could certainly be pushed further, but indicates that a long initial segment of κ^{++} consists of nice α.

Lemma 2: If { γ < α | γ is a repeat } has order type less than κ^{+}, then α is nice.

Proof: We show that all β ≤ α are of the form j(F)(u ↾ α). Defining F (v) = lh(v) handles α, so let β < α.

If β is not a repeat, let X ∈ u(β) witness this. Define F (v) to be the least η < lh(v) such that X ∩ V_{v(0)} ∈ v(η), then j(F)(u ↾ α) = β.

If β is a repeat, let it be the γ^{th} repeat where γ < κ^{+}, and find X ⊆ κ×κ such that γ is the order type of X. Define F (v) to be the η^{th} repeat point of v, where η is the order type of X ∩ (v(0) × v(0)).

♦

Claim 3: If { γ < α | γ is a repeat } has order type less than κ^{+}, then for X ⊆ V_{κ}

X ∈ u(α) ⟺ u ↾ α ∈ j_{E⃗(κ,β_{α})}(X),

where β_{α} = (κ^{++})_{Ult(V,u(α))}.

Proof: α is nice, so that if we factor j through the ultrapower by u(α) then we get a commutative triangle

[figure: M_{α}; k; V; j_{α}; M; j]

where (by niceness) u ↾ α ∈ M_{α}, and k(u ↾ α) = u ↾ α. As usual we may show crit(k) = (κ^{++})_{M_{α}}, and if we denote this by β_{α} then we know that β_{α} > α. It is interesting to notice that if ν < α then u(ν) ∈ M_{α}, so that we get β_{ν} < β_{α}.

As in the treatment of u(1) we define a formula ϕ, this time a slightly more complex version; ϕ(γ, δ, F⃗) means that 1. F⃗ is coherent in L[F⃗].

2. In the model L[F⃗]

(a) (γ, γ^{++}) ∈ dom(F⃗).

(b) Let F = F⃗ (γ, γ^{++}), let j_{F} : V ⟶ M_{F} be the ultrapower by F, and define a sequence v from F by the recursion v(0) = γ, v(η) = { X ⊆ V_{γ} | v ↾ η ∈ j_{F} (X) }. Then there is X ∈ v(δ) such that v ↾ δ ∉ j_{F⃗ (γ,ϵ)}(X), where ϵ = (γ^{++})_{Ult(V,v(δ))}.

For a contradiction, assume that ϕ(κ, α, E⃗). We may take it, without loss of generality, that

1. ϕ(γ, δ, G⃗) is false for all γ, δ, and G⃗ any proper initial segment of E⃗.

2. κ is minimal such that ∃δ ϕ(κ, δ, E⃗).

3. α is minimal such that ϕ(κ, α, ϵ).

Exactly as in the analysis of u(1), we may now argue that we have a commutative diagram

[figure: N; M_{α}; l; k; V; i; j_{α}; M; j]

where i : V ⟶ N and l : M_{α} ⟶ N are the co-iteration maps. We also know that in the comparison iteration the first step is to apply E⃗(κ, β_{α}) to V and nothing to M_{α}, and that all subsequently applied extenders have critical points greater than β_{α}.

Now let X ∈ u(α). Then u ↾ α ∈ j(X), and since k(u ↾ α) = u ↾ α we have u ↾ α ∈ j_{α}(X). l has critical point greater than β_{α}, which in turn is greater than α, so u ↾ α ∈ i(X). Finally, the analysis of i gives us u ↾ α ∈ j_{E⃗(κ,β_{α})}(X).

The claim is proved.

♦

3 A direct definition

In this section we will construct Radin forcing directly from a coherent sequence E⃗ and a sequence of ordinals β⃗. We start with a motivating example, then do a general construction.

3.1 An example

Retain from the last section the assumptions that

1. E⃗ is a coherent sequence, V = L[E⃗].

2. (κ, κ^{++}) ∈ dom(E⃗).

3. u is the Radin sequence generated from the embedding j_{E} : V ⟶ M_{E}, where E is the extender E⃗(κ, κ^{++}).

Let v = u ↾ 3, then we will use what we learned in the last section to analyse the Radin forcing ℝ_{v} in terms of the coherent sequence E⃗ and the sequence of ordinals ⟨κ, β_{1}, β_{2}⟩.

Definition 4: The Radin forcing ℝ_{v} is defined in the following way.

1. A condition is a finite sequence ⟨(v_{0}, A_{0}), …, (v_{n}, A_{n})⟩ where (a) v_{n} = v, and A_{n} ∈ v(1) ∩ v(2). Let β_{n} = κ.

(b) If i < n then either v_{i} = ⟨β_{i}⟩ for some β_{i} < κ and A_{i} = ∅, or v_{i} = ⟨β_{i}, V_{i}⟩ with V_{i} a measure on [β_{i}]^{1} and A_{i} ∈ V_{i}.

(c) If i < n then v_{i} ∈ V_{β_{i+1}}.

2. If p and q are conditions then p ≤ q means that

(a) Whenever a pair (s, A) appears in q, then a pair (s, B) appears in p with B ⊆ A.

(b) If a pair (s, B) appears in p then either a pair of form (s, A) appears in q with B ⊆ A, or for (w, A) the first pair appearing in p with w(0) > s(0) we have s ∈ A and B ⊆ A.

What does it mean to say that A ∈ v(1) ∩ v(2)? A ∈ v(1) means

⟨κ⟩ ∈ j_{E⃗(κ,β_{1})}(A),

that is there are many ⟨α⟩ for α < κ in A. A ∈ v(2) means

⟨κ, β_{1}⟩ ∈ j_{E⃗(κ,β_{2})}(A^{∗}),

which is to say that for many pairs of ordinals (γ, δ) ∈ dom(E⃗) we have ⟨γ, μ_{1}(E⃗, γ, δ)⟩ ∈ A.

Now observe that from the sequence ⟨κ, u(1)⟩ we may recover β_{1}, because β_{1} is (κ^{++})_{Ult(V,u(1))}. This recovery can be done inside Ult(V, E⃗(κ, β_{2})), so that v(2) concentrates on sequences ⟨γ, W ⟩ with

W = μ_{1}(E⃗, γ, ν)

for ν = (γ^{++})_{Ult(V,W)}.

The upshot of all this is that (below a suitable condition ⟨(v, A)⟩) ℝ_{v} is equivalent to the following forcing notion ℝ(E⃗, ⟨κ, β_{1}, β_{2}⟩).

Definition 5: The forcing ℝ(E⃗, ⟨κ, β_{1}, β_{2}⟩) is defined in the following way.

1. Let μ_{1} = μ_{1}(E⃗, κ, β_{1}), that is

{ X ⊆ V_{κ} | ⟨κ⟩ ∈ j_{E⃗(κ,β_{1})}(X) }.

2. Let μ_{2} be

{ X ⊆ V_{κ} | ⟨κ, β_{1}⟩ ∈ j_{E⃗(κ,β_{2})}(X) }.

3. A condition is a finite sequence ⟨(γ⃗_{i}, A_{0}), …, (γ⃗_{n}, A_{n})⟩ where

(a) γ_{n} = ⟨κ, β_{1}, β_{2}⟩, and A_{n} ∈ μ_{1} ∩ μ_{2}.

(b) If i < n then either γ⃗_{i} = ⟨β_{i}⟩ for some β_{i} < κ and A_{i} = ∅, or γ⃗_{i} = ⟨β_{i}, δ_{i}⟩ with (β_{i}, δ_{i}) ∈ dom(E⃗) and A_{i} ∈ μ_{1}(E⃗, β_{i}, δ_{i}).

(c) If i < n then γ⃗_{i} ∈ V_{β_{i+1}}.

4. If p and q are conditions then p ≤ q means that

(a) Whenever a pair (s, A) appears in q, then a pair (s, B) appears in p with B ⊆ A.

(b) If a pair (s, B) appears in p then either a pair of form (s, A) appears in q with B ⊆ A, or for (w, A) the first pair appearing in p with w(0) > s(0) we have s ∈ A and B ⊆ A.

The difference here is simply that in ℝ_{v} we had sequences of measures, each measure concentrating on sequences of measures, where now we have sequences of ordinals, each of which codes (via E⃗) a sequence of measures on sequences of ordinals.

This example motivates the general definition which follows.

3.2 The general definition

We will define a forcing notion ℝ(F⃗ , γ⃗) from a coherent sequence F⃗ and a suitable sequence of ordinals γ.

Definition 6: Let F⃗ be a coherent sequence. γ⃗ is a good sequence for F⃗ if

1. γ⃗ is an increasing sequence of ordinals.

2. If 0 < δ < lh(γ⃗) then

(a) (γ_{0}, γ_{δ}) ∈ dom(F⃗).

(b) γ⃗ ↾ δ ∈ Ult(V, F⃗ (γ_{0}, γ_{δ})).

Definition 7: If F⃗ is coherent, γ⃗ is good for F⃗, and 0 < δ < lh(γ⃗) then define

μ_{δ}(F⃗ , γ⃗) = { X ⊆ V_{γ_{0}} | γ⃗ ↾ δ ∈ j_{F⃗ (γ_{0},γ_{δ})}(X) }.

Definition 8: If F⃗ is coherent and γ⃗ is good for F⃗, with lh(γ⃗) ≥ 2, then

F(F⃗ , γ⃗) = { X ⊆ V_{γ_{0}} | 0 < δ < lh(γ⃗) ⟹ X ∈ μ_{δ}(F⃗ , γ⃗) }.

Lemma 3: F(F⃗ , γ⃗) is a γ_{0}-complete filter on V_{γ_{0}}, and concentrates on the set of sequences which are good for F⃗.

Proof: F(F⃗ , γ⃗) is the intersection of γ_{0}-complete ultrafilters, so is certainly a γ_{0}-complete filter. For the other claim we need to show that if 0 < δ < lh(γ⃗) and we define j_{δ} : V ⟶ M_{δ} to be the ultrapower by F⃗ (γ_{0}, γ_{δ}) then γ⃗ ↾ δ is good for the sequence j_{δ}(F⃗) in the model M_{δ} = Ult(V, F⃗ (γ_{0}, γ_{δ})).

To show this let 0 < β < δ = lh(γ⃗ ↾ δ). By coherence we get

(γ_{0}, γ_{β}) ∈ dom(j_{δ}(F⃗)),

and moreover

j_{δ}(F⃗)(γ_{0}, γ_{β}) = F⃗ (γ_{0}, γ_{β}).

Let F denote this extender. Ult(V, F) and Ult(M_{δ}, F) agree past rank γ_{β}, γ⃗ ↾ β ∈ Ult(V, F) because γ⃗ is good for F⃗, so γ⃗ ↾ β ∈ Ult(M_{δ}, F).

♦

The definition of ℝ(F⃗ , γ⃗) now follows a familiar pattern.

Definition 9: Let γ⃗ be good for the coherent sequence F⃗. The forcing ℝ(F⃗ , γ⃗) is defined in the following way.

1. A condition is a finite sequence ⟨(γ⃗_{i}, A_{0}), …, (γ⃗_{n}, A_{n})⟩ where (a) γ⃗_{n} = γ⃗, and each γ⃗_{i} is an increasing sequence of ordinals which is

good for F⃗.

(b) Let β_{i} = γ⃗_{i}(0). For i ≤ n either γ⃗_{i} = ⟨β_{i}⟩ and A_{i} = ∅, or lh(γ_{i}) > 1 and A_{i} ∈ F(F⃗ , γ⃗_{i}).

(c) If i < n then γ⃗_{i} ∈ V_{β_{i+1}}.

2. If p and q are conditions then p ≤ q means that

(a) Whenever a pair (s, A) appears in q, then a pair (s, B) appears in p with B ⊆ A.

(b) If a pair (s, B) appears in p then either a pair of form (s, A) appears in q with B ⊆ A, or for (w, A) the first pair appearing in p with w(0) > s(0) we have s ∈ A and B ⊆ A.

We will not give a complete analysis of this forcing, as the results and proofs are so similar to those for the Radin forcing ℝ_{v} as expounded in [5]. We will however prove a few key facts, to give the flavour of the proofs; the reader will notice how the coherence of the sequence F⃗ is used here at points where, in the situation of [5], we would be using the uniform generation of the measures via a single embedding.

Lemma 4 (The addability lemma): Let γ⃗ be good for F⃗, and suppose that A ∈ F(F⃗ , γ⃗). Then if

B = { δ⃗ ∈ V_{γ_{0}} | A ∩ V_{δ_{0}} ∈ F(F⃗ , δ⃗) }

we have B ∈ F(F⃗ , γ⃗).

Proof: If 0 < ν < lh(γ) then we need to show that, defining j_{ν} : V ⟶ M_{ν} to be the ultrapower by F⃗ (γ_{0}, γ_{ν}), γ⃗ ↾ ν ∈ j_{ν}(B). This amounts to showing that in M_{ν} = Ult(V, F⃗ (γ_{0}, γ_{ν})) we have

A ∈ F(j_{ν}(F⃗), γ ↾ ν).

If 0 < β < ν then j_{ν}(F⃗)(γ_{0}, γ_{β}) = F⃗ (γ_{0}, γ_{β}), and we denote this extender by F. Let

j_{F}^{V} : V ⟶ Ult(V, F)
j_{F}^{M_{ν}} : M_{ν} ⟶ Ult(M_{ν}, F)

be the ultrapowers by F as computed in V, M_{ν}. There is agreement between Ult(V, F) and Ult(M_{ν}, F) to a rank greater than γ_{β}. We also have j_{F}^{V} (A) = j_{F}^{M_{ν}}(A). Now A ∈ F(F⃗ , γ⃗), so certainly A ∈ μ_{β}(F⃗, γ⃗), that is γ⃗ ↾ β ∈ j_{F}^{V} (A) = j_{F}^{M_{ν}}(A), which is what we needed to prove.

♦

The addability lemma shows that it is possible to extend a condition in a non-trivial fashion.

Definition 10: Let p = ⟨(γ⃗_{0}, A_{0}), …, (γ⃗_{n}, A_{n})⟩ be a condition, then the lower part of p is the sequence ⟨(γ⃗_{0}, A_{0}), …, (γ⃗_{n−1}, A_{n−1})⟩. Let LP be the set of lower parts.

Lemma 5 (The diagonal intersection lemma): Let ⟨A_{x} : x ∈ LP ⟩ be a family of subsets of V_{γ_{0}} such that A_{x} ∈ F(F⃗ , γ⃗) for all x. Then there is A ∈ F(F⃗ , γ⃗) such that

x ⌢ (γ⃗, A) ≤ x ⌢ (γ⃗, A_{x})

for all x.

Proof: If we define

A = { δ⃗ | x ∈ LP ∩ V_{δ_{0}} ⟹ δ⃗ ∈ A_{x} },

then it suffices to show A ∈ F(F⃗ , γ⃗). Let 0 < β < lh(γ), and define j_{β} : V ⟶ M_{β} to be the ultrapower by F⃗ (γ_{0}, γ_{β}). Then we want

γ ↾ β ∈ j_{β}(A).

j_{β}(LP)∩V_{γ_{0}} = LP, and if x ∈ LP then A_{x} ∈ μ_{β}(F⃗ , γ⃗), so that γ⃗ ↾ β ∈ j_{β}(A)_{x} and we are done.

The set A defined here is called the diagonal intersection of the function x ⟼ A_{x}.

♦

4 The Prikry lemma

In this section we will sketch a proof that ℝ(F⃗, γ⃗) has a certain property, namely that questions about the forcing extension can be decided by “direct” extensions. It is this property, which is characteristic of Prikry forcing and its generalisations, that legitimates ℝ(F⃗ , γ⃗) as a kind of Radin forcing.

Fix for the rest of this section a coherent sequence F⃗ and a sequence γ⃗ which is good for F, and define ℝ = ℝ(F⃗ , γ⃗).

Definition 11: Let p and q be conditions in ℝ. p is a direct extension of q (we write p ≤_{d} q) if and only if p ≤ q and lh(p) = lh(q).

The direct extensions of q are those conditions that can be obtained from p by shrinking the measure one sets which appear as the second entries in the pairs comprising q. The idea is that a direct extension restricts the commitments we may make, while making no (positive) commitments itself.

Theorem 1: Let p ∈ ℝ, let b ∈ RO(ℝ). Then there exists q such that q ≤_{d} p and q ∥ b.

Proof: Suppose for simplicity that p is of the form ⟨(γ⃗, C)⟩. The general case is an easy variation on this one.

By taking an appropriate diagonal intersection, we may assume that for all lower parts x

(∃D x ⌢ (γ⃗, D) ∥ b) ⟺ x ⌢ (γ⃗, C) ∥ b.

For each x such that x ⌢ (u, C) ∦ b, define

C_{1}^{x} = { δ⃗ ∈ C | ∃D x ⌢ (δ⃗, D) ⌢ (γ⃗, C) ⊩ b },
C_{2}^{x} = { δ⃗ ∈ C | ∃D x ⌢ (δ⃗, D) ⌢ (γ⃗, C) ⊩ ¬b },
C_{3}^{x} = C \ (C_{1}^{x} ∪ C_{2}^{x}).

For α with 0 < α < lh(γ⃗) let C^{x}(α) = C_{i}^{x} for the unique i such that C_{i}^{x} ∈ μ_{α}(F⃗ , γ⃗). Define

C^{x} = ⋃_{0<α<lh(γ⃗)} C^{x}(α),

and let E be the diagonal intersection of the function x ↦ C_{x}. We claim that (γ⃗, E) ∥ b.

Suppose (towards a contradiction) that this is not so, and let

⟨(γ^{0}, A^{0}), …, (γ^{n}, A^{n}), (γ⃗, E)⟩

be an extension of ⟨(γ⃗, E)⟩ deciding b and having minimal length. Without loss of generality this condition forces b (else we may replace b by ¬b). Let x = ⟨(γ^{0}, A^{0}), …, (γ^{n−1}, A^{n−1})⟩. γ^{n} ∈ E, so by construction γ^{n} ∈ C^{x}, that is γ^{n} ∈ C^{x}(α) for some α < lh(γ⃗). By the construction of C, x ⌢ (γ^{n}, A_{n}) ⌢ (γ⃗, C) ⊩ b, so γ^{n} ∈ C_{1}^{x} and C^{x}(α) must have been chosen as C_{1}^{x}. Therefore ∀δ⃗ ∈ C^{x}(α) ∃D x ⌢ (δ⃗, D) ⌢ (γ⃗, C) ⊩ b. Fix some choice function h which chooses appropriate D, so that we have

∀δ⃗ ∈ C^{x}(α) x ⌢ (δ⃗, h(δ⃗)) ⌢ (γ⃗, C) ⊩ b.

Notice that h(δ⃗) ∈ F(F⃗ , δ⃗) for all δ⃗.

The contradiction will be almost immediate once we have the following lemma.

Lemma 6: There is B ∈ F(F⃗ , γ⃗) such that if p ≤ ⟨(γ⃗, B)⟩, then p is compatible with ⟨(δ⃗, h(δ⃗)), (γ⃗, B)⟩ for some δ⃗ ∈ dom(h).

Proof: We construct B as B_{1} ∪ B_{2} ∪ B_{3}, where

B_{1} ∈ ⋂ { μ_{β}(F⃗ , γ⃗) | 0 < β < α },
B_{2} ∈ μ_{α}(F⃗ , γ⃗),
B_{3} ∈ ⋂ { μ_{β}(F⃗ , γ⃗) | α < β < lh(γ) }.

The reader who finds the definitions of the B_{i} puzzling is encouraged to look ahead to the end of the proof and see how they are used. For each ν let

j_{ν} : V ⟶ M_{ν} = Ult(V, F⃗ (γ_{0}, γ_{ν}))

be the standard ultrapower map, and let μ_{ν} = μ_{ν}(F⃗ , γ⃗).

Define

B_{1}^{0} = { δ⃗ | { ϵ⃗ ∈ dom(h) | δ⃗ ∈ h(ϵ⃗) } ∈ μ_{α} } We claim that B_{1}^{0} ∈ μ_{β} for 0 < β < α. Fix δ for the moment, and consider

A = { ϵ⃗ ∈ dom(h) | δ⃗ ∈ h(ϵ⃗) }. By the definition of μ_{α} and the elementarity of j_{α},

A ∈ μ_{α}(F⃗ , γ⃗) ⟺ γ⃗ ↾ α ∈ j_{α}(A) ⟺ δ⃗ ∈ j_{α}(h)(γ⃗ ↾ α). We have shown that B_{1}^{0} = j_{α}(h)(γ⃗ ↾ α), so that by elementarity we have B_{1}^{0} ∈ F(j_{α}(F⃗), γ⃗ ↾ α). But by the coherence of F⃗ we see that

F(j_{α}(F⃗), γ⃗ ↾ α) = F(F⃗ , γ⃗ ↾ α), and since μ_{β}(F⃗ , γ⃗) = μ_{β}(F⃗ , γ⃗ ↾ α) for β < α we are done.

Now define

B_{1}^{1} = { δ⃗ | { ϵ⃗ ∈ dom(h) | h(ϵ⃗) ∩ V_{δ_{0}} ∈ F(F⃗ , δ⃗) } ∈ μ_{α} }. By an argument very similar to that in the last paragraph for B_{1}^{0},

B_{1}^{1} = { δ⃗ | B_{1}^{0} ∩ V_{δ_{0}} ∈ F(F⃗ , δ⃗) }. We claim that B_{1}^{1} ∈ μ_{β} for 0 < β < α. To show that B_{1}^{1} ∈ μ_{β} is to show γ⃗ ↾ β ∈ j_{β}(B_{1}^{1}), which is to say that

j_{β}(B_{1}^{0}) ∩ V_{γ_{0}} ∈ F(j_{β}(F⃗), γ⃗ ↾ β). This is immediate since j_{β}(B_{1}^{0}) ∩ V_{γ_{0}} = B_{1}^{0}, and

F(j_{β}(F⃗), γ⃗ ↾ β) = F(F⃗ , γ⃗ ↾ β) using coherence again.

Having defined B_{1}^{0} and B_{1}^{1}, let

B_{1} = B_{1}^{0} ∩ B_{1}^{1}. B_{2} is now defined by a kind of diagonal intersection. For δ⃗ ∈ B_{1} define

g(δ⃗) = { ϵ⃗ ∈ dom(h) | δ⃗ ∈ h(ϵ⃗), h(ϵ⃗) ∩ V_{δ_{0}} ∈ F(F⃗ , δ⃗) }. By the definition of B_{1}, g(δ⃗) ∈ μ_{α}. Let

B_{2} = { ϵ⃗ ∈ dom(h) | δ⃗ ∈ B_{1} ∩ V_{ϵ_{0}} ⟹ ϵ⃗ ∈ g(δ⃗) }.

We need to show B_{2} ∈ μ_{α}, that is to say that γ⃗ ↾ α ∈ j_{α}(B_{2}). Now j_{α}(B_{1})∩V_{γ_{0}} = B_{1}, and for δ⃗ ∈ B_{1} we know g(δ⃗) ∈ μ_{α}, hence γ⃗ ↾ α ∈ j_{α}(g)(δ⃗), using again the fact that for δ⃗ ∈ V_{γ_{0}} we have j_{α}(δ⃗) = δ⃗. This confirms that B_{2} ∈ μ_{α}.

Finally let

B_{3} = { δ⃗ | ∃ν < lh(δ⃗) ∀C ∈ F(F⃗ , δ⃗) { ϵ⃗ ∈ B_{2} ∩ C | h(ϵ⃗) ∩ C ∈ F(F⃗ , ϵ⃗) } ∈ μ_{ν}(F⃗ , δ⃗) }.

We need to show that B_{3} ∈ μ_{β} for every β with α < β < lh(γ⃗). That is, we need to show that for all such β we have γ⃗ ↾ β ∈ j_{β}(B_{3}). We will use α as the ν < β demanded by the definition and show that, for each C ∈ F(j_{β}(F⃗), γ⃗ ↾ β),

{ ϵ⃗ ∈ j_{β}(B_{2}) ∩ C | j_{β}(h)(ϵ⃗) ∩ C ∈ F(j_{β}(F⃗), ϵ⃗) } ∈ μ_{α}(j_{β}(F⃗), γ⃗ ↾ β)

As usual we know that C ∈ F(F⃗ , γ⃗ ↾ β) and that μ_{α}(j_{β}(F⃗), γ⃗ ↾ β) = μ_{α}. C ⊆ V_{γ_{0}}, so j_{β}(B_{2}) ∩ C = B_{2} ∩ C. For ϵ⃗ ∈ B_{2} ∩ C we know ϵ⃗ and h(ϵ⃗) are in V_{γ_{0}}, hence h(ϵ⃗) = j_{β}(h(ϵ⃗)) = j_{β}(h)(ϵ⃗). What is more

F(j_{β}(F⃗), ϵ⃗) = j_{β}(F(F⃗ , ϵ⃗)) = F(F⃗ , ϵ⃗).

So we are required to show that

{ ϵ⃗ ∈ B_{2} ∩ C | h(ϵ⃗) ∩ C ∈ F(F⃗ , ϵ⃗) } ∈ μ_{α}.

B_{2} ∩ C ∈ μ_{α} because β > α. All that remains to be seen is that

j_{α}(h)(γ⃗ ↾ α) ∩ j(C) ∈ F(j_{α}(F⃗), γ⃗ ↾ α).

This is easy as j_{α}(h)(γ⃗ ↾ α) = B_{1}^{0}, and B_{1}^{0} ∩ j_{α}(C) = B_{1}^{0} ∩ C which is clearly a member of F(F⃗ , γ⃗ ↾ α).

This concludes the construction of the set B.

We now show that B = B_{1} ∪ B_{2} ∪ B_{3} has the property desired. Let

p = ⟨(γ^{0}, A^{0}), …, (γ^{n}, A^{n}), (γ⃗, B̄)⟩ ≤ ⟨(γ⃗, B)⟩.

For each i ≤ n there is j < 3 with γ^{i} ∈ B_{j}. Let δ⃗ be the first sequence occurring in p with δ⃗ ∉ B_{1}. Then δ⃗ ∈ B_{2}, δ⃗ ∈ B_{3}, or δ⃗ = γ⃗; we take each case in turn.

1. δ⃗ = γ^{j} ∈ B_{2}. By construction γ^{j} ∈ dom(h). By the definition of B_{2} and the fact that for i < j we have γ^{i} ∈ B_{1}, we have for all i < j that γ^{i} ∈ h(γ^{j}) and h(γ^{j}) ∩ V_{γ_{0}^{i}} ∈ F(F⃗ , γ^{i}).

Therefore

⟨(γ^{0}, A^{0} ∩ h(γ^{j})), …, (γ^{j}, A^{j} ∩ h(γ^{j})), (γ^{j+1}, A^{j+1}), …, (γ⃗, B̄)⟩

is a common extension of p and ⟨(γ^{j}, h(γ^{j})), (γ⃗, B)⟩.

2. δ⃗ = γ^{j} ∈ B_{3}. By construction it is possible to find ϵ⃗ such that ϵ_{0} > γ_{0}^{j−1}, ϵ⃗ ∈ A^{j} ∩ B^{2}, h(ϵ⃗) ∩ A^{j} ∈ F(F⃗ , ϵ⃗). As before, if i < j then γ^{i} ∈ h(ϵ⃗) and f(ϵ⃗) ∩ V_{κ(γ^{i})} ∈ F(F⃗ , γ^{i}), so

⟨(γ^{1}, A^{1} ∩ h(ϵ⃗)), …, (γ^{j−1}, A^{j−1} ∩ h(ϵ⃗)), (ϵ⃗, A^{j} ∩ h(ϵ⃗)), (γ^{j}, A^{j}) …, (γ⃗, B̄)⟩

is a common extension of p and ⟨(ϵ⃗, h(ϵ⃗)), (γ⃗, B)⟩.

3. δ⃗ = γ⃗. The proof is similar to that of Case 2. We claim that

{ ϵ⃗ ∈ B_{2} ∩ B̄ | h(ϵ⃗) ∩ B̄ ∈ F(F⃗ , ϵ⃗) } ∈ μ_{α}.

To see this, observe that

j_{α}(h)(γ⃗ ↾ α) ∩ j_{α}(B̄) = B_{1}^{0} ∩ B̄ ∈ F(F⃗ , γ⃗ ↾ α).

Using this we may find ϵ⃗ ∈ B̄ ∩ B_{2} with ϵ_{0} > γ_{0}^{n}, we have γ^{i} ∈ B_{1} for all i ≤ n, and we may take as our common extension

⟨(γ^{1}, A^{1} ∩ h(ϵ⃗)), …, (γ^{n}, A^{n} ∩ h(ϵ⃗)), (ϵ⃗, h(ϵ⃗) ∩ B̄), (γ⃗, B̄)⟩.

This ends the proof of the lemma.

♦

To finish the proof of the theorem, apply the lemma we just proved. This produces E^{∗} ∈ F(F⃗ , γ⃗), E^{∗} ⊆ E, such that every extension of the condition ⟨(γ⃗, E^{∗})⟩ is compatible with some condition ⟨(ϵ⃗, h(ϵ⃗)), (γ⃗, E^{∗})⟩. In fact more is true; by the uniform definition of the forcing, every extension of x ⌢ (γ⃗, E^{∗}) is compatible with some x ⌢ (ϵ⃗, h(ϵ⃗)) ⌢ (γ⃗, E^{∗}), that is to say with a condition that forces b. So x ⌢ (γ⃗, E^{∗}) ⊩ b, contradicting the assumption that x ⌢ (γ^{n}, A^{n}) ⌢ (γ⃗, E) had minimal length.

It remains for us to extend the result to general conditions. Let

p = ⟨(γ⃗_{0}, A_{0}), …, (γ⃗_{n}, A_{n}), (γ⃗, A)⟩

be an arbitrary condition. Construct E as above and refine p to

p̄ = ⟨(γ⃗_{0}, A_{0}), …, (γ⃗_{n}, A_{n}), (γ⃗, A ∩ E)⟩

The argument used above shows that if q is an extension of p̄ deciding b and having minimal length then q is of the form x̄ ⌢ (γ⃗, Z) with x̄ ∈ ℝ(F⃗ , γ⃗_{n}), x̄ ≤ ⟨(γ⃗_{1}, A_{1}), …, (γ⃗_{n}, A_{n})⟩. Now an easy induction on length of conditions gives the result.

♦

Using similar ideas, we can show that ℝ(F⃗ , γ⃗) shares all the basic properties of Radin forcing as set out in [5] and [3].

5 Conclusion

In the light of what we have proved, we can see that Radin’s version of Radin forcing in [5] corresponds to ℝ(F⃗ , γ⃗) where F⃗ has some long extenders and γ⃗ grows rapidly. Mitchell’s version in [3] corresponds to the situation in which F⃗ has only short extenders (measures) and γ⃗ grows slowly. We hope that this formulation may be helpful in understanding the exact strength of the assumptions that are needed to make various applications of Radin forcing in cardinal arithmetic and choiceless set theory.

We may also speculate that there is some similarly direct way of extracting interesting forcing notions from overlapping extender sequences, such as are used in the inner model theory for large cardinals past a strong cardinal.

References

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

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

[3] W. Mitchell, How weak is a closed unbounded ultrafilter? In Logic Colloquium 80, edited by D. van Dalen, D. Lascar, J. Smiley (North-Holland).

[4] W. Mitchell, Hypermeasurable cardinals. In Logic Colloquium 78,, edited by M. Boffa, D. van Dalen and K. McAloon (North-Holland).

[5] L. B. Radin, Adding closed cofinal sequences to large cardinals. Annals of Mathematical Logic 22 (1982) 243–261.