A MODEL IN WHICH GCH HOLDS AT

SUCCESSORS BUT FAILS AT LIMITS

James Cummings

Mathematical Sciences Research Institute

Abstract. Starting with GCH and a P_{3}κ-hypermeasurable cardinal, a model is produced in which 2^{λ} = λ^{+} if λ is a successor cardinal and 2^{λ} = λ^{++} if λ is a limit cardinal. The proof uses a Reverse Easton extension followed by a modified Radin forcing.

1. INTRODUCTION

1. Historical remarks and statement of the main result. The continuum problem is an old one, dating back to Cantor and his

statement of the Continuum Hypothesis in [Ca]. Put in a modern form which might have puzzled Cantor, the problem is to determine which behaviours of the continuum function κ ⟼ 2^{κ} are consistent with ZFC. Throughout this paper ZFC will be the base set theory, though as we see below strong set-theoretic hypotheses will play an essential rôle in the result.

Before Gödel progress on the continuum problem was made by the descriptive set theorists, who showed that certain easily definable sets of reals could not be counterexamples to CH. Gödel [G] took the major step forward of showing that in a certain submodel of V, the constructible universe L, the following Generalised Continuum Hypothesis is true (here CARD denotes the class of cardinals):

∀κ ∈ CARD 2^{κ} = κ^{+}  (GCH)

Then in 1963 Cohen [Co] initiated the method of forcing and showed that the negation of CH was also consistent with ZFC, making CH the

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1991 Mathematics Subject Classification. Primary 03E35; Secondary 03E55..

Key words and phrases. Radin forcing, singular cardinals problem, large cardinals..

I would like to thank Adrian Mathias and Hugh Woodin, who supervised my doctoral thesis from which this result is taken; also the Science and Engineering Research Council of Great Britain for their support.

first statement of “natural” mathematics to be shown undecidable in set theory. This breakthrough was quickly followed by the work of Easton [E], who showed that if GCH holds in V and F : REG ⟶ CARD (where REG is the class of regular cardinals) is a class function such that

∀κ ∈ REG F (κ) > κ
∀κ, λ ∈ REG κ < λ ⟹ F (κ) ≤ F (λ)
∀κ ∈ REG cf(F (κ)) > κ

then there exists a class forcing preserving all cardinals and cofinalities such that in the extension

∀κ ∈ REG 2^{κ} = F (κ)

That is, the values of the continuum function at regular cardinals are highly independent of each other, and can be “anything reasonable”.

It is worth noticing that all the consistency results mentioned so far are relative to ZFC, that is are results of the form “Con(ZFC) implies Con(ZFC + ϕ)” where ϕ is some statement of set theory.

The first connection between large cardinals and the continuum function was made slightly prior to the invention of forcing by Scott [Sc], who used his technique of taking ultrapowers of V to show that if κ is measurable, U is a normal measure on κ, and

{ α < κ | 2^{α} = α^{+} } ∈ U

then 2^{κ} = κ^{+}. So sufficiently strong large cardinals can impose some structure on the behaviour of the continuum function.

For some time after Easton’s work it was felt that Easton’s result should generalise to any “reasonable” function F : CARD ⟶ CARD, and that all that was needed were more sophisticated techniques for forcing over models of ZFC; but then Silver [Si1] proved some theorems showing that if κ is a singular strong limit cardinal of uncountable cofinality then there is a strong connection between the behaviour of the continuum function below κ and the value of 2^{κ}. For example if GCH fails at κ then it already fails on a closed unbounded set of cardinals less than κ (the parallel with Scott’s result is not coincidence; Silver’s proof goes via a generic ultrapower by an ultrafilter extending the club filter).

More surprises were in store; let SCH (Singular Cardinals Hypothesis) be the assertion that if κ is singular then 2^{κ} is the least cardinal λ ≥ 2^{<κ} with cf(λ) > κ. This would imply that the continuum function was determined by its behaviour on the class of regular cardinals. Jensen [DJ] was inspired by seeing Silver’s result to show that if 0^{♯} does not exist then SCH is true: however by work of Silver and Prikry (see [Si2], [P]) it was known that if the existence of a supercompact cardinal is consistent then so is the failure of SCH (use Reverse Easton forcing to get a measurable κ such that 2^{κ} = κ^{++}, then use Prikry forcing to make κ have cofinality ω. κ becomes a rather large singular strong limit cardinal at which GCH fails). So large cardinals are essential for an understanding of the continuum function.

It is natural to ask about the situation at small singular cardinals. Magidor showed in [Ma1], [Ma2] that (among other things)

1) Con(ZFC +GCH+ there is κ κ^{++}-supercompact) implies Con(ℵ_{ω} is strong limit and 2^{ℵ_{ω}} = ℵ_{ω+2})

2) Con(ZFC + there exist κ < λ with κ supercompact and λ huge) implies Con(GCH holds below ℵ_{ω} and 2^{ℵ_{ω}} = ℵ_{ω+2}) From 1) SCH can fail at the smallest singular cardinal of them all;

from 2) no direct analogue of the theorem from [Si1] mentioned above can hold.

There is a large gap in strength between 0^{♯} and a supercompact cardinal. At one end Dodd and Jensen [D] invented the core model and showed that failure of SCH has consistency strength greater than that of a measurable cardinal, and following this line of research Mitchell [Mi2] got the consistency of measurable cardinals of high order from the failure of SCH. At the other end Woodin [W] showed how to get Magidor’s result 1) from the consistency of a P_{2}κ-hypermeasurable cardinal. Shelah got some bounds ([Sh1], Chap. XIII) for singular cardinals of cofinality ω: for example if ℵ_{ω} is strong limit then 2^{ℵ_{ω}} < ℵ_{(2^{ℵ_{0}} )^{+}}. Shelah also showed [Sh2] that ℵ_{ω} being strong limit was consistent with 2^{ℵ_{ω}} being any successor cardinal below ℵ_{ω_{1}}, using a supercompact, and Magidor extended this in unpublished work to get a similar result with GCH holding below ℵ_{ω}.

Another line of development has involved global results. Foreman and Woodin [FW] showed that starting from slightly more than a supercompact it is consistent that the GCH fail everywhere, and Woodin then produced a model where ∀κ 2^{κ} = κ^{++}. Woodin later showed that hypermeasurability-type hypotheses suffice for these results.

This paper, which is basically a revised version of Chapter Two of my doctoral thesis [C], also proves a global result: if a P_{3}κ-hypermeasurable cardinal is consistent then so is

ZFC + GCH holds at all successors and fails at all limits

The final model is of the form V_{κ}^{∗} where κ is a P_{2}κ-hypermeasurable in V^{∗}, so the model contains many smallish large cardinals.

The proof combines Radin forcing with ideas from unpublished work of Woodin, in which he shows that Magidor’s result 2) mentioned above can also be done from a P_{2}κ-hypermeasurable.

Added in proof: Magidor and Gitik [GM] have recently determined the exact strength of the failure of SCH. A sample result is that failure at ℵ_{ω} is equiconsistent with o(κ) = κ^{++}. Their methods will also do Magidor’s result 2) from this hypothesis (and indeed the stronger result with GCH up to ℵ_{ω} and a countable jump at ℵ_{ω} from less than a strong cardinal) but are very specific to cardinals of cofinality ω and do not seem to mix with Radin forcing in any straightforward way.

Also Shelah has proved the surprising theorem that if 2^{ℵ_{0}} < ℵ_{ω} then ℵ_{ω}^{ℵ_{0}} < ℵ_{ω_{4}}, and shown that some new idea is needed to make a model in which ℵ_{ω} is strong limit and 2^{ℵ_{ω}} > ℵ_{ω_{1}}.

2. Background material.

In this section we establish our terminology and notation; we also give proofs for some facts which we need in the Reverse Easton construction of Section Two. These facts are part of the folk literature of set theory, but we give the proofs here to make the exposition self-contained.

The set-theoretic notation used is mostly that of [J], with the caveat that the forcing terminology is that of Kunen’s reference [K], in particular “κ-closed” means “having lower bounds for decreasing sequences of length less than κ”. “κ-dense” means that the intersection of fewer than κ dense open sets is dense.

(set theoretic term)_{M} means the result of computing the term in the model M, and when we write for example

M ⊨ cf(λ) = κ^{++}

we mean that cf_{M}(λ) = κ_{M}^{++}. If P is a poset and p ∈ P then P ↾ p is used to denote { q ∈ P | q ≤_{P} p } with the inherited partial ordering. When we say that two posets are isomorphic we mean there is an order preserving map from one to the other with dense domain and range (this is enough to make them equivalent from the point of view of forcing).

Certain forcing notions will be used very heavily; if κ is regular, λ ≥ κ is a cardinal and μ > κ is inaccessible then

(1) Add(κ, λ) is the poset of partial functions from κ × λ to 2 with supports of size less than κ. We regard it as adding λ generic subsets of κ. It is κ-closed and (2^{<κ})^{+}-c.c., and will make 2^{<κ} have cardinality κ without adding bounded subsets to κ.

(2) Coll(κ, λ) is the poset of partial functions from κ to λ with supports of size less than κ. It will collapse λ to be of cardinality κ.

(3) Coll(κ, < μ) is the poset of partial functions f : κ×μ ⇝ μ where f has support of size less than κ and ∀α ∀β f(α, β) < β. It is κ-closed and μ-c.c., and makes μ into the successor of κ.

Many well-known facts about iterated forcing will be used without comment; we refer the reader to Baumgartner’s article [B].

¶1. Extenders. An extender is a device for describing an elementary embedding between models of set theory; there is a good account in [MS] but our conventions differ rather from those of Martin and Steel. The reader of [MS] should have no trouble in providing proofs for the assertions we make here.

Definition. Let a, b be finite sets of ordinals with a ⊆ b. Suppose

\begin{aligned}b & = \{α_{1}, …, α_{n}\}_{<} \\ a & = \{α_{l_{1}}, …, α_{l_{m}}\}_{<} \end{aligned}

Define f_{ab} : [ON]^{n} ⟶ [ON]^{m} by

f_{ab} : {λ_{1}, …, λ_{n}}_{<} ⟼ {λ_{l_{1}}, …, λ_{l_{m}}}_{<}

Definition. Let M be a transitive model of ZFC, let κ be a cardinal of M, and λ > κ an ordinal. Then E is a (κ, λ)-M-extender iff

E = ⟨E_{a} : a ∈ [λ]^{<ω}⟩

where for some ordinal μ

(1) For all a ∈ [λ]^{<ω} E_{a} is a M-κ-complete M-ultrafilter on [μ]^{|a|}.

(2) For all a, b ∈ [λ]^{<ω}, if a ⊂ b and f_{ab} : [μ]^{|b|} ⟶ [μ]^{|a|} is defined as above, then

∀A A ∈ E_{a} ⟺ { x | f_{ab}(x) ∈ A } ∈ E_{b}

(3) The limit ultrapower of M by the system E (which exists by (1) and (2) and standard facts about limit ultrapowers) is well-founded.

(4) If j_{E} : M ⟶ N is the limit ultrapower embedding then (a) crit(j_{E}) = κ.

(b) λ ≤ j_{E}(μ).

(c) For all a ∈ [λ]^{<ω} X ∈ E_{a} ⟺ a ∈ j_{E}(X).

(d) N = { j_{E}(f)(a) | a ∈ [λ]^{<ω} ∧ f ∈ M ∧ f : [μ]^{|a|} ⟶ M }.

Definition. If E is an extender as above then the critical point of E is κ, the length of E is λ and the width of E is μ.

The motivation for this definition is that if we have transitive models of ZFC M, N, and an elementary embedding i : M ⟶ N such that λ ≤ i(μ), then we can define a (crit(i), λ)-M-extender of width μ by

X ∈ E_{a} ⟺ a ∈ i(X)

for a ∈ [λ]^{<ω}.The embedding i will factor as k ∘ j_{E} where crit(k) ≥ λ and k : ⟨f, a⟩_{E} ⟼ i(f)(a). i_{E} is an approximation to i which improves as λ increases.

We list some facts.

Fact 1. If GCH holds and j : V ⟶ M is an elementary embedding into a transitive inner model M, with critical point κ, such that ^{κ}M ⊂ M and V_{κ+n} ⊂ M (a so-called P_{n}κ-hypermeasurable embedding) then the extender of length κ^{+n} approximating j has width κ and gives rise to a P_{n}κ-hypermeasurable embedding.

Fact 2. Let GCH hold, and let j : V ⟶ M be a P_{n}κ-hypermeasurable embedding generated by a (κ, κ^{+n})-V -extender. Factor j as k ∘ j_{0} where j_{0} : V ⟶ M_{0} is the ultrapower of V by { X | κ ∈ j(X) }. Then k is generated by a (κ_{M_{0}}^{++}, κ^{+n})-M_{0}-extender of width κ_{M_{0}}^{+n}.

Fact 3. If E ∈ V is a V -extender then for every ordinal α the α^{th} iterate of V by E exists and is well-founded.

We make the remark that GCH was critical to the truth of Facts 1 and 2; later on we will construct a model in which κ is P_{2}κ-hypermeasurable and 2^{κ} = κ^{++}, and there we will need a (κ, κ^{+3})-extender to describe the associated elementary embedding. This is natural enough, since in this instance we have |V_{κ+2}| = κ^{+3}.

¶2. Extending embeddings. We collect some information about forcing and elementary embeddings, for use in the Reverse Easton constructions of Section 2.

Fact 1. Let M, N be inner models of ZFC, j : M ⟶ N an elementary embedding, P ∈ M a notion of forcing. Let G be P -generic over M, H j(P)-generic over N, and suppose

∀p ∈ P p ∈ G ⟹ j(p) ∈ H  (∗)

Then j can be extended to j^{∗} : M[G] ⟶ N[H], where

j^{∗} : i_{G}(τ̇) ⟼ i_{H}(j(τ̇))

and j^{∗} is still elementary.

Proof. Let ϕ(x_{1}, …, x_{n}) be a formula of set theory with free variables among x_{1}, …, x_{n}.

M[G] ⊨ ϕ(i_{G}(τ̇_{1}), …, i_{G}(τ̇_{n}))
⟹ ∃p ∈ G p ⊩_{P}^{M} ϕ(τ̇_{1}, …, τ̇_{n})
⟹ ∃q ∈ H q ⊩_{j(P)}^{N} ϕ(j(τ̇_{1}), …, j(τ̇_{n}))
⟹ N[H] ⊨ ϕ(i_{H}(j(τ̇_{1})), …, i_{H}(j(τ̇_{n})))

Letting ϕ be “x_{1} = x_{2}”, j^{∗} is well-defined; replacing ϕ by ¬ϕ j^{∗} is fully elementary. □

Fact 2. Let M, N, j, P ∈ M be as in the statement of Fact 1. Let j be generated by an extender with width μ, and let M ⊨ “P is μ^{+}−dense”. Let G be P -generic over M. Then

H = { q ∈ j(P) | ∃p ∈ G j(p) ≤ q }

is j(P)-generic over N, and as (*) holds j extends to an elementary embedding j^{∗} : M[G] ⟶ N[H].

Proof. Let D ∈ N, D ⊆ j(P) be a dense open subset. D = j(f)(a), where a ∈ [ON]^{<ω}, f ∈ M, and

∀x ∈ [μ]^{|a|} f(x) is a dense open subset of P

As M ⊨ P μ^{+}−dense,

D^{∗} = ⋂_{x∈[μ]^{|a|}} f(x)

is a dense open subset of P. D^{∗} ∈ M, so find p such that p ∈ D^{∗} ∩ G. M ⊨ ∀x p ∈ f(x), thus N ⊨ j(p) ∈ j(f)(a) = D. So D ∩ H ≠ ∅, hence H is generic. □

Fact 3. Let M, N, j, P, G, H be as in the statement of Fact 1. Let (*) hold, and let j be generated by a (κ, λ)-M-extender. Then j^{∗} is so generated.

Proof. Let x ∈ N[H]. x = i_{H}(τ̇) for some τ̇ ∈ N^{j(P)}. τ̇ = j(f)(a), where a ∈ [λ]^{<ω}, f ∈ M, f : [μ]^{|a|} → M^{P}. Define f^{∗} ∈ M[G], dom(f^{∗}) = [μ]^{|a|}, f^{∗} : y ⟼ i_{G}(f(y)). Clearly x = j^{∗}(f^{∗})(a), so

N[H] = { j^{∗}(g)(b) | g ∈ M[G], b ∈ [λ]^{<ω} } □

¶3. Building generics. We collect some facts pertinent to constructing generics over one model inside another.

Fact 1. Let M, N be inner models of ZFC, M ⊆ N. Let P ∈ M, N ⊨ “P is λ−closed” N ⊨ |{ A | A is a maximal antichain in P, A ∈ M }| ≤ λ

Then there is in N a P -generic filter over M.

Proof. In N, enumerate the set of maximal antichains of P lying in M as ⟨A_{α} : α < λ⟩. Build by induction a sequence ⟨p_{α} : α < λ⟩ such that

α < β ⟹ p_{β} ≤ p_{α}

∀α ∃x ∈ A_{α} p_{α} ≤ x The closure enables the construction to be continued at limit stages.

G = { p ∈ P | ∃α p_{α} ≤ p }

is clearly a filter on P which is generic over M. □

Fact 2. Let M, N be inner models of ZFC, M ⊆ N, κ ∈ ON. Then

N ⊨ ^{κ}M ⊆ M ⟺ N ⊨ ^{κ}ON ⊆ M

Proof. By AC , every set in M is in bijection with some ordinal. □

Fact 3. Let M, N be inner models of ZFC, M ⊆ N, N ⊨ ^{κ}M ⊆ M. Let P ∈ M be a forcing notion, N ⊨ “P is κ^{+}−c.c.”, and let G be P -generic over N. Then N[G] ⊨ ^{κ}M[G] ⊆ M[G].

Proof. By Fact 2 it is enough to show that N[G] ⊨ ^{κ}ON ⊆ M[G]. But it is immediate from the chain condition of P in N that to any name in N^{P} for a κ-sequence of ordinals there corresponds an equivalent name in M^{P}. □

¶4. Mutual genericity. We give some results which guarantee the mutual genericity of generic objects under certain circumstances.

Fact 1 (Easton’s Lemma). Let P be a κ-c.c notion of forcing, and Q be a κ-closed one. Then ⊩_{P} “ Q̂ is κ̂-dense”, ⊩_{Q} “P̂ is κ̂-c.c”, and generics for P and Q are mutually generic.

Proof. Let Ȧ ∈ V^{Q}, q ∈ Q be such that

q ⊩ “Ȧ : κ̂ ⟶ P̂ enumerates an antichain”

Define a descending sequence ⟨q_{α} : α < κ⟩ of elements of Q, and a sequence ⟨p_{α} : α < κ⟩ of elements of P such that q_{0} = q and ∀α < κ q_{α+1} ⊩ Ȧ(α̂) = p̂_{α}. Then clearly ⟨p_{α} : α < κ⟩ enumerates an antichain in P lying in V; this is a contradiction, so we conclude that ⊩_{Q} P̂ κ̂−c.c..

Let G be P -generic, and H Q-generic. The mutual genericity of G and H is immediate, for if A ∈ V [H] is a maximal antichain in P then |A| < κ so A ∈ V and A ∩ G ≠ ∅.

Finally we show that V [G] and V [G][H] have the same < κ-sequences of ordinals. V [G][H] = V [H][G], so let τ̇ ∈ V [H]^{P} be a term for such a sequence. By a chain condition argument we can assume τ̇ ⊂ P × ON and |τ̇| < κ, so τ̇ ∈ V by closure. □

The following fact is due to Woodin, and plays a central role in [W].

Fact 2. Let κ be a measurable cardinal, U some normal measure on κ, j : V ⟶ M the ultrapower by U. Let P be a κ-closed partial ordering, let Q = j(P). ^{κ}M ⊆ M, and j(κ) > κ, so Q is κ^{+}-closed; let G be Qgeneric over V, and transfer it along j to get j^{∗} : V [G] ⟶ M[j^{∗}(G)]. Then G is in fact Q-generic over M[j^{∗}(G)].

Proof. Let Ḋ ∈ M^{j(Q)} be a name for a dense subset of Q. As M is an ultrapower, Ḋ is of the form j(f)(κ), where for each α f(α) ∈ V^{Q} is a name for a dense subset of P. Define a subset of Q,

E = { q ∈ Q | ∃g : κ ⟶ P q = j(g)(κ) ∧ ∀α q ⊩ ĝ(α) ∈ f(α) }

The claim is that E is a dense subset of Q. Fix r in Q, and let r be represented as j(h)(κ) for some h : κ → P. Construct by induction on α < κ functions h_{α} and elements of P p_{α} such that

h_{α} : κ ⟶ P, and if α < β then ∀γ h_{β}(γ) ≤_{P} h_{α}(γ)

j(h_{α})(κ) ⊩ p̂_{α} ∈ f(α)

∀γ ≤ α h_{α}(γ) = p_{γ}

j(h_{0})(κ) ≤ r

Construction. At stage α, let p ∈ P be some lower bound for the sequence ⟨h_{β}(α) : β < α⟩, and let q ∈ Q be a lower bound for the sequence ⟨j(h_{β})(κ) : β < α⟩. f(α) names a dense subset of P, so find q^{∗} ≤ q, p^{∗} ≤ p, such that q^{∗} ⊩ p̂^{∗} ∈ f(α). Let p_{α} be p^{∗}, and alter some function representing q^{∗} to get an appropriate h_{α}. (This is possible because we only need alter the function on a set of measure zero). Define h^{∗} by h^{∗}(α) = p_{α}; then r^{∗} =_{def} j(h^{∗})(κ) is an extension of r lying in E.

\begin{matrix}h(0) & h(1) & … & h(α) & … \\ p_{0} & h_{0}(1) & … & h_{0}(α) & … \\ p_{0} & p_{1} & … & h_{1}(α) & … \\ ⋮ & ⋮ & ... & ⋮ & … \\ p_{0} & p_{1} & … & p_{α} & … \end{matrix}

10 JAMES CUMMINGS

As E lies in V and is dense, find q = j(h)(κ) such that q ∈ E ∩ G, where h is chosen to witness q ∈ E. q ⊩ ∀α < κ ĥ(α) ∈ f(α), so applying j and letting α be κ, j(q) ⊩ { \atop j(h)(κ)} ∈ j(f)(κ). j(q) ∈ j^{∗}(G), so q ∈ i_{j^{∗}(G)}(Ḋ) hence G is Q-generic over M[j^{∗}(G)]. □

¶5. Term forcing. We describe a forcing trick which is needed for the Reverse Easton construction of Section 2.

Let P be a forcing notion, Q̇ ∈ V^{P}, ⊩_{P} Q̇ is a forcing notion. Define a forcing Q̇/P ∈ V by

τ̇ ∈ Q̇/P ⟺ τ̇ ∈ V^{P} ∧ ⊩_{P} τ̇ ∈ Q̇
τ̇ ≤_{Q̇/P} σ̇ ⟺ ⊩_{P} τ̇ ≤Q̇ σ̇

Q̇/P is technically a class, but can be treated as a set, as in the standard treatment of iterated forcing (see [B] or [K]).

Forcing with Q̇/P adds a “universal generic object” for Q̇. More precisely we have Fact 1. Let H be Q̇/P -generic over V, and let G be P -generic over V. Then F = { i_{G}(τ̇) | τ̇ ∈ H } is i_{G}(Q̇)-generic over V [G]. Proof. Claim 1. F is a filter. Proof of Claim 1.

ρ̇, σ̇ ∈ H ⟹ ∃τ̇ ∈ H ⊩_{P} τ̇ ≤Q̇ ρ̇, σ̇ ⟹ i_{G}(τ̇) ≤_{i_{G}(Q̇)} i_{G}(ρ̇), i_{G}(σ̇)

ρ̇ ∈ H∧i_{G}(ρ̇) ≤ i_{G}(σ̇) ⟹ ∃τ̇ i_{G}(τ̇) = i_{G}(σ̇)∧ ⊩ ρ̇ ≤ τ̇ ⟹ τ̇ ∈ H □

Claim 2. F is generic. Proof of Claim 2. Let Ḋ be a name for a dense subset of Q̇, and define

E = { τ̇ ∈ Q̇/P | ⊩_{P} τ̇ ∈ Ḋ }

E is dense in Q̇/P; for if ρ̇ ∈ Q̇/P,

⊩_{P} ∃x x ∈ Ḋ ∧ x ≤ ρ̇

so by the maximum principle there is τ̇ such that

⊩_{P} τ̇ ∈ Ḋ ∧ τ̇ ≤ ρ̇

As H is generic and E ∈ V, for some τ̇ ∈ H i_{G}(τ̇) ∈ i_{G}(Ḋ). □

The proof is done. □

There are a couple of interesting cases in which, if Q is some forcing notion defined in the ground model and Q̇ names the forcing defined by working that definition in V^{P}, then Q is isomorphic to Q̇/P. This is useful in Reverse Easton forcing as another way of transferring generics between models.

Fact 2. Let P be a forcing notion of cardinality κ with the κ-c.c., where κ is such that κ^{<κ} = κ. Then

a) For any λ Add(κ, λ)_{V^{P}} /P is isomorphic to Add(κ, λ).

b) If λ ≥ κ is such that λ^{<κ} = λ then Coll(κ, λ)_{V^{P}} /P is isomorphic to Coll(κ, λ).

Proof. If β is some cardinal let P_{β} denote the partial ordering which has a top element ⊤ with β incomparable extensions (which we identify with the elements of β).

For a), recall that Add(κ, λ) is the product of λ copies of P_{2} with < κsupport. As P is κ−c.c. it is clear that Add(κ, λ)_{V^{P}} /P is just the product of λ copies of P_{2}/P with < κ-support. If τ̇ ∈ P_{2}/P, τ̇ is determined up to equivalence by the pair of Boolean values ([τ̇ = 0], [τ̇ = 1]), and any pair (b_{1}, b_{2}) such that b_{1}∧b_{2} = 0 corresponds to a name: moreover the ordering is just the restriction of the product order on R.O.(P)×R.O.(P). This ordering has a dense subset isomorphic with P_{κ}, where the κ incomparable elements are the pairs (b, ¬b) for b ∈ R.O.(P).

So Add(κ, λ)_{V^{P}} /P is isomorphic to the product of λ copies of P_{κ} with < κ-support, i.e. isomorphic to Add(κ, λ) as claimed. The proof for b) is exactly similar with P_{λ} playing the role of P_{2}; λ^{<κ} = λ guarantees that P_{λ}/P is isomorphic to P_{λ}. □

¶6. Two facts about Cohen forcing. The following facts are also needed in Section 2.

Fact 1. Let κ = cf(κ) > ω and let P be a κ-closed notion of forcing with |P | = κ. Then P is isomorphic to the Cohen forcing Add(κ, 1).

Proof. The proof is just like the proof (see [J] p.277) that a forcing which collapses λ and has size λ is isomorphic to Coll(ω, λ). □

Fact 2. Let j : V ⟶ M be the ultrapower of V by a normal measure U on some cardinal κ, where 2^{κ} = κ^{+}. Then j(Add(κ, κ^{++})) is isomorphic in V to Add(κ^{+}, κ^{++}).

Proof. As 2^{κ} = κ^{+} and j is an ultrapower embedding, j(κ^{++}) = κ^{++} and the set S defined by

S = { α < κ^{++} | j(α) = α }

is unbounded in κ^{++}. Let S̄, the closure of the set S, be enumerated in increasing order as ⟨λ_{ν} : ν < κ^{++}⟩. As S̄ is club in κ^{++},

κ^{++} = ⋃_{ν<κ^{++}}[λ_{ν}, λ_{ν+1})

In V the forcing Add(j(κ), [λ_{ν}, λ_{ν+1}))_{M} is κ^{+}−closed and cardinality κ^{+}, hence isomorphic by Fact 1 to the forcing Add(κ^{+}, [λ_{ν}, λ_{ν+1})). Choose for each ν an isomorphism π_{ν}, and define a function π with domain Add(j(κ), j(κ^{++}))_{M} by

π : p ⟼ ⋃_{ν<κ^{++}} π_{ν}(p ↾ j(κ) × [λ_{ν}, λ_{ν+1}))

Then we claim π is a bijection onto Add(κ^{+}, κ^{++}).

First we show that π maps into Add(κ^{+}, κ^{++}). Let p be a condition in Add(j(κ), κ^{++})_{M}. It is necessary to show that π(p) is a condition in Add(κ^{+}, κ^{++}), which will be immediate once we check that the support of π(p) has cardinality ≤ κ. ∃ν < κ^{++} support(p) ⊆ j(κ) × λ_{ν}, and therefore support(π(p)) ⊆ κ^{+} × λ_{ν}. Do an induction on μ ≤ ν with the hypothesis that π(p) ↾ κ^{+} × λ_{μ} has support of cardinality ≤ κ. The induction step is trivial at successor μ and those limit μ with cf_{V} (μ) ≤ κ. So assume cf_{V} (μ) > κ; λ_{μ} ∈ S, because cf_{V} (λ_{μ}) = cf_{V} (μ) > κ and so λ_{μ} is a continuity point of j. So

cf_{M}(λ_{μ}) = cf_{M}(j(λ_{μ})) = j(cf_{V} (λ_{μ})) > j(κ)

Hence ∃ρ < μ support(p ↾ j(κ) × λ_{μ}) ⊆ j(κ) × λ_{ρ} and the induction hypothesis goes through.

To show that π is onto, let q ∈ Add(κ^{+}, κ^{++}). Because ^{κ}M ⊂ M, and q has support of cardinality ≤ κ, the sequence

⟨π_{ν}^{−1}(q ↾ κ^{+} × [λ_{ν}, λ_{ν+1})) : ν < κ^{++} ∧ support(q) ∩ κ^{+} × [λ_{ν}, λ_{ν+1}) ≠ ∅⟩

is in M. As κ < j(κ) and j(κ) is regular in M, the union of this sequence is a condition which maps to q under π. □

3. The structure of the proof.

As the proof is rather long and complicated we give a brief overview of what is going on.

We start with the hypotheses

(1) κ is P_{3}κ-hypermeasurable, as witnessed by j : V ⟶ M with κM ⊂ M, V_{κ+3} ⊂ M, crit(j) = κ.

(2) GCH.

(3) j is generated by a (κ, κ^{+3})-extender.

(3) is really superfluous as given a model of (1), (2) we can factor through an extender to get (3).

In Section 2 we do two Reverse Easton extensions, and at the end we have a model of the hypotheses

(1) κ is P_{2}κ-hypermeasurable, as witnessed by j : V ⟶ M with κM ⊂ M, V_{κ+2} ⊂ M, crit(j) = κ.

(2) For each strong inaccessible α ≤ κ 2^{α} = α^{++} and for all n 0 < n < ω ⟹ 2^{α^{+n}} = α^{+n+1}.

(3) M ⊨ 2^{κ} = κ^{++} ∧ ∀n 0 < n < ω ⟹ 2^{κ^{+n}} = κ^{+n+1}

(4) There is F ∈ V which is P -generic over N where

P = Coll(κ^{+4}, < i(κ)) × Coll(i(κ), i(κ^{+}))_{N}

and i : V ⟶ N is the ultrapower of V by the ultrafilter

U = { X ⊂ κ | κ ∈ j(X) }

(5) j is generated by a (κ, κ^{+3})-extender

Then in Section 3 we use these hypotheses to define a modified Radin forcing; this forcing will shoot a closed unbounded set of V -measurables C through κ, and in addition if α, β are successive points of C it will add a generic for the collapsing algebra Coll(α^{+4}, < β) × Coll(β, β^{+}).

We prove analogues of the Prikry Lemma and Mathias’ criterion for Prikry genericity, and use these to show that the forcing has produced the right behaviour for the continuum function below κ and that κ has retained its hypermeasurability.

2. PREPARING THE MODEL

1. First stage.

Recall that the initial hypotheses are

(1) κ is P_{3}κ-hypermeasurable, as witnessed by j : V ⟶ M with κM ⊂ M, V_{κ+3} ⊂ M, crit(j) = κ.

(2) GCH.

(3) j is generated by a (κ, κ^{+3})-extender.

Factor j as k ∘ i where i : V ⟶ N is the ultrapower by the ultrafilter U = {X ⊆ κ|κ ∈ j(X)} and k : [f]_{U} = i(f)(κ) ⟼ j(f)(κ). Let λ be the critical point of k. GCH implies that λ = κ_{N}^{++}, and κ^{+} < λ < κ^{++}. As we remarked in the introduction k is generated by a (λ, κ^{+3})-N-extender of width κ_{N}^{+3}.

We do the Reverse Easton iteration of Add(α^{+}, α^{++}) for strong inaccessible α ≤ κ. Let P_{κ} denote the forcing up to stage κ, and let G_{κ} be P_{κ}-generic over V. Let Q_{κ} = (Add(κ^{+}, κ^{++}))_{V [G_{κ}]} be the forcing we do at stage κ, and let g be Q_{κ}-generic over V [G_{κ}]. Factor Add(κ^{+}, κ^{++}) as

(Add(κ^{+}, λ) × Add(κ^{+}, κ^{++} \ λ))_{V [G_{κ}]}

and split up g as g_{1} × g_{2} correspondingly.

Now consider the iterations j(P_{κ} ∗ Q̇_{κ}), i(P_{κ} ∗ Q̇_{κ}) computed by M and N respectively. As the three models V, M, N agree to rank κ they compute the same iteration below κ, so defining (P_{κ})_{M} and (P_{κ})_{N} in the obvious way

P_{κ} = (P_{κ})_{M} = (P_{κ})_{N}

P_{κ} is κ-c.c. and therefore

\begin{aligned}V [G_{κ}] & ⊨ ^{κ}M[G_{κ}] ⊂ M[G_{κ}] \\ V [G_{κ}] & ⊨ ^{κ}N[G_{κ}] ⊂ N[G_{κ}] \end{aligned}

So κ^{+} = κ_{V [G_{κ}]}^{+} = κ_{M[G_{κ}]}^{+} = κ_{N[G_{κ}]}^{+}. Because V_{κ+2} ⊂ M we have κ^{++} = κ_{M}^{++} = κ_{M[G_{κ}]}^{++}, and so if (Q_{κ})_{M} is the forcing M[G_{κ}] computes at κ then

(Q_{κ})_{M} = (Add(κ^{+}, κ^{++}))_{M[G_{κ}]} = (Add(κ^{+}, κ^{++}))_{V [G_{κ}]} = Q_{κ}

On the other hand if (Q_{κ})_{N} is the forcing which N[G_{κ}] computes at κ then because κ_{N[G_{κ}]}^{++} = κ_{N}^{++} = λ we have

(Q_{κ})_{N} = (Add(κ^{+}, κ^{++}))_{N[G_{κ}]} = (Add(κ^{+}, λ))_{V [G_{κ}]}

Now k(P_{κ}) = P_{κ} and ∀p ∈ G_{κ} k(p) = p, so we can extend k (using Fact I.2.3.1) to

k : N[G_{κ}] ⟶ M[G_{κ}]

k((Q_{κ})_{N}) = (Q_{κ})_{M} = Q_{κ}, and since crit(k) = λ and g_{1} ⊂ g we have that ∀p ∈ g_{1} k(p) = p ∈ g. So we can extend k further to get

k : N[G_{κ}][g_{1}] ⟶ M[G_{κ}][g]

Because the forcing notion (Q_{κ})_{N} is κ^{+} closed it adds no κ-sequences of ordinals, so

V [G_{κ}][g_{1}] ⊨ ^{κ}N[G_{κ}][g_{1}] ⊂ N[G_{κ}][g_{1}]

Also, because N is an ultrapower and GCH holds in V,

V [G_{κ}][g_{1}] ⊨ |i(κ)| = κ^{+}

Consider the iteration R which N[G_{κ}][g_{1}] does between κ + 1 and i(κ). Standard facts about Reverse Easton forcing imply that

N[G_{κ}][g_{1}] ⊨ |R| = i(κ), R is i(κ)-c.c. and κ^{+4}-closed

We exploit these facts to build H ∈ V [G_{κ}][g_{1}] which is R-generic over N[G_{κ}][g_{1}]. The closure of R in N[G_{κ}][g_{1}] and the width of the extender for k allow us to transfer H along k, and get k(H) k(R)-generic over M[G_{κ}][g].

So we can build maps

\begin{aligned}i & : V [G_{κ}] ⟶ N[G_{κ}][g_{1}][H] \\ j & : V [G_{κ}] ⟶ M[G_{κ}][g][k(H)] \\ k & : N[G_{κ}][g_{1}][H] ⟶ M[G_{κ}][g][k(H)] \end{aligned}

such that j = k ∘ i.

The map i : V [G_{κ}] ⟶ N[G_{κ}][g_{1}][H] is defined in V [G_{κ}][g_{1}]. Since Q_{κ} is κ^{+}-closed and i is generated by an extender of width κ we can transfer g_{1} along i and get an internal ultrapower

i : V [G_{κ}][g_{1}] ⟶ N[G_{κ}][g_{1}][H][i(g_{1})] = N[i(G_{κ} ∗ g_{1})]

Now let Q_{1} =_{def} (Add(κ, κ^{++}))_{V [G_{κ}][g_{1}]}. As V [G_{κ}][g_{1}] is a model of 2^{κ} = κ^{+} Fact I.2.6.2 implies that in V [G_{κ}][g_{1}]

i(Q_{1}) ≃ Add(κ^{+}, κ^{++})

Let Q_{2} =_{def} (Coll(κ, κ^{+}))_{V [G_{κ}][g_{1}]}. In V [G_{κ}][g_{1}] we also know that |i(κ^{+})| = κ^{+} and i(Q_{2}) is κ^{+}-closed so

i(Q_{2}) ≃ Add(κ^{+}, 1)

The upshot of all this is that in V [G_{κ}][g_{1}]

i(Q_{1} × Q_{2}) ≃ Add(κ^{+}, κ^{++} \ λ)

Recall that g_{2} is generic over V [G_{κ}][g_{1}] for this latter forcing (by closure, this has the same definition in V [G_{κ}] and V [G_{κ}][g_{1}]) so letting

Q =_{def} i(Q_{1} × Q_{2})

we can rearrange g_{2} as ḡ a Q-generic over V [G_{κ}][g_{1}].

Transferring g_{2} along i, j we get maps

\begin{aligned}i & : V [G_{κ}][g] ⟶ N[i(G_{κ} ∗ g)] \\ j & : V [G_{κ}][g] ⟶ M[j(G_{κ} ∗ g)] \\ k & : N[i(G_{κ} ∗ g)] ⟶ M[j(G_{κ} ∗ g)] \end{aligned}

with j = k ∘ i.

Let V^{∗} = V [G_{κ}][g], N^{∗} = N[i(G_{κ} ∗ g)], M^{∗} = M[j(G_{κ} ∗ g)]. From Fact I.2.4.2 we have that ḡ is Q-generic over N^{∗}. Closure implies that

Q_{1} × Q_{2} = (Add(κ, κ^{++}) × Coll(κ, κ^{+}))_{V^{∗}}

and hence that

Q = (Add(i(κ), i(κ^{++})) × Coll(i(κ), i(κ^{+})))_{N^{∗}}

It is also worth observing that i : V^{∗} ⟶ N^{∗} is the ultrapower of V^{∗} by the normal measure on κ induced by j.

We claim that j : V^{∗} ⟶ M^{∗} is a P_{3}κ-hypermeasurable embedding. As GCH holds in V^{∗} it is enough to show that

V^{∗} ⊨ P(κ^{++}) ⊂ M

This follows immediately from the fact that the forcing P_{κ} ∗ Q̇_{κ} has size κ^{++} and is κ^{++}-c.c., so that every subset of κ^{++} in V^{∗} has a canonical name in M.

We now have a model of the hypotheses

(1) κ is P_{3}κ-hypermeasurable, as witnessed by j : V ⟶ M with κM ⊂ M, V_{κ+3} ⊂ M, crit(j) = κ.

(2) GCH.

(3) j is generated by a (κ, κ^{+3})-extender.

(4) There is in V an object ḡ which is i(P)-generic over N, where P = Add(κ, κ^{++})×Coll(κ, κ^{+}) and i : V ⟶ N is the ultrapower by { X ⊆ κ | κ ∈ j(X) }.

2. Second stage.

We are given the hypotheses from the end of the first stage. Factor j in two steps through the models

N = the collapse of { j(f)(κ) | f : κ ⟶ V }
N̄ = the collapse of { j(f)(a) | a ∈ [κ^{++}]^{<ω} ∧ f : [κ]^{|a|} ⟶ V }

N is the familiar ultrapower approximating M, while N̄ corresponds to the extender of length κ^{++}. We have maps

\begin{aligned}i & : V ⟶ N \\ k & : N ⟶ M \\ ī & : N ⟶ N̄ \\ k̄ & : N̄ ⟶ M \end{aligned}

such that

\begin{aligned}j & = k ∘ i \\ k & = k̄ ∘ ī \end{aligned}

Let λ = κ_{N}^{++}, and argue as before that κ^{+} < λ < κ^{++}.

Do the reverse Easton iteration where we force at inaccessible λ ≤ κ with Add(λ, λ^{++}). Let P_{κ} be the forcing up to κ, and let G_{κ} be P_{κ}- generic over V. Let

\begin{aligned}Q_{κ} & = (Add(κ, κ^{++}))_{V [G_{κ}]} \\ P & = (Add(κ, κ^{++}) × Coll(κ, κ^{+}))_{V} \end{aligned}

Let g be Q_{κ}-generic over V [G_{κ}], and let ḡ be i(P)-generic over N as guaranteed by the hypotheses.

The same argument as in stage one shows that all four models compute the same forcing up to κ, and it is easy to extend the maps k, ī, k̄ to get

\begin{aligned}k & : N[G_{κ}] ⟶ M[G_{κ}] \\ ī & : N[G_{κ}] ⟶ N̄[G_{κ}] \\ k̄ & : N̄[G_{κ}] ⟶ M[G_{κ}] \end{aligned}

where

k = k̄ ∘ ī

N̄[G_{κ}], M[G_{κ}] are closed under κ-sequences from V [G_{κ}] and compute the true κ^{++}, so in the obvious notation

Q_{κ} = (Q_{κ})N̄ = (Q_{κ})_{M}

Also we have

(Q_{κ})_{N} = (Add(κ, λ))_{V [G_{κ}]}

Letting g factor as g_{1} × g_{2} where g_{1} is (Q_{κ})_{N}-generic we build further extensions

\begin{aligned}k & : N[G_{κ}][g_{1}] ⟶ M[G_{κ}][g] \\ ī & : N[G_{κ}][g_{1}] ⟶ N̄[G_{κ}][g] \\ k̄ & : N̄[G_{κ}][g] ⟶ M[G_{κ}][g] \end{aligned}

still preserving the relation

k = k̄ ∘ ī

The same argument as in stage one allows us to build H ∈ V [G_{κ}][g_{1}] which is R-generic over N[G_{κ}][g_{1}], where R is the iteration between κ+1 and i(κ) as computed by N[G_{κ}]. We transfer H along ī, k to get

\begin{aligned}i & : V [G_{κ}] ⟶ N[i(G_{κ})] \\ k & : N[i(G_{κ})] ⟶ M[j(G_{κ})] \\ ī & : N[i(G_{κ})] ⟶ N̄[ī∘ i(G_{κ})] \\ k̄ & : N̄[ī∘ i(G_{κ})] ⟶ M[j(G_{κ})] \end{aligned}

where all the maps are defined in V [G_{κ}][g]. Let l =_{def} ī∘ i.

|P_{κ}| = κ and P_{κ} is κ-c.c., so by Fact I.2.6.2 the term forcing

(Add(κ, κ^{++}) × Coll(κ, κ^{+}))_{V^{Pκ}} /P_{κ}

is isomorphic in V to P. Applying i, we use the i(P)-generic ḡ to define a generic g_{a} × g_{c}^{1} for the forcing

(Add(i(κ), i(κ^{++})) × Coll(i(κ), i(κ^{+})))_{N[i(G_{κ})]}

over the model N[i(G_{κ})]. Using the fact that

V [G_{κ}][g_{1}] ⊨ ^{κ}N[i(G_{κ})] ⊂ N[i(G_{κ})]

we also build g_{c}^{0} generic over N[i(G_{κ})] for the forcing

(Coll(κ^{+4}, < i(κ)))_{N[i(G_{κ})]}

and observe that automatically g_{c}^{0} is mutually generic with g_{a} × g_{c}^{1}.

We transfer g_{c}^{0}, g_{c}^{1}, g_{a} along ī to get new generics ḡ_{c}^{0}, ḡ_{c}^{1}, ḡ_{a}. We need to alter the generic ḡ_{a} to get a generic h_{a} such that

∀p ∈ g l(p) ∈ h_{a}  (†)

Recall that conditions in Q_{κ} are partial functions p : κ^{++} × κ ⇝ 2 with |dom(p)| < κ. Therefore l(p) = l“p, and so if q ∈ h_{a} we need to alter its values on the set l“κ^{++} × κ.

Claim 1. If q ∈ l(Q_{κ}) then |dom(q) ∩ (l“κ^{++} × κ)| ≤ κ in the model V [G_{κ}][g].

Proof. Let q = l(f)(a), where

f : [κ]^{|a|} ⟶ Q_{κ} ∧ f ∈ V [G_{κ}] ∧ a ∈ [κ^{++}]^{<ω}

Define

X_{x} = { β < κ^{++} | ∃γ < κ (β, γ) ∈ dom(f(x)) }

and let X = ⋃_{x∈[κ]^{|a|}} X_{x}. Clearly |X| ≤ κ and

{ β < κ^{++} | ∃γ < κ (i(β), γ) ∈ dom(q) } ⊆ X □

Since

V [G_{κ}][g] ⊨ ^{κ}N̄[l(G_{κ})] ⊂ N̄[l(G_{κ})]

we can alter each condition without getting out of the model N̄[l(G_{κ})], and the set of altered conditions h_{a} is in V [G_{κ}][g]. It remains to be shown that h_{a} is still l(Q_{κ})-generic. Let D ∈ N̄[l(G_{κ})] be a dense open subset of l(Q_{κ}).

Claim 2. Define an equivalence relation on l(Q_{κ}) by

p ∽ q ⟺ dom(p) = dom(q) ∧ |{ z | p(z) ≠ q(z) }| ≤ κ

Let

E = { q ∈ l(Q_{κ}) | ∀p ∈ l(Q_{κ}) p ∽ q ⟹ p ∈ D }

Then E is dense in l(Q_{κ}).

Proof. Work in N̄[l(G_{κ})].We first prove the following fact.

Subclaim 1. If r ∈ l(Q_{κ}) then there exists s ≤ r such that

∀p p ∽ r ⟹ p ∪ (s \ r) ∈ D

Proof. Enumerate { X ⊆ dom(r) | |X| ≤ κ } as ⟨X_{α} : α < μ⟩ where μ < l(κ). Define by induction a decreasing sequence of conditions ⟨r_{α} : α < μ⟩. r_{0} = r, and r_{β+1} is defined as follows; let q be given by

q(z) = \{ \begin{matrix}r_{β}(z) \\ 1 \ r_{β}(z) \end{matrix} \begin{matrix}z ∈ & dom(r_{β}) \ X_{β} \\ z ∈ & X_{β} \end{matrix}

let q̄ ≤ q be in D, and let r_{β+1} = r_{β} ∪ (q̄ \ q). The forcing l(Q_{κ}) is l(κ)-closed so define s as a lower bound for ⟨r_{α} : α < μ⟩. s clearly works. □

To finish the proof of the claim start with r ∈ l(Q_{κ}) and apply the subclaim κ^{+} times to produce a decreasing sequence ⟨s_{α} : α < κ^{+}⟩ such that if p ∽ s_{α} then p ∪ (s_{α+1} \ s_{α}) ∈ D. Let q be a lower bound for ⟨s_{α} : α < κ^{+}⟩. We claim this works. If dom(p) = dom(q) and X = {z|p(z) ≠ q(z)} has cardinality ≤ κ then for some α X ⊆ dom(s_{α}). So p ∈ D. □

This suffices to show that h_{a} is generic, and a small variation on the argument shows that it is mutually generic with ḡ_{c}^{0} × ḡ_{c}^{1}.

We now transfer h_{a} along k̄ to get H_{a} j(Q_{κ})-generic over M. (†) implies that ∀p ∈ g j(p) ∈ H_{a}, so we can build maps

\begin{aligned}j & : V [G_{κ}][g] ⟶ M[j(G_{κ} ∗ g)] \\ k̄ & : N̄[l(G_{κ} ∗ g)] ⟶ M[j(G_{κ} ∗ g)] \\ l & : V [G_{κ}][g] ⟶ N̄[l(G_{κ} ∗ g)] \end{aligned}

where

j = k̄ ∘ l

Define V^{∗} = V [G_{κ}][g], M^{∗} = M[j(G_{κ} ∗g)], N^{∗} = N̄[l(G_{κ} ∗g)]. We argue that l : V^{∗} ⟶ N^{∗} is just the ultrapower approximating j : V^{∗} ⟶ M^{∗}. To do this factor l through l^{†} : V ⟶ N^{†} the ultrapower approximating l, say l = k^{†} ∘ l^{†}. P_{V^{∗}}(κ) ⊂ N^{†} and N^{†} ⊨ 2^{κ} = κ^{++} so κ_{N^{†}}^{++} = κ^{++} and crit(k^{†}) > κ^{++}. Since κ^{++} ⊂ rge(k^{†}) and N^{∗} is generated by a (κ, κ^{++})-extender N^{†} = N^{∗} and we are done.

It remains to be argued that ḡ_{c}^{0} × ḡ_{c}^{1} is generic for an appropriate collapse ordering, but this is immediate by the closure of l(Q_{κ}).

So we have produced a model of the hypotheses

(1) κ is P_{2}κ-hypermeasurable, as witnessed by j : V ⟶ M with κM ⊂ M, V_{κ+2} ⊂ M, crit(j) = κ.

(2) For each strong inaccessible α ≤ κ 2^{α} = α^{++} and for all n 0 < n < ω ⟹ 2^{α^{+n}} = α^{+n+1}.

(3) M ⊨ 2^{κ} = κ^{++} ∧ ∀n 0 < n < ω ⟹ 2^{κ^{+n}} = κ^{+n+1}

(4) There is F ∈ V which is P -generic over N where

P = Coll(κ^{+4}, < i(κ)) × Coll(i(κ), i(κ^{+}))_{N}

and i : V ⟶ N is the ultrapower of V by the ultrafilter

U = { X ⊂ κ | κ ∈ j(X) }

(5) j is generated by a (κ, κ^{+3})-extender We make some incidental remarks

(1) F ∈ M, as it can be coded by an element of V_{κ+2}.

(2) κ_{N}^{++} = κ^{++} and κ_{M}^{+3} = κ^{+3}

(3) crit(k̄) = κ_{N}^{+3} < κ^{+3}

(4) F can be transferred along k̄ because it is generic for sufficiently closed forcing.

3. THE RADIN FORCING

1. Measure sequences.

We use the results of the last section to define measure sequences, which will be the building blocks for the modified Radin forcing.

Definition. If α and β are inaccessibles,

P (α, β) =_{def} Coll(α^{+4}, < β) × Coll(β, β^{+})

Retaining the notation used in the last paragraph of the preceding section, let i_{12} : N ⟶ i(N) be i(i), and let i_{02} : V ⟶ i(N) be i(i) ∘ i. Since N ⊨ ^{i(κ)}i(N) ⊂ i(N)

P = i(P)(κ, i(κ)) = i_{02}(P)(κ, i(κ))

and so we can sensibly define

\begin{aligned}P^{∗∗} & = \{ f : [κ]^{2} ⟶ V_{κ} | dom(f) ∈ U × U ∧ ∀α, β f(α, β) ∈ P (α, β) \} \\ F^{∗∗} & = \{ f ∈ P^{∗∗} | i_{02}(f)(κ, i(κ)) ∈ F \} \end{aligned}

Notice that U can be read off from F^{∗∗} as

U = { X ⊆ κ | ∃f ∈ F^{∗∗} dom(f) = X × X }

We use j and F^{∗∗} to define our prototypical measure sequence u:

u(0) =_{def} crit(j) = κ
u(1) =_{def} F^{∗∗}
u(β) =_{def} { X ∈ V_{κ+1} | u ↾ β ∈ j(X) } (β > 1)

where the construction continues for as long as u ↾ β ∈ M. If the reader is familiar with Radin forcing she will notice that u has in second place the object F^{∗∗}, where putting

Ū =_{def} { X ⊂ V_{κ} | { α | ⟨α⟩ ∈ X } ∈ U }

would give a standard Radin sequence as in [R].

The idea behind the definitions that follow is that we use the strong reflection properties of the cardinal κ to show that V_{κ} contains many objects which resemble u. In fact each measure on the sequence u should concentrate on the set of u-like sequences.

Definition. (k, g) is a constructing pair iff

(1) k : V ⟶ k(V) is an elementary embedding into a transitive inner model, and if α is the critical point then ^{α}k(V) ⊂ k(V).

(2) g is k_{0}(P)(α, k_{0}(α))-generic over k_{0}(V), where k_{0} is the ultrapower approximating k (as usual we factor k through k_{0}, say k = l ∘ k_{0}).

(3) g ∈ k(V).

(4) g can be transferred along l to give a k(P)(α, k(α))-generic over k(V).

Definition. If (k, g) is a constructing pair as above then g^{∗∗} is defined from g in the same way as F^{∗∗} was defined from F, that is

g^{∗∗} =_{def} { f | k_{02}(f)(α, k_{0}(α)) ∈ g }

where k_{02} = k_{0}(k_{0}) ∘ k_{0}.

Definition. If (k, g) is a constructing pair as above, then a sequence w is constructed by (k, g) iff

(1) w ∈ k(V).

(2) w(0) = α = crit(k).

(3) w(1) = g^{∗∗}.

(4) w(β) = { X ∈ V_{α+1} | w ↾ β ∈ k(X) } (1 < β < lh(w)).

(5) k(V) ⊨ |lh(w)| ≤ w(0)^{++}.

(j, F) is the archetypal constructing pair, and for every β < κ^{+3} it constructs w ↾ β.

Definition. If w is constructed by some (k, g) then define

κ_{w} = w(0)

If lh(w) ≥ 2 define also

g_{w}^{∗∗} =_{def} w(1)
μ_{w} =_{def} { X ⊂ κ_{w} | ∃f ∈ g_{w}^{∗∗} dom(f) = [X]^{2} }
μ̄_{w} =_{def} { X ⊂ V_{κ_{w}} | { α | ⟨α⟩ ∈ X } ∈ μ_{w} }
g_{w} =_{def} { [f]_{μ_{w}×μ_{w}} | f ∈ g_{w}^{∗∗} }
F_{w} =_{def} μ̄_{w} ∩ ⋂{ w(α) | 1 < α < lh(w) }

where [f]_{μ_{w}×μ_{w}} is the set represented by the function f in the ultrapower V^{κ_{w}×κ_{w}}/μ_{w} × μ_{w}.

Of course, given a particular (k, g) which constructs w then g_{w} = g and so forth, but the point is that these definitions are independent of the choice of a constructing pair for w.

Definition. Define inductively classes of sequences

U_{0} =_{def} { w | ∃(k, g) (k, g) constructs w }
U_{n+1} =_{def} { w ∈ U_{n} | U_{n} ∩ V_{κ_{w}} ∈ F_{w} }
U_{∞} =_{def} ⋂_{n∈ω} U_{n}

Lemma 1. For every α < κ^{+3} u ↾ α exists and is a member of U_{∞}.

Proof. It is routine to check that if α < κ^{+3} then u ↾ α can be coded by an element of V_{κ+2}, and hence u ↾ α ∈ M and u(α) exists.

Before proving the second half we unravel the definition of U_{∞}.

Claim 1.

∀α < κ^{+3} u ↾ α ∈ U_{∞}

if and only if

∀α < κ^{+3} ∀n < ω M_{n} ⊨ u ↾ α ∈ U_{0}

where M_{n} is the n^{th} iterate of V by j.

Proof. This is quite easy by induction.

∀α < κ^{+3} u ↾ α ∈ U_{n+1}
⟺ ∀α < κ^{+3} U_{n} ∩ V_{κ} ∈ u(α)
⟺ ∀α < κ^{+3} u ↾ α ∈ j(U_{n})
⟺ ∀α < κ^{+3} M ⊨ u ↾ α ∈ U_{n}
⟺ ∀α < κ^{+3} M_{n+1} ⊨ u ↾ α ∈ U_{0} □

Fix α < κ^{+3}. We prove that M ⊨ u ↾ α ∈ U_{0}, and then use the agreement between M and M_{n} for n > 1 to argue that this is enough.

Fix β such that, if j_{β} : V ⟶ M_{β} is the elementary embedding had by restricting the extender E which describes j to E_{β} = E ↾ [β]^{<ω}, then

(1) κ^{++} < β < κ^{+3} and β > α.

(2) ^{κ}M_{β} ⊂ M_{β}.

(3) u ↾ α and F are in M_{β}.

By the choice of β κ_{M_{β}}^{++} = κ^{++} and α < κ_{M_{β}}^{+++}. Also E_{β} ∈ M, and if j_{β}^{∗} : M ⟶ j_{β}^{∗}(M) is the embedding which M computes from E_{β} then j_{β}^{∗} = j_{β} ↾ M and the models M_{β}, j_{β}^{∗}(M) agree up to rank β.

Using this information it is routine to check that

M ⊨ “ (j_{β}^{∗}, F) constructs u ↾ α”

To finish the proof observe that if j_{1n} : M ⟶ M_{n} is the map arising from the iteration of j then

crit(j_{1n}) = j(κ) > κ^{+3} > rank(u ↾ α)

so

M_{n} ⊨ u ↾ α ∈ U_{0} □

Because the measures on a measure sequence are countably complete, if w ∈ U_{∞} and lh(w) ≥ 2 then U_{∞} ∩ V_{κ} ∈ F_{w}.

2. The forcing poset.

Given w ∈ U_{∞} with lh(w) ≥ 2 we define a poset Q_{w}. The forcing which we eventually use to establish the theorem will be Q_{u↾β} for a wellchosen value of β, but it is advantageous to do the proofs uniformly for all Q_{w}.

Definition. p is a condition in Q_{w} iff

p = ⟨X_{i} : i < n⟩

where n > 1 and

1) X_{0} = ⟨κ_{0}⟩ for some κ_{0} < κ.

2) For 0 < i < n − 1, X_{i} is

EITHER a pair (P_{i}, κ_{i}) where κ_{i} < κ. P_{i} ∈ P (κ_{i−1}, κ_{i}).

OR a quadruple (w_{i}, A_{i}, H_{i}, h_{i}), where w_{i} ∈ U_{∞} and lh(w_{i}) > 1. κ_{i} =_{def} κ_{w_{i}}. A_{i} ∈ F_{w_{i}}. H_{i} ∈ g_{w_{i}}^{∗∗}. h_{i} is a function, dom(h_{i}) = { κ_{v} | v ∈ A_{i} } \ (κ_{i−1} + 1). ∀α ∈ dom(h_{i}) h_{i}(α) ∈ P (κ_{i−1}, α). dom(H_{i}) = [dom(h_{i})]^{2}.

3) κ_{0} < κ_{1} < ⋯ < κ_{n−1} < κ.

4) X_{n} is a quadruple (w_{n}, A_{n}, H_{n}, h_{n}) with the same properties as above, such that w_{n} = w.

Before giving the formal definition of the ordering on the conditions we give some informal motivation. The intended generic object is a pair of sequences (v⃗, F⃗) where

(1) lh(v⃗) = lh(F⃗).

(2) ∀α < lh(v⃗) v(α) ∈ U_{∞}. (Let κ_{α} = κ_{v(α)})

(3) α ⟼ κ_{α} is monotone increasing, and C = { κ_{α} | α < lh(v⃗) } is closed unbounded in κ_{w}.

(4) For α < lh(v⃗) F (α) is a P (κ_{α}, κ_{α+1})-generic filter. The typical condition p = ⟨X_{i} : i < n⟩ of the last definition is intended

put the following constraints on (v⃗, F⃗):

(1) v_{0} = κ_{0}.

(2) If 0 < i < n − 1 and X_{i} is a pair (P_{i}, κ_{i}) then for some α < lh(v⃗) κ_{i−1} = κ_{α}, κ_{i} = κ_{α+1}, and P_{i} ∈ F (α).

(3) If 0 < i < n and X_{i} is a quadruple (w_{i}, A_{i}, H_{i}, h_{i}) then κ_{i} is a limit point κ_{λ} in C. Let κ_{i−1} = κ_{α}. Then h_{i}(κ_{α+1}) ∈ F (α), and if α < β < λ we have v(β) ∈ A_{i} and H(κ_{β}, κ_{β+1}) ∈ F (β).

Definition. Let X be a tuple.

(1) u_{X} = ⟨δ⟩ if X = ⟨δ⟩ or X = ⟨P, δ⟩.

(2) u_{X} = w if X = ⟨w, A, H, h⟩.

(3) κ_{X} = κ_{u_{X}}

Definition. Let

\begin{matrix}p = ⟨X_{i} : i < m⟩ \\ q = ⟨Y_{j} : j < n⟩ \end{matrix}

p extends q iff

1) m ≥ n

2) {κ_{X_{0}}, … κ_{X_{m−1}}} ⊇ {κ_{Y_{0}}, … κ_{Y_{n−1}}}

3) κ_{X_{0}} = κ_{Y_{0}}, X_{0} = Y_{0}

4) For 0 < i < m, if X_{i} is a pair ⟨P, α⟩ then EITHER there is j < n such that Y_{j} = ⟨Q, α⟩, and P ≤ Q (this will make sense because other clauses of the definition will ensure that in this case κ_{X_{i−1}} = κ_{Y_{j−1}}, so P and Q are in the same partial ordering) OR there is no j with α = κ_{Y_{j}} and for the least j such that α < κ_{Y_{j}}

Y_{j} is a quadruple ⟨u, A, H, h⟩

α ∈ A

IF κ_{X_{i−1}} = κ_{Y_{j−1}} P ≤ h(α)

IF κ_{X_{i−1}} > κ_{Y_{j−1}} P ≤ H(κ_{X_{i−1}}, α)

5) For 0 < i < m, if X_{i} is a quadruple ⟨u, A, H, h⟩ then

EITHER there is j < n Y_{j} = ⟨u, B, I, i⟩, in which case A ⊆ B ∀(α, β) ∈ dom(h) H(α, β) ≤ I(α, β) and IF κ_{X_{i−1}} = κ_{Y_{j−1}} ∀α ∈ dom(h) h(α) ≤ i(α) IF κ_{X_{i−1}} > κ_{Y_{j−1}} ∀α ∈ dom(h) h(α) ≤ I(κ_{X_{i−1}}, α)

OR there is no j with κ_{X_{i}} = κ_{Y_{j}}, and for the least j such that

κ_{X_{i}} < κ_{Y_{j}}
Y_{j} is a quadruple ⟨v, C, J, j⟩
u ∈ C
A ⊆ C
∀(α, β) ∈ dom(H) H(α, β) ≤ J(α, β)
IF κ_{X_{i−1}} = κ_{Y_{j−1}} ∀α ∈ dom(h) h(α) ≤ j(α)
IF κ_{X_{i−1}} > κ_{Y_{j−1}} ∀α ∈ dom(h) h(α) ≤ J(κ_{X−1}, α)

Remark. Q_{w} has κ_{w}^{++}-c.c.

Proof. Let ⟨p_{α} : α < κ_{w}^{++}⟩ be an antichain in Q_{w}, and let us write

p_{α} = x_{α} ⌢ ⟨w, A_{α}, H_{α}, h_{α}⟩

Since x_{α} ∈ V_{κ_{w}} we may assume that x_{α} is some fixed sequence x. Incompatibility can only arise from disagreement between the h_{α}, and p_{α} ⊥ p_{β} implies h_{α} and h_{β} represent incompatible conditions in the ultrapower by μ_{w}. Since they represent conditions in a forcing notion of size κ_{w}^{++} we have a contradiction and the claim follows. □

3. The generic object.

Definition. A sequence u ∈ U_{∞} occurs in a condition p = ⟨X_{i} : i < m⟩ iff there is i < m − 1 such that one of the following holds:

(1) i = 0 and u = X_{0}

(2) X_{i} = ⟨P, δ⟩ and u = ⟨δ⟩

(3) X_{i} = ⟨u, A, H, h⟩

Definition. P occurs in p = ⟨X_{i} : i < m⟩ iff there is i with 0 < i < m such that X_{i} = ⟨P, δ⟩ for some δ.

Notice that w does not occur in the conditions of Q_{w}.

Definition. Let G be a Q_{w}-generic filter. Define a pair of sequences

(⟨v(α) : α < λ⟩, ⟨F (α) : α < λ⟩)

(1) ⟨v(α) : α < λ⟩ enumerates { v ∈ U_{∞} | ∃p ∈ G v occurs in p } so that α ⟼ κ_{α} = κ_{v(α)} is increasing.

(2) F (α) = { P ∈ P (κ_{α}, κ_{α+1}) | ∃p ∈ G P occurs in p }

It is routine to check that the generic filter G and the pair of sequences (v⃗, F⃗) are each definable from the other.

We briefly develop the basic properties of the forcing; the proofs are very similar to those for Radin forcing(see [R]).

Lemma 1. Let G be Q_{w}-generic and let (v⃗, F⃗) and κ⃗ be defined as above. Then C = { κ_{α} | α < lh(v⃗) } is a closed unbounded subset of κ_{w}.

Proof.

1. C is closed.

Proof. Let p ∈ Q_{w} be a condition, α < κ_{w} an ordinal such that p ⊩ α ∉ C. No sequence v with κ_{v} = α occurs in p, so consider X the first entry of p with κ_{X} > α. If X is a pair then p forces α is not a limit point of C. If X is a quadruple ⟨u, A, H, h⟩ then define p′ ≤ p by shrinking A to A′ = A \ V_{α}, so that p′ forces α is not a limit point of C.

2. C is unbounded.

Proof. Let β < κ_{w}, p ∈ Q_{w}, say p = x ⌢ ⟨w, A, H, h⟩. Since A ∈ μ̄_{w} there is ⟨α⟩ ∈ A with α > β, so p can be extended to

p′ = x ⌢ ⟨h(α), α⟩ ⌢ ⟨w, A, H, H(α)⟩

and p′ forces that β is not a bound for C.

The lemma is proved. □

We make the remark that if lh(w) ≥ 3 then we could also have done the “unbounded” clause by showing that many quadruples could be added to the condition p. (See the next Lemma)

Lemma 2. With the same assumptions as lemma 1

(1) If μ < λ is such that lh(v(μ)) > 1 then the pair (v⃗ ↾ μ, F⃗ ↾ μ) is generic for the forcing Q_{v(μ)}.

(2) ∀A ∈ V_{κ_{w}+1} (A ∈ F_{w} ⟺ ∃β ∀α > β v(α) ∈ A)

(3) ∀H ∈ g_{w}^{∗∗} ∃β ∀α > β H(κ_{α}, κ_{α+1}) ∈ F (α)

Proof. (1) follows immediately from the observation that if p is a condition in which u_{μ} appears then Q_{w} ↾ p factors into two parts, of which the first is Q_{u_{μ}} ↾ p′ for p′ ∈ Q_{u_{μ}} an initial segment of p.

For (2), first assume that B ∈ F_{w}. Let p = x ⌢ ⟨w, A, H, h⟩ be some condition; then the condition

p′ = x ⌢ ⟨w, A ∩ B, H, h⟩

is an extension of p forcing that a final segment of v⃗ lies in B. For the other direction, assume towards a contradiction that B contains a final segment of v⃗ but B ∉ F_{w}. The “unbounded” clause of the last lemma shows that V_{κ} \ B ∉ μ̄_{w}, so we must have V_{κ} \ B ∈ w(β) for β > 1.

Subclaim 1. If β > 1 and B ∈ w(β) then v(α) ∈ B for an unbounded set of α < λ.

Proof. We show that if α < κ_{w} and q = x ⌢ ⟨w, A, H, h⟩ is a condition then the set of v for which κ_{v} > α and there exist B, I, i such that (v, B, I, i) can be added to q is a measure one set for w(β). This clearly suffices.

Fix (k, g) a constructing pair for w. By definition, it is enough to show that in k(V) there are are B, I, i such that ⟨w ↾ β, B, I, i⟩ can be added to

k(q) = x ⌢ ⟨k(w), k(A), k(H), k(h)⟩

We claim that A, H, h will work (they exist in k(V) because they are all elements of V_{κ_{w}+1}).

(1) A ∈ w(β), so w ↾ β ∈ k(A).

(2) A = i(A) ∩ V_{κ_{w}} ⊂ i(A).

(3) H = i(H) ↾ κ_{w} × κ_{w}.

(4) h = i(h) ↾ κ_{w}. The subclaim is proved. □

This gives a contradiction, so (2) is proved.

Finally, let I ∈ g_{w}^{∗∗}. From (2) (κ_{α}, κ_{α+1}) ∈ dom(I) for large enough α, and an argument parallel to that for a) shows that any condition refines to one which forces I(κ_{α}, κ_{α+1}) ∈ F_{α} on a final segment. The lemma is proved. □

4. The Prikry property.

We prove that questions about the generic extension by some Q_{w} can be decided in a simple way; this will enable us to prove that the forcing does not do too much damage to the universe.

Definition. If p and q are conditions in Q_{w} then q is a direct extension of p iff q ≤_{Q_{w}} p and lh(q) = lh(p). We write q ≤_{d} p in this case.

Theorem 1. Let b ∈ RO(Q_{w}) be a Boolean value, and let p ∈ Q_{w} be some condition. Then there is q ≤_{d} p such that q ∥ b.

Proof. The strategy of the proof is to write p as z ⌢ ⟨w, A, H, h⟩, show that b is decided by some extension z′ ⌢ ⟨w, A′, H′, h′⟩ ≤ p in which no sequences have been added past the top of z, and then make an appeal to induction and the uniform definition of the forcing. So fix p = p^{0} = z^{0} ⌢ ⟨w, A^{0}, H^{0}, h^{0}⟩. Also fix (k, g) a constructing pair for w.

We factor k through k_{0} the ultrapower by μ_{w}, to get

k = l ∘ k_{0}

Definition.

(1) P_{1}^{∗} = {f ∈ k_{0}(V)|dom(f) ∈ k_{0}(μ_{w})∧∀α f(α) ∈ k_{0}(P)(κ_{w}, α)}.

(2) g^{∗} = { f ∈ P_{1}^{∗} | [f]_{k_{0}(μ_{w})} ∈ g }.

(3) P^{∗} = l(P_{1}^{∗}).

(4) P^{∗∗} = { F ∈ V | ∀(α, β) ∈ dom(F) F (α, β) ∈ P (α, β) }.

Equip P_{1}^{∗} with the pre-ordering

f ≤ g ⟺ { α | f(α) ≤ g(α) } ∈ k_{0}(μ_{w})

and equip P^{∗} similarly. By the definition of constructing pair g^{∗} is P_{1}^{∗}-generic over k_{0}(V) and l“g^{∗} generates a P^{∗}-generic over k(V).

Definition. x is a lower part iff x is a proper initial segment of some condition in Q_{w}.

Definition. If x is a lower part then κ_{x} =_{def} κ_{X} for X the last tuple in x.

Definition. Let F, G be elements of P^{∗∗}.

(1) F ≤ G iff ∀(α, β) ∈ dom(F) F (α, β) ≤ G(α, β).^{1}

(2) F ≤^{∗} G iff { (α, β) | F (α, β) ≤ G(α, β) } ∈ μ_{w} × μ_{w}. In the first case we say that F refines G, in the second that F refines G almost everywhere.

Similarly, let f, g be functions with domain in μ_{w} such that for some δ < κ ∀α f(α), g(α) ∈ P (δ, α).

(1) f ≤ g iff ∀α ∈ dom(f) f(α) ≤ g(α).

(2) f ≤^{∗} g iff { α | f(α) ≤ g(α) } ∈ μ_{w}.

Step One. We find A^{1} ⊆ A^{0}, H^{1} ≤ H^{0}, h^{1} ≤ h^{0}, such that

p^{1} = z^{0} ⌢ ⟨w, A^{1}, H^{1}, h^{1}⟩

has the following property; if z ⌢ X ⌢ ⟨w, B, I, i⟩ is an extension of p^{1} deciding b and having κ_{X} > κ_{z^{0}} then there exist Ā ⊂ A^{1}, H̄ ≤ H^{1} such that z ⌢ X ⌢ ⟨w, Ā, H̄, H̄(κ_{X})⟩ decides b.

Construction. Consider conditions in k(Q_{w}) of the form

z ⌢ X ⌢ ⟨k(w), A, H, h⟩

where z is a lower part, X is a tuple such that u_{X} = w ↾ α for some 0 < α < lh(w). Notice that h ∈ P^{∗}.

1So implicitly we are demanding dom(F) ⊆ dom(G).

Say that h ∈ P^{∗} is good if for all lower parts z ∈ V_{κ_{w}} and all X such that ∃α < lh(w) u_{X} = w ↾ α

∃A ∃H ∃h′ ≤ h z ⌢ X ⌢ ⟨k(w), A, H, h′⟩ ∥ k(b) ⟹
⟹ ∃A′ ∃H′ z ⌢ X ⌢ ⟨k(w), A′, H′, h⟩ ∥ k(b)

Claim 1. The set of good h is a dense subset of P^{∗} lying in k(V).

Proof of Claim. Work in k(V). There are κ_{w} possibilities for z. From the definition of constructing pair w ∈ k(V) and

k(V) ⊨ |lh(w)| ≤ κ_{w}^{++}

Also for each α there are only 2^{κ_{w}} = κ_{w}^{++} X such that u_{X} = w ↾ α. So enumerate the possible pairs (z, X) as ⟨(z_{γ}, X_{γ}) : γ < κ_{w}^{++}⟩. Fix f ∈ P^{∗}, and define a descending sequence in P^{∗} by setting f_{0} = f and defining f_{γ+1} to be some g ≤ f_{γ} such that

∃A ∃H z_{γ} ⌢ X_{γ} ⌢ ⟨k(w), A, H, g⟩ ∥ k(b)

if such exists, and f_{γ+1} = f_{γ} otherwise. Let f′ ∈ P^{∗} be a lower bound for the sequence ⟨p_{γ} : γ < κ_{w}^{++}⟩. Now let z, X, A, H, h′ ≤ f′ be such that

z ⌢ X ⌢ ⟨k(w), A, H, h′⟩ ∥ k(b)

(z, X) is enumerated as (z_{γ}, X_{γ}) for some γ. f_{γ+1} was defined, so for some B, I

z ⌢ X ⌢ ⟨k(w), B, I, f_{γ+1}⟩ ∥ k(b)

f′ ≤ f_{γ+1}, so B, I witness that f′ is good. □

Since g^{∗} is generic we can find F ∈ g^{∗∗} such that F ≤ H^{0} and k(F)(κ_{w}) is good. Now the set R of those u such that

∀z ∀X [u_{X} = u ∧ ∃A ∃H ∃h′ ≤ F (κ_{u}) z ⌢ X ⌢ ⟨w, A, H, h′⟩ ∥ b ⟹

⟹ ∃A′ ∃H′ z ⌢ X ⌢ ⟨w, A′, H′, F (κ_{u})⟩ ∥ b

is by construction a member of the filter F_{w}.

Define

A^{1} = A^{0} ∩ R
h^{1} = h^{0} ↾ { κ_{v} | v ∈ A^{1} } \ (κ_{z^{0}} + 1)
H^{1} = F ↾ [dom(h^{1})]^{2}

We prove that p^{1} has the right behaviour. If z ⌢ X ⌢ ⟨w, B, I, i⟩ is an extension of p^{1} deciding b and having κ_{X} > κ_{z^{0}} then u_{X} is chosen from R and i ≤ F (κ_{X}) so we can find A′ and H′ as in the definition of R. It is easy to alter A′, H′ to Ā, H̄ with the right properties. □

Step Two. We find A^{2} ⊆ A^{1}, H^{2} ≤ H^{1}, h^{2} ≤ h^{1}, such that

p^{2} = z^{0} ⌢ ⟨w, A^{2}, H^{2}, h^{2}⟩

has the following property; if z ⌢ X ⌢ ⟨w, B, I, i⟩ is an extension of p^{2} deciding b and having κ_{X} > κ_{z^{0}} then z ⌢ X ⌢ ⟨w, A^{2}, H^{2}, H^{2}(κ_{X})⟩ decides b.

Construction. For each α < κ_{w} there are fewer than κ_{w} lower parts z with κ_{z} = α. Fix α for the moment, and for each z such that κ_{z} = α and

∃B ⊆ A^{1} ∃G ≤ H^{1} z ⌢ ⟨w, B, G, H^{1}(κ_{z})⟩ ∥ b

choose an appropriate pair (B^{z}, G^{z}). Using genericity find F_{α} ∈ g^{∗∗} such that F_{α} ≤ G^{z} for all z with κ_{z} = α. Find C_{α} ∈ with [C_{α}]^{2} ⊂ dom(F_{α}).

E_{α} =_{def} A^{1} ∩ ⋂_{z,κ_{z}=α} B^{z} ∩ { v | ⟨κ_{v}⟩ ∈ C_{α} }

By the usual argument using the constructing pair E_{α} ∈ F_{w}. By genericity again find F ∈ g^{∗∗}, F ≤ H^{1}, such that

∀α < κ_{w} F ≤^{∗} F_{α}

Find D_{α} ∈ μ_{w} such that

∀(γ, δ) ∈ [D_{α}]^{2} F (γ, δ) ≤ F_{α}(γ, δ)

Let

\begin{aligned}E′_{α} & = E_{α} ∩ \{ v | ⟨κ_{v}⟩ ∈ D_{α} \} \\ A^{2} & = \{ v ∈ A^{1} | ∀α < κ_{w} v ∈ E′_{α} \} \\ h^{2} & = h^{1} ↾ \{ κ_{v} | v ∈ A^{2} \} \\ H^{2} & = F ↾ [dom(h^{2})]^{2} \\ p^{2} & = z^{0} ⌢ ⟨w, A^{2}, H^{2}, h^{2}⟩ \end{aligned}

We claim that p^{2} has the property we demanded. To show this let z ⌢ X ⌢ ⟨w, B, I, i⟩ be an extension of p^{2} deciding b and having κ_{X} > κ_{z^{0}}. Let x = z ⌢ X. Now by the work we did in Step One and the definitions above

z ⌢ X ⌢ ⟨w, B^{x}, G^{x}, H^{2}(κ_{X})⟩ ∥ b

Therefore it will be enough to show that

z ⌢ X ⌢ ⟨w, A^{2}, H^{2}, H^{2}(κ_{X})⟩ ≤ z ⌢ X ⌢ ⟨w, B^{x}, G^{x}, H^{2}(κ_{X})⟩ which amounts to showing that anything which can be added between X and the top on the LHS can also be added on the RHS.

If v ∈ A^{2}, and κ_{v} > κ_{X} = κ_{x}, then v ∈ E′_{κ_{x}} and therefore v ∈ B^{x}. If v, w ∈ A^{2}, and κ_{x} < κ_{v} < κ_{w}, then v, w ∈ E′_{κ_{x}} and so by definition

κ_{v}, κ_{w} ∈ C_{κ_{x}} ∩ D_{κ_{x}}. Therefore

H^{2}(κ_{v}, κ_{w}) = F (κ_{v}, κ_{w})
≤ F_{κ_{x}}(κ_{v}, κ_{w}) because (κ_{v}, κ_{w}) ∈ [D_{κ_{x}}]^{2}
≤ G^{x}(κ_{v}, κ_{w}) because (κ_{v}, κ_{w}) ∈ [C_{κ_{x}}]^{2}

So the constraint imposed by A^{2} and H^{2} is more stringent than that imposed by B^{x} and G^{x}, and the claim is proved. □

Step Three. We find A^{3} ⊆ A^{2}, H^{3} ≤ H^{2}, h^{3} ≤ h^{2} such that if

p^{3} = z^{0} ⌢ ⟨w, A^{3}, H^{3}, h^{3}⟩

and

z′ ⌢ X ⌢ ⟨P, δ⟩ ⌢ ⟨w, A′, H′, h′⟩ ≤ p^{3}

is a condition deciding b, κ_{z^{0}} < κ_{X} < δ, then

z′ ⌢ X ⌢ ⟨H^{3}(κ_{X}, δ), δ⟩ ⌢ ⟨w, A^{3}, H^{3}, H^{3}(δ)⟩ ∥ b

Construction. We define embeddings k^{12} = k(k), k^{02} = k(k) ∘ k. Consider conditions in k^{02}(Q_{w}) of the form

z ⌢ X ⌢ ⟨P, k(κ_{w})⟩ ⌢ ⟨k^{02}(w), k^{02}(A^{2}), k^{02}(H^{2}), k^{02}(H^{2})(k(κ_{w}))⟩

where z is a lower part and u_{X} = w ↾ α for some α with 0 < α < lh(w). P ∈ P (κ_{w}, k(κ_{w}))_{k^{02}(V)}, and by the definition of constructing pair l“g generates a generic over k(V) for this forcing.

An argument exactly parallel to that of Step One enables us to define A^{3}, H^{3}, h^{3}. □

Step Four. We find A^{4} ⊆ A^{3}, H^{4} ≤ H^{3}, h^{4} ≤ h^{3} such that if

p^{4} = z^{0} ⌢ ⟨w, A^{4}, H^{4}, h^{4}⟩

and

z′ ⌢ X ⌢ ⟨w, A′, H′, h′⟩ ≤ p^{4}

is a condition deciding β, lh(u_{X}) > 1, and κ_{X} > κ_{z^{0}} then for some β and for w(β)-many u

∃B ∃h z′ ⌢ ⟨u, B, H^{4} ↾ [κ_{u}]^{2}, h⟩ ⌢ ⟨w, A^{4}, H^{4}, H^{4}(κ_{u})⟩ ∥ β

Construction. For each lower part x and β < lh(w) let

X^{x,1} = { ⟨α⟩ | ⟨α⟩ ∈ A^{3} }
X_{0}^{x,β} = { v | ∃B ∃I ∃i x ⌢ ⟨v, B, I, i⟩ ⌢ ⟨w, A^{3}, H^{3}, H^{3}(κ_{v})⟩ ∥ β }
X_{1}^{x,β} = (X_{0}^{x,β})^{c}
X^{x,β} = X_{i}^{x,β} for the unique i such that X_{i}^{x,β} ∈ w(β)  β > 1

Now for each x let

X^{x} = ⋃_{0<β<lh(w)} X^{x,β}

and define

X = { v | ∀x κ_{x} < κ_{v} ⟹ v ∈ X^{x} }

The point of this definition is that X ∈ F_{w}, and if v ∈ X, lh(v) > 1, and

∃B ∃I ∃i x ⌢ ⟨v, B, I, i⟩ ⌢ ⟨w, A^{3}, H^{3}, H^{3}(κ_{v})⟩ ∥ β  (∗)

then for some β < lh(w) the relation (*) holds for w(β)-many v.

For each x, if there exists such a v, find one such that the value of β associated with it is minimal. By the definition of w(β) there are B_{x}, I_{x}, i_{x} in k(V)

x ⌢ ⟨w ↾ β, B_{x}, I_{x}, i_{x}⟩ ⌢ ⟨k(w), k(A^{3}), k(H^{3}), k(H^{3})(κ_{w})⟩ ∥ k(b)

Let I ∈ g^{∗∗} be such that

∀x I ≤^{∗} I_{x}

and find C ∈ μ_{w} such that

∀(α, β) ∈ [C]^{2} ∀x κ_{x} < α < β ⟹ I(α, β) ≤ I_{x}(α, β)

Define A^{4} ⊂ A^{3} ∩ { u | κ_{u} ∈ C }, H^{4} ≤ H^{3}, I, and h^{4} ≤ h^{3} appropriately. By construction p^{4} has the right property. □

34 JAMES CUMMINGS

Step Five. We find A^{5} ⊆ A^{4}, H^{5} ≤ H^{4}, h^{5} ≤ h^{4} such that if

p^{5} =_{def} z^{0} ⌢ ⟨w, A^{5}, H^{5}, h^{5}⟩

and z′ ⌢ ⟨v, A, H^{5} ↾ [κ_{v}]^{2}, H^{5}(κ_{z′}) ↾ κ_{v}⟩ ⌢ ⟨w, A^{5}, H^{5}, H^{5}(κ_{v})⟩ ≤ p^{5} lh(v) > 1, κ_{v} > κ_{z^{0}}, is a condition which forces b then for some β ∃A z′ ⌢ ⟨v, A, H^{5} ↾ [κ_{v}]^{2}, H^{5}(κ_{z′}) ↾ κ_{v}⟩ ⌢ ⟨w, A^{5}, H^{5}, H^{5}(κ_{v})⟩ ∥ b

for w(β)-many v (and similarly for ¬b) Construction. Define

C^{x,1} = { ⟨α⟩ | ⟨α⟩ ∈ A^{4} }

Then for 1 < β < lh(w) let

\begin{matrix}B_{0}^{x} = \{ v | ∃A x ⌢ ⟨v, A, H^{4} ↾ [κ_{v}]^{2}, H^{4}(κ_{x}) ↾ κ_{v}⟩ & ⌢ \\ ⌢ ⟨w, A^{4}, H^{4}, H^{4}(κ_{v})⟩ & ⊩ b \} \end{matrix}

\begin{matrix}B_{1}^{x} = \{ v | ∃A x ⌢ ⟨v, A, H^{4} ↾ [κ_{v}]^{2}, H^{4}(κ_{x}) ↾ κ_{v}⟩ & ⌢ \\ ⌢ ⟨w, A^{4}, H^{4}, H^{4}(κ_{v})⟩ & ⊩ ¬b \} \end{matrix}

\begin{matrix}B_{2}^{x} = \{ v | ∃A x ⌢ ⟨v, A, H^{4} ↾ [κ_{v}]^{2}, H^{4}(κ_{x}) ↾ κ_{v}⟩ & ⌢ \\ ⌢ ⟨w, A^{4}, H^{4}, H^{4}(κ_{v})⟩ & ∦ b \} \end{matrix}

C^{x,β} = B_{i}^{x} for the unique i such that B_{i}^{x} ∈ w(β)

Finally define

\begin{aligned}C^{x} & = ⋃_{0<β<lh(w)} C^{x,β} \\ C & = \{ v | ∀x κ_{x} < κ_{v} ⟹ v ∈ C^{x} \} \end{aligned}

Let A^{5} = A^{4} ∩ C and refine H^{4}, h^{4} appropriately. □

Step Six. We find A^{6} ⊆ A^{5}, H^{6} ≤ H^{5}, h^{6} ≤ h^{5} such that if

p^{6} = z^{0} ⌢ ⟨w, A^{6}, H^{6}, h^{6}⟩

and

z′ ⌢ ⟨H^{6}(κ_{z′}, β), β⟩ ⌢ ⟨w, A^{6}, H^{6}, H^{6}(β)⟩ ≤ p^{6}

forces b, and β > κ_{z^{0}}, then

z′ ⌢ ⟨H^{6}(κ_{z′}, γ), γ⟩ ⌢ ⟨w, A^{6}, H^{6}, H^{6}(γ)⟩ ⊩ b

for all γ with ⟨γ⟩ ∈ A^{6}. Notice that this implies that

z′ ⌢ ⟨w, A^{6}, H^{6}, H^{6}(κ_{z′})⟩ ⊩ b

as any extension of this condition is comparable with a condition forcing b. (Of course the same claim is also true with ¬b in the place of b.)

Construction.

B_{0}^{x} = { β | x ⌢ ⟨H^{5}(κ_{x}, β), β⟩ ⌢ ⟨w, A^{5}, H^{5}, H^{5}(β)⟩ ⊩ b }
B_{1}^{x} = { β | x ⌢ ⟨H^{5}(κ_{x}, β), β⟩ ⌢ ⟨w, A^{5}, H^{5}, H^{5}(β)⟩ ⊩ ¬b }
B_{2}^{x} = { β | x ⌢ ⟨H^{5}(κ_{x}, β), β⟩ ⌢ ⟨w, A^{5}, H^{5}, H^{5}(β)⟩ ∦ b }
B^{x} = B_{i}^{x} for the unique i with B_{i}^{x} ∈ μ_{w}
B = { v | ∀x κ_{x} < κ_{v} ⟹ κ_{v} ∈ B^{x} }

Let A^{6} = A^{5} ∩B and define H^{6}, h^{6} by an appropriate shrinking of the domains of H^{5}, h^{5}. This finishes the construction phase of the proof. □

Step Seven. We prove that p^{6} is an extension which decides b “up to lower parts”. That is, b can be decided by an extension in which no object X with κ_{X} > κ_{z^{0}} is inserted.

Proof. Find an extension r of p^{6} deciding b such that the lower part x has a minimal number of entries Y with κ_{Y} > κ_{z^{0}}, and such that among such conditions κ_{x} has the minimal value.

Claim 1. κ_{x} = κ_{z^{0}} (i.e the minimal number of entries Y with κ_{Y} > κ_{z^{0}} is in fact zero ).

Proof of claim. Suppose that κ_{x} > κ_{z^{0}}, and let x = z′ ⌢ X where X is a single entry. By the work done in Steps One and Two

z′ ⌢ X ⌢ ⟨w, A^{6}, H^{6}, H^{6}(κ_{X})⟩ ∥ b

Suppose that in fact it forces b (the argument for ¬b will be identical ).

Subclaim 1. X is a pair, rather than a quadruple.

Proof of subclaim. Let X be a quadruple ⟨v, B, I, i⟩. The construction of Step Five guarantees

{ v | ∃B ∃i z′ ⌢ ⟨v, B, H^{6} ↾ [κ_{v}]^{2}, i⟩ ⌢ ⟨w, A^{6}, H^{6}, H^{6}(κ_{v})⟩ ⊩ b } ∈ w(β)

for some β > 1. For each such v choose appropriate B, i, and let ⟨P, α⟩ be some pair such that ⟨α⟩ ∈ B and P ≤ i(α). An easy normality argument (v ↦ α < κ_{v}, so fix α on a large set; now use completeness to fix P) gives that the choice of ⟨P, α⟩ can be stabilised on a set in w(β). Fix some ⟨P, α⟩ such that for w(β) many v there exists B

z′ ⌢ ⟨P, α⟩ ⌢ ⟨v, B, H^{6} ↾ [κ_{v}]^{2}, H^{6}(α) ↾ κ_{v}⟩ ⌢ ⟨w, A^{6}, H^{6}, H^{6}(κ_{v})⟩ ⊩ b

Fix f a choice function giving for each such v an appropriate B. The situation is now essentially the same as that in the last stage of the proof that ordinary Radin forcing has the Prikry property (see [R] ). Define sets

A_{1}^{0} = { v | { t ∈ dom(f) | v ∈ f(t) } ∈ w(β) }
A_{1}^{1} = { v | { t ∈ dom(f) | f(t) ∩ V_{κ_{v}} ∈ F_{v} } ∈ w(β) }
A^{1} = A_{1}^{0} ∩ A_{1}^{1}
A_{2} = { t ∈ dom(f) | ∀v ∈ A_{1} κ_{v} < κ_{t} ⟹ v ∈ f(t) }
A_{3} = { v | ∀C ∈ F_{v} ∃μ < lh(v) { t ∈ A^{2} ∩ C | f(t) ∩ C ∈ F_{v} } ∈ v(μ) }

Using the constructing pair argument it is routine to check that

(1) A_{1} ∈ F_{w↾β}.

(2) A_{2} ∈ w(β).

(3) A_{3} ∈ ⋂{ w(α) | β < α < lh(w) }.

So A_{1} ∪A_{2} ∪A_{3} ∈ F_{w}. Define A′ = A^{6} ∩(A_{1} ∪A_{2} ∪A_{3}). The following subclaim gives us what we need.

Subclaim 2. Any extension of

z′ ⌢ ⟨P, α⟩ ⌢ ⟨w, A′, H^{6}, H^{6}(α)⟩

is comparable with

z′ ⌢ ⟨P, α⟩ ⌢ ⟨v, f(v), H^{6} ↾ [κ_{v}]^{2}, H^{6}(α) ↾ κ_{v}⟩ ⌢ ⟨w, A^{6}, H^{6}, H^{6}(κ_{v})⟩

for some v ∈ dom(f).

Proof of subclaim. Let the extension be

p = z̄ ⌢ X_{1} ⌢ X_{2} ⋯ ⌢ X_{n} ⌢ ⟨w, A^{∗}, H^{∗}, h^{∗}⟩

where κz̄ = α. ∀i ∃j u_{X_{i}} ∈ A_{j}. Let v be the first sequence with v ∉ A_{1}. Then v ∈ A_{2}, v ∈ A_{3}, or v = w. Case 1) v ∈ A_{2}. Say v = u_{X_{j}}. By construction

\begin{matrix}v ∈ dom(f) \\ ∀i < j f(v) ∩ V_{κ_{X_{i}}} ∈ F_{u_{X_{i}}} \end{matrix}

So defining (for i ≤ j)

\begin{matrix}X′_{i} = ⟨t, A ∩ f(v), H, h⟩ & if X_{i} = ⟨t, A, H, h⟩ \\ ⟨Q, δ⟩ & if X_{i} = ⟨Q, δ⟩ \end{matrix}

the condition

z̄ ⌢ X′_{1} ⌢ ⋯ ⌢ X′_{j} ⌢ X_{j+1} ⋯ ⌢ X_{n} ⌢ ⟨w, A^{∗}, H^{∗}, h^{∗}⟩

is a common extension of p and

z′ ⌢ ⟨P, α⟩ ⌢ ⟨v, f(v), H^{6} ↾ [κ_{v}]^{2}, H^{6}(α) ↾ κ_{v}⟩ ⌢ ⟨w, A^{6}, H^{6}, H^{6}(κ_{u})⟩

Case 2) v = u_{X_{j}} ∈ A_{3}. Say X_{j} = ⟨v, A, H, h⟩. By construction find v′ with κ_{v′} > κ_{X_{j−1}}, v′ ∈ A ∩ A_{2}, f(v′) ∩ A ∈ F_{v′}. As before ∀i < j f(v′) ∩ V_{κ_{X_{i}}} ∈ F_{u_{X_{i}}}, so defining X′_{i} as before for i < j

\begin{aligned}z̄ ⌢ X′_{1} ⌢ ⋯ ⌢ X′_{j−1} & ⌢ ⟨v′, f(v′), H^{6} ↾ [κ_{v′}]^{2}, H^{6}(κ_{X_{j−1}}) ↾ κ_{v′}⟩ ⌢ \\ & ⌢ X_{j} ⌢ ⋯ ⌢ X_{n} ⌢ ⟨w, A^{∗}, H^{∗}, h^{∗}⟩ \end{aligned}

is the sought-after common extension. Case 3) Like Case 2). Exploit the fact that

{ v ∈ A^{2} | f(v) ∩ B^{∗} ∈ F_{w} } ∈ w(β)

to produce an appropriate v′ with κ_{v′} > κ_{X_{n}}. The subclaim is proved. □

So the condition

r′ = z′ ⌢ ⟨P, α⟩ ⌢ ⟨w, A′, H^{6}, H^{6}(α)⟩

has the property that any extension of it is compatible with some condition which forces b. Thus by elementary facts about forcing the condition itself forces b. However this contradicts the assumption that r was chosen to minimise the last ordinal mentioned in the lower part, amongst conditions extending p^{5}, deciding b, and having the same number of entries Y with κ_{Y} > κ_{z^{0}} as does x. The subclaim is therefore proved. □

If κ_{x} > κ_{z^{0}} then by the subclaim x = z′ ⌢ ⟨P, β⟩, where κ_{z′} = κ_{z^{0}}. For if not then

z′ ⌢ ⟨H(κ_{z′}, β), β⟩ ⌢ ⟨w, A^{6}, H^{6}, H^{6}(β)⟩ ⊩ b

by the work done in Step Three, then

z′ ⌢ ⟨w, A^{6}, H^{6}, H^{6}(κ_{z′})⟩ ⊩ b

by Step Six. This contradicts the choice of x to contain as few entries past κ_{z^{0}} as possible.

Now define

C = { β | ∃z_{β} ∃P ≤ h^{6}(β) z_{β} ⌢ ⟨P, β⟩ ⌢ ⟨w, A^{6}, H^{6}, H^{6}(β)⟩ ∥ b }

Subclaim 3. C ∈ μ_{w}.

Proof. Suppose not, and let A^{∗} = A^{6} \ { w | κ_{w} ∈ C }, define H^{∗}, h^{∗} by restriction of H^{6}, h^{6}, and argue as above that extension of the condition z^{0} ⌢ ⟨w, A^{∗}, H^{∗}, h^{∗}⟩ in the same “minimal” way to decide b demands adding at worst one pair Y with κ_{Y} > κ_{z^{0}}; this gives an immediate contradiction. □

Pick for each β ∈ C an appropriate z_{β}. Find C′ ∈ μ_{w} on which z_{β} is constant, with value z^{∗} say, and such that every element chosen decides b the same way. Shrinking A^{6} to

A′ = A^{6} ∩ { w | κ_{w} ∈ C′ }

defining h′ so that for each β in C

z^{∗} ⌢ ⟨h′(β), β⟩ ⌢ ⟨w, A^{6}, H^{6}, H^{6}(β)⟩ ∥ b

and shrinking H^{6} to H′ appropriately the proof is done. For the condition

z^{∗} ⌢ ⟨w, A^{6}, H^{6}, h′⟩

decides b. □

Step Eight. We conclude the proof by showing that b can be decided by a direct extension of p^{0}.

Proof. We just use induction on the length of conditions and the following obvious fact about the forcing Q_{w};

Fact 1. If p ∈ Q_{w} is a condition of the form

x ⌢ ⟨v, B, I, i⟩ ⌢ ⟨P_{0}, δ_{0}⟩ … ⟨P_{n}, δ_{n}⟩ ⌢ ⟨w, A, H, h⟩

then Q_{w} ↾ p factors as P_{1} × P_{2} × P_{3}, where (1) P_{1} = Q_{v} ↾ x ⌢ ⟨v, B, I, i⟩

(2) P_{2} = P (κ_{v_{0}}, δ_{0}) ↾ P_{0} × ⋯ × P (δ_{n−1}, δ_{n}) ↾ P_{n}

(3) P_{3} = Q_{w} ↾ ⟨δ_{n}⟩ ⌢ ⟨w, A, H, h⟩

The theorem is proved. □

Using this theorem we can go some way towards showing that the cardinals of the extension by Q_{w} are exactly those which we have not taken steps to collapse.

Theorem 2. Let v⃗, F⃗ be Q_{w}-generic and as usual define κ_{α} = κ_{v(α)}. Then for every α the V -cardinals κ_{α}, κ_{α}^{++}, κ_{α}^{+3}, κ_{α}^{+4}, are cardinals in the generic extension.

Proof. We divide the proof into two cases. Case 1) α is a limit. One application of the theorem shows that if X

is a bounded subset of κ_{α} in the extension then X is generic for forcing of cardinality less than κ_{α}; hence κ_{α} survives as a cardinal. Similarly if X is a bounded subset of κ_{α}^{+4} then X is in the extension by Q_{v_{α}}, which is κ_{α}^{++}-c.c. and therefore preserves κ_{α}^{+2}, κ_{α}^{+3}, κ_{α}^{+4}.

Case 2) α is a successor. Say α = λ+n for λ limit. As before it is easy to argue that κ_{α} is preserved. Now we have that bounded subsets of κ_{α}^{+4} are in the extension by

Q_{v(λ)} × P (κ_{λ}, κ_{λ+1}) × … P (κ_{λ+n−1}, κ_{α})

which is κ_{α}^{++}-c.c. forcing. The theorem is proved. □

Of course we have explicitly collapsed κ_{α}^{+} for α successor, so the question left open here is the fate of κ_{α}^{+} when α is limit. The answer to this must await the fine analysis of Q_{w}-genericity which we undertake in the next section.

We also make the remark that the same kind of analysis will show that we have not added too many subsets to any cardinal of the extension; in fact, once we have shown that κ_{α}^{+} is preserved for α limit it will be east to show that in the extension GCH holds at all successors and fails at all limits.

5. The geometric condition.

We prove a simple criterion for Q_{w} genericity, analogous to Mathias’ criterion for Prikry genericity [M]. Recall that if (u⃗, F⃗) is generic with lh(u⃗) = lh(F⃗) = λ then

a) If μ < λ is such that lh(u_{μ}) > 1 then the pair (u⃗ ↾ μ, F⃗ ↾ μ) is generic for the forcing Q_{u_{μ}}.

b) ∀A ∈ V_{κ_{w}+1} (A ∈ F_{w} ⟺ ∃β ∀α > β u_{α} ∈ A).

c) ∀H ∈ w(1) ∃β ∀α > β H(κ_{α}, κ_{α+1}) ∈ F_{α}.

Theorem 1. The pair (⟨u_{α} : α < λ⟩, ⟨F_{α} : α < λ⟩) is generic if and only if it satisfies the conditions a), b), c).

Proof.

Lemma 1. Let D be an open set, let A, H, β be such that D is dense below

⟨β⟩ ⌢ ⟨w, A, H, H(β)⟩

By the methods of the Prikry Property proof, Steps One and Two it may be assumed that if

x ⌢ ⟨w, A′, H′, h′⟩ ≤ ⟨β⟩ ⌢ ⟨w, A, H, H(β)⟩

is a member of D and κ_{x} > β then

x ⌢ ⟨w, A, H, H(κ_{x})⟩ ∈ D

Then there exists j < ω such that ( introducing a quantifier ∀_{μ}^{∗}, where ∀_{μ}^{∗}x ϕ(x) is to be interpreted as “ϕ(x) holds on a set in the ultrafilter μ” )

∃n_{0} ∀_{U^{n_{0}}}^{∗} (δ_{1}^{0}, … δ_{n_{0}}^{0}) ∃(P_{1}^{0}, … P_{n_{0}}^{0}) ∃β_{0} > 1 ∀_{w(β_{0})}^{∗}u_{0} ∃A_{0} ∃H_{0}

…

∃n_{j} ∀_{U^{n_{j}}}^{∗} (δ_{1}^{j}, … δ_{n_{j}}^{j}) ∃(P_{1}^{j}, … P_{n_{j}}^{j}) ∃β_{j} > 1 ∀_{w(β_{j})}^{∗}u_{j} ∃A_{j} ∃H_{j}

⟨β⟩ ⌢ ⟨P_{1}^{0}, δ_{1}^{0}⟩ ⌢ ⋯ ⌢ ⟨P_{n_{0}}^{0}, δ_{n_{0}}^{0}⟩ ⌢ ⟨u_{0}, A_{0}, H_{0}, H_{0}(δ_{n_{0}}^{0})⟩ ⌢
…
⌢ ⟨P_{1}^{j}, δ_{1}^{j}⟩ ⌢ ⋯ ⌢ ⟨P_{n_{j}}^{j}, δ_{n_{j}}^{j}⟩ ⌢ ⟨u_{j}, A_{j}, H_{j}, H_{j}(δ_{n_{j}}^{j})⟩ ⌢
⌢ ⟨w, A, H, H(κ_{u_{j}})⟩
extends ⟨β⟩ ⌢ ⟨w, A, H, H(β)⟩ and is in D

A MODEL IN WHICH GCH HOLDS AT SUCCESSORS BUT FAILS AT LIMITS 41

Proof of lemma. Suppose not. For the sake of brevity abbreviate the formula

“⟨β⟩ ⌢ ⟨P_{1}^{0}, δ_{1}^{0}⟩ ⌢ ⋯ ⌢ ⟨P_{n_{0}}^{0}, δ_{n_{0}}^{0}⟩ ⌢ ⟨u_{0}, A_{0}, H_{0}, H_{0}(δ_{n_{0}}^{0})⟩ ⌢
…
⌢ ⟨P_{1}^{j}, δ_{1}^{j}⟩ ⌢ ⋯ ⌢ ⟨P_{n_{j}}^{j}, δ_{n_{j}}^{j}⟩ ⌢ ⟨u_{j}, A_{j}, H_{j}, H_{j}(δ_{n_{j}}^{j})⟩ ⌢
⌢ ⟨w, A, H, H(κ_{u_{j}})⟩
extends ⟨β⟩ ⌢ ⟨w, A, H, H(β)⟩ and is in D”

as ϕ_{j}(δ⃗, P⃗ , w⃗, A⃗, H⃗), and the quantifier sequence

∀n_{i} ∀_{U^{n_{i}}}^{∗} (δ_{1}^{i}, … δ_{n_{i}}^{i}) ∀(P_{1}^{i}, … P_{n_{i}}^{i}) ∀β_{i} > 1 ∀_{w(β_{i})}^{∗}u_{i} ∀A_{i} ∀H_{i} as Q_{i}. So for all j

Q_{0} Q_{1} … Q_{j} ¬ϕ_{j}(δ⃗, P⃗ , w⃗, A⃗, H⃗)

Fix j and i ≤ j for the moment, and define for each

s = ⟨δ_{1}^{0}, … δ_{n_{0}}^{0}, P_{1}^{0}, … P_{n_{0}}^{0}, u_{0}, A_{0}, h_{0}, … P_{1}^{i}, … P_{n_{i}}^{i}⟩

such that ∀β_{i} > 1 ∀_{w(β_{i})}^{∗}u_{i} ∀A_{i} ∀H_{i} Q_{i+1} … Q_{j} ¬ϕ_{j}(δ⃗, P⃗ , w⃗, A⃗, H⃗)

\begin{matrix}Y_{i,j}^{s} = \{ u_{i} ∈ A′ | lh(u_{i}) = 1 ∨ ∀A_{i} ∀H_{i} Q_{i+1} … Q_{j} ¬ϕ_{j}(δ⃗, P⃗ , w⃗, A⃗, H⃗) \} \\ Y_{i,j} = \{ u | ∀s δ_{n_{i}}^{i} < κ_{u} ⟹ u ∈ Y_{i,j}^{s} \} \end{matrix}

Now let i, j vary and define

Y =_{def} ⋂_{j<ω} ⋂_{i<ω} Y_{i,j}

For each

t = ⟨δ_{1}^{0}, …, A_{i}, H_{i}⟩

such that ∀n_{i+1} ∀_{U^{n_{i+1}}} ⟨δ_{1}^{i+1}, …, δ_{n_{i+1}}^{i+1}⟩ … ∀H_{i+1} Q_{i+2} … Q_{j}¬ϕ(δ⃗, P⃗ , w⃗, A⃗, H⃗) define for each n_{i+1} Z^{t,n_{i+1}} to be

{ δ⃗^{i+1} ∈ [κ]^{n_{i+1}} | ∀P⃗^{i+1} ∀H_{i+1} Q_{i+2} … Q_{j}¬ϕ(δ⃗, P⃗ , w⃗, A⃗, H⃗) } and find X^{t} ∈ U such that ∀n_{i+1} [X^{t}]^{n_{i+1}} ⊂ Z^{t,n_{i+1}}. Find Y′ ⊂ Y, Y′ ∈ F_{w} such that

⟨α⟩ ∈ Y′ ∧ rank(t) < α ⟹ α ∈ X^{t}

⟨β⟩ ⌢ ⟨w, Y, H, H(β)⟩ has some extension in D, say

y_{0} ⌢ ⟨u_{0}, A_{0}, H_{0}, h_{0}⟩ ⌢
…
⌢ y_{n} ⌢ ⟨u_{n}, A_{n}, H_{n}, h_{n}⟩ ⌢
⌢ y_{n+1} ⌢ ⟨w, Y, H, H(κ_{y_{n+1}})⟩ ∈ D

As D is open it is possible to extend this condition to

y_{0} ⌢ ⟨u_{0}, A_{0}, H_{0}, h_{0}⟩ ⌢
…
⌢ y_{n} ⌢ ⟨u_{n}, A_{n}, H_{n}, h_{n}⟩ ⌢
⌢ y_{n+1} ⌢ ⟨u_{n+1}, A_{n+1}, H_{n+1}, h_{n+1}⟩ ⌢
⌢ ⟨w, Y, H, H(κ_{u_{n+1}})⟩ ∈ D

contradicting the definition of Y′. □

Fix j with the property guaranteed by the lemma above. Using the same idea as in Step Three of the proof of the Prikry property refine A, H, to A^{∗}, H^{∗} such that

∃n_{0} ∀_{U^{n_{0}}}^{∗} (δ_{1}^{0}, … δ_{n_{0}}^{0}) ∃β_{0} > 1 ∀_{w(β_{0})}^{∗}u_{0} ∃A_{0} ∈ F_{u_{0}} ∃H_{0} ∈ h_{u_{0}}

…

∃n_{j} ∀_{U^{n_{j}}}^{∗} (δ_{1}^{j}, … δ_{n_{j}}^{j}) ∃β_{j} > 1 ∀_{w(β_{j})}^{∗}u_{j} ∃A_{j} ∈ F_{u_{j}} ∃H_{j} ∈ h_{u_{j}}

⟨β⟩ ⌢ ⟨H^{∗}(β, δ_{1}^{0}), δ_{1}^{0}⟩ ⌢ ⋯ ⌢ ⟨H^{∗}(δ_{n_{0}−1}^{0}, δ_{n_{0}}^{0}), δ_{n_{0}}^{0}⟩ ⌢
⌢ ⟨u_{0}, A_{0}, H_{0}, H_{0}(δ_{n_{0}}^{0})⟩ ⌢ …
⋯ ⌢ ⟨H^{∗}(κ_{u_{j−1}}, δ_{1}^{j}), δ_{1}^{j}⟩ ⌢ ⋯ ⌢ ⟨H^{∗}(δ_{n_{j}−1}^{j}, δ_{n_{j}}^{j}), δ_{n_{j}}^{j}⟩ ⌢
⌢ ⟨u_{j}, A_{j}, H_{j}, H_{j}(δ_{n_{j}}^{j})⟩ ⌢ ⟨w, A^{∗}, H^{∗}, H^{∗}(κ_{u_{j}})⟩
extends ⟨β⟩ ⌢ ⟨w, A^{∗}, H^{∗}, H^{∗}(β)⟩ and is in D

Abbreviate this formula as

R_{0}R_{1} … R_{j}ψ(δ⃗, w⃗, A⃗, H⃗)

At the same time arrange that the δ_{b}^{a} drawn from A^{∗} all come from appropriate measure one sets, as in the proof of the previous lemma.

Lemma 2. With the same hypotheses and notation as above there exist A^{∗∗}, H^{∗∗} refining A^{∗}, H^{∗}, such that if (⟨v_{γ} : γ < μ⟩, ⟨G_{γ} : γ < μ⟩) satisfies the conditions a), b), c), v_{0} = ⟨β⟩, and in addition

forallγ \{ \begin{matrix}v_{γ} ∈ A^{∗∗} \\ (κ_{v_{γ}} , κ_{v_{γ+1}}) ∈ dom(H^{∗∗}) \\ H^{∗∗}(κ_{v_{γ}} , κ_{v_{γ+1}}) ∈ G_{γ} \end{matrix}

then D meets the filter on Q_{w} generated by (⟨v_{γ} : γ < μ⟩, ⟨G_{γ} : γ < μ⟩).

Proof. For each

t = ⟨δ_{1}^{0}, … δ_{n_{0}}^{0}, u_{0}, A_{0}, H_{0}, … δ_{1}^{i}, … δ_{n_{i}}^{i}⟩

such that

∃β_{i} > 1 ∀_{w(β_{i})}^{∗}u_{i} ∃A_{i} ∈ F_{u_{i}} ∃H_{i} ∈ h_{w_{i}} R_{i+1} … R_{j}ψ(δ⃗, w⃗, A⃗, H⃗)

fix an appropriate β^{t} and functions f^{t}, g^{t} defined on a set X^{t} ∈ w(β^{t}) such that

∀u_{i} ∈ X^{t} A_{i} = f^{t}(u_{i}) ∧ H_{i} = g^{t}(u_{i}) ⟹ R_{i+1} … R_{j}ψ(δ⃗, w⃗, A⃗, H⃗)

Now perform the following construction (which is reminiscent of the proof of a subclaim in the proof of the Prikry property, and indeed would prove a slightly stronger result in which “H-parts” are dependent on the sequence chosen in a more arbitrary way)

tA_{1}^{0} = { v | { u | v ∈ f(u) } ∈ w(β^{t}) }
= j(f)(w ↾ β^{t})
tA_{2}^{0} = { v | { u | f(u) ∩ V_{κ_{v}} ∈ F_{v} } ∈ w(β^{t}) }
= { v | A_{1}^{0} ∩ V_{κ_{v}} ∈ F_{v} }
tA_{3}^{0} = { v | { u | g(u) ↾ [κ_{v}]^{2} ∈ h_{v} } ∈ w(β^{t}) }
= { v | j(g)(w ↾ β^{t}) ↾ [κ_{v}]^{2} ∈ h_{v} }
H^{t} = j(g^{t})(w ↾ β^{t})
tA^{0} = ⋃_{0<i<3}^{t}A_{i}^{0}
tA^{1} = { u | ∀v ∈ ^{t}A^{0} κ_{v} < κ_{u} ⟹

⟹ { u | v ∈ f(u) ∧ f(u) ∩ V_{κ_{v}} ∈ F_{v} ∧ g(u) ↾ [κ_{v}]^{2} ∈ h_{v} } ∈ w(β^{t}) }

tA^{2} = { x | ∀C ∈ F_{x} ∃μ < lh(x) C ∩^{t}A^{1} ∈ x(μ) }

As in the Prikry property proof it is easy to show

tA^{0} ∈ μ_{w} ∩ ⋂_{1<γ<β^{t}} w(γ)

tA^{1} ∈ w(β^{t})

tA^{3} ∈ ⋂_{β^{t}<γ<lh(w)} w(γ)

tA =_{def} ⋃_{i<2}^{t}A^{i} ∈ F_{w}

Define A^{∗∗}, H^{∗∗} by taking the diagonal intersection of ^{t}A, H^{t} over suitable t. To show that this construction works, select an n_{0} such that

∀δ⃗^{n_{0}} ∈ [{α|⟨α⟩ ∈ A^{∗∗} }]^{n_{0}} ∃β_{0} ∀_{w(β_{0})}^{∗}u_{0} ∃A_{0}H_{0} R_{1} … R_{j}ψ(δ⃗, w⃗, A⃗, H⃗)

Define δ_{i}^{0} by v_{i} = ⟨δ_{i}^{0}⟩ for 1 ≤ i ≤ n_{0}. Set t_{0} = ⟨δ_{1}^{0}, … δ_{i}^{0}⟩. Find μ_{0} the least μ with v_{μ} ∈ ^{t_{0}}A^{1}. The existence of such a μ is guaranteed by conditions a), b), c), for if v⃗ avoids ^{t_{0}}A on a final segment then by b) (^{t_{0}}A)^{c} ∈ F_{w} which is absurd. Set u_{0} = v_{μ}.

Claim 1. The condition

\begin{aligned}⟨β⟩ & ⌢ ⟨H^{∗∗}(β, δ_{1}^{0}), δ_{1}^{0}⟩ ⌢ ⋯ ⌢ ⟨H^{∗∗}(δ_{n_{0}−1}^{0}, δ_{n_{0}}^{0}), δ_{n_{0}}^{0}⟩ ⌢ \\ & ⌢ ⟨u_{0}, f^{t_{0}}(u_{0}), g^{t_{0}}(u_{0}), g^{t_{0}}(u_{0})(δ_{n_{0}}^{0})⟩ ⌢ ⟨w, A^{∗∗}, H^{∗∗}, H^{∗∗}(κ_{u_{0}})⟩ \end{aligned}

is in the filter generated by (v⃗, G⃗).

Proof of Claim. It will suffice to show that for ν < μ

\begin{matrix}v_{ν} ∈ f^{t_{0}}(u_{0}) \\ g^{t_{0}}(u_{0})(κ_{v_{ν}} , κ_{v_{ν+1}}) ∈ G_{ν} \end{matrix}

By construction v_{ν} ∉ ^{t_{0}}A^{1}. If v_{ν} ∈ ^{t_{0}}A^{0} then v_{ν} ∈ f^{t_{0}}(u_{0}) by the definition of ^{t_{0}}A^{1} . Suppose that v_{ν} ∈ ^{t_{0}}A^{2}. Then for some σ

f^{t_{0}}(v_{ν}) ∩^{t_{0}}A^{2} ∈ v_{ν}(σ)

but the sequence ⟨v_{δ} : δ < ν⟩ avoids this set contradicting the local genericity condition a).

So v_{ν} ∈ f^{t_{0}}(u_{0}). To get the second clause of the claim notice that by construction

H^{∗∗}(κ_{v_{ν}} , κ_{v_{ν+1}}) ≤ g^{t_{0}}(u_{0})(κ_{v_{ν}} , κ_{v_{ν+1}})

but H^{∗∗}(κ_{v_{ν}} , κ_{v_{ν+1}}) ∈ G_{ν} so g^{t_{0}}(u_{0})(κ_{v_{ν}} , κ_{v_{ν+1}}) ∈ G_{ν}. The claim follows. □

Now select n_{1} and so on … after j stages this process will produce a condition in the filter which also lies in D. □

Now fix E a dense subset of Q_{w}. Let x be a lower part of the form z ⌢ ⟨v, B, I, i⟩, i.e a lower part which is a member of some Q_{v}.

If q = ⟨κ_{v}⟩ ⌢ y ⌢ ⟨w, C, J, j⟩ then define

x ⊕ q =_{def} x ⌢ y ⌢ ⟨w, C, J, j⟩

Let

E_{x} =_{def} { q ∈ Q_{w} | x ⊕ q ∈ E }

Fix ν, and use the Prikry property to find A_{ν}, H_{ν} such that

⟨ν⟩ ⌢ ⟨w, A_{ν}, H_{ν}, H_{ν}(ν)⟩ ∥ ∃q x ⊕ q ∈ E

for each x with κ_{x} = ν. Say that x is good or bad depending on whether q is forced to exist or not. If x is good it is easy to see that E_{x} is dense below ⟨κ_{x}⟩ ⌢ ⟨w, A_{κ_{x}}, H_{κ_{x}}, H_{κ_{x}}(κ_{x})⟩. Apply lemma 6.3 to produce appropriate A_{x}^{∗∗}, H_{x}^{∗∗} for each good x, and define A^{†}, H^{†} by taking the diagonal intersection over all good x. Because of a), b), c) there is some β < λ such that lh(u_{β}) > 1 and for all α > β

u_{α} ∈ A^{†}
(κ_{u_{α}}, κ_{u_{α+1}}) ∈ dom(H^{†})
H^{†}(κ_{u_{α}}, κ_{u_{α+1}}) ∈ F_{α}

Lemma 3. Good x are dense in Q_{u_{β}}.

Proof. Fix x ∈ Q_{u_{β}}, and extend

x ⌢ ⟨w, A_{κ_{u_{β}}} , H_{κ_{u_{β}}} , H_{κ_{u_{β}}} (κ_{u_{β}})⟩

to

x′ ⌢ s ⌢ ⟨w, A_{κ_{u_{β}}} , H_{κ_{u_{β}}} , H_{κ_{u_{β}}} (κ_{s})⟩ ∈ D

where x′ ∈ Q_{u_{β}}. x′ must be good. □

So find a good x in the filter on Q_{u_{β}} generated by (u⃗ ↾ β, F⃗ ↾ β). By construction there is q ∈ E_{x} which also lies in the filter generated by (⟨u_{γ} : β ≤ γ < λ⟩, ⟨F_{γ} : β ≤ γ < λ⟩). So x ⊕ q ∈ E, and this also lies in the filter generated by (⟨u_{γ} : γ < λ⟩, ⟨F_{γ} : γ < λ⟩). So the latter filter meets every dense set and is therefore generic. □

6. The forcing preserves successors.

We are now able to answer the question which was left unsolved by the Prikry lemma, namely whether the successors of limit points on the generic sequence are collapsed. By the Prikry lemma it is enough to decide whether Q_{w} collapses κ_{w}^{+}. The proof given here was suggested by Hugh Woodin, and replaces a much clumsier one which used the canonical generic object described in the next section.

Theorem 1. If G is Q_{w}-generic then κ_{w}^{+} is a cardinal in V [G].

Proof. Suppose not, and let τ̇ be a term, p ∈ Q_{w} a condition such that

p ⊩ τ̇ : κ_{w} ↠ κ_{w}^{+}

Find θ some large regular cardinal such that p,τ̇,Q_{w} are in H_{θ}. Find X ≺ H_{θ} such that V_{κ_{w}} ⊆ N and p, τ̇, Q_{w} are in X, and such that |X| = κ_{w}. Let π : X ⟶ N be the Mostowski collapse of X to a transitive set N. Force over H_{θ} with Q_{w} ↾ p to get a generic (v⃗, F⃗).

Claim 1. The pair (v⃗, F⃗) is π(Q_{w})-generic over the model N.

Proof. This is immediate using the geometric criterion for genericity and the observation that we have

\begin{aligned}π(F_{w}) & = F_{w} ∩ N \\ π(w(1)) & = w(1) ∩ N □ \end{aligned}

Let G be the generic filter on Q_{w} corresponding to (v⃗, F⃗), and let Ḡ be the filter on π(Q_{w}) similarly defined. Observing that π^{−1} maps elements of Ḡ to elements of G, we build an elementary embedding

k : N[Ḡ] ⟶ H_{θ}[G]

But π(p) ∈ Ḡ so the interpretation of τ̇ by Ḡ must be f : κ_{w} ↠ (κ_{w})_{N}^{+}. But (κ_{w})_{N}^{+} is the critical point of k so k(f) = f, while by elementarity k(f) : κ_{w} ↠ κ_{w}^{+}. This gives a contradiction. □

7. A canonical generic sequence.

Radin showed in [R] that generics for Radin forcing can be generated by iterated elementary embeddings; the same is true for Q_{w}, with a similar proof (Radin’s iterative scheme for choosing the ordinals to which sequences are restricted has the same effect as the one given here).

Fix w ∈ U_{∞}, with constructing pair (k, F), where lh(w) < k(κ_{w}). Iterate k to get a sequence of models

⟨M_{α} : α ∈ ON⟩

(where M_{0} = V) and embeddings k_{αβ} for α < β in the usual way. Define F_{α} = k_{0α}(F), κ_{α} = k_{0α}(κ_{w}) for each α ∈ ON. It is easily seen that F_{α} is P (κ_{α}, κ_{α+1})_{M_{β}}-generic over M_{β} for β > α; also if α < α′ < β then F_{α} and F_{α′} are mutually generic over M_{β}. Define by induction a sequence ⟨u_{α}⟩

u_{α} = ⟨κ_{α}⟩ u_{α} = k_{0α}(w) ↾ λ_{α}

α 0 or a successor

α limit

where in the limit case λ_{α} is the least λ < lh(k_{0α}(w)) such that for some X ∈ k_{0α}(w)(λ) the sequence ⟨u_{β} : β < α⟩ avoids X, if such λ exists, and λ_{α} = lh(k_{0α}(w)) otherwise. The construction ends at α in the latter case.

Lemma 1. For all α such that u_{α} is defined and lh(u_{α}) > 1 the pair (⟨u_{β} : β < α⟩, ⟨F_{β} : β < α⟩) is generic over M_{α} for the forcing Q_{u_{α}}.

Proof. By induction on α. The non-trivial part is to show that the sequence ⟨u_{β} : β < α⟩ generates the filter F_{u_{α}}. One direction is easy; suppose that X ∈ M_{α}, X contains a final segment of u⃗ ↾ α, but X ∉ F_{u_{α}}. So for some γ < λ_{α} X^{c} ∈ k_{0α}(w)(γ). But by the minimal choice of λ_{α} { β < α | u_{β} ∈ X^{c} } is unbounded in α and this is absurd.

For the other direction let X ∈ F_{u_{α}}. Fix Y ∈ k_{0α}(w)(λ_{α}) such that ∀β < α u_{β} ∉ Y, and define

Z = { v | ∃μ Y ∩ V_{κ_{v}} ∈ v(μ) }

A familiar style of argument using k_{αα+1} (which generates the sequence k_{0α}(w)) shows

T = X ∪ Y ∪ Z ∈ F_{k_{0α}(w)}

α is limit, so by the definition of direct limit for all large β < α T = k_{βα}(T_{β}) where T_{β} ∈ F_{k_{0β}(w)}. But k_{ββ+1} generates k_{0β}(w), therefore for such β

∀γ < lh(k_{0β}(w)) k_{0β}(w) ↾ γ ∈ T_{β+1}

k_{β+1α} fixes the sequence k_{0β}(w) ↾ γ, because

lh(k_{0β}(w)) < k_{0β+1}(κ_{0}) = crit(k_{0β+1})

So

∀γ < lh(k_{0β}(w)) k_{0β}(w) ↾ γ ∈ T

As the construction did not halt at β u_{β} is a proper initial segment of k_{0β}(w), so

u_{β} ∈ T

But now it must be the case that u_{β} ∈ X; for certainly u_{β} ∉ Y, while if u_{β} ∈ Z then (as by induction ⟨u_{γ} : γ < β⟩ is Q_{u_{β}}-generic over M_{β}) there is γ < β u_{γ} ∈ Y. □

Theorem 1. The construction of the u_{α} terminates at some stage.

Proof. Suppose not. Fix μ a large enough regular cardinal and select for each limit α ≤ μ X_{α} ∈ k_{0α}(w)(λ_{α}) such that u⃗ ↾ α avoids X_{α}. A familiar argument from the theory of inner models (see e.g. lemma 4 of Mitchell’s paper [Mi1]) shows that for some stationary set S ⊂ μ

∀(α, β) ∈ [S]^{2} k_{0α}(λ_{α}) = λ_{β} ∧ k_{0α}(X_{α}) = X_{β}

Fix such a pair (α, β). X_{α} ∈ k_{0α}(w)(λ_{α}), so arguing as above

u_{α} = k_{0α}(w) ↾ λ_{α} ∈ k_{αβ}(X_{α}) = X_{β}

contradicting the choice of X_{β}. □

8. Repeat points and the preservation of large cardinals.

Definition. Let w ∈ U_{∞}. α < lh(w) is a repeat point for w iff

∀X X ∈ w(α) ⟹ ∃β < α X ∈ w(β)

Lemma 1. If lh(w) ≥ (2^{κ_{w}})^{+} then w has a repeat point.

Proof. For each ν < lh(w) which is not a repeat, fix (by the definition of repeat point) X_{ν} ∈ w(ν)\ ⋃_{μ<ν} w(μ). There are only 2^{κ_{w}} possible X_{ν}. □

Lemma 2. If β is a repeat point for w and A ∈ F_{w↾β} then A ∈ w(β).

Proof. Suppose not: then A^{c} ∈ w(β), so as β is a repeat point there is α < β with A^{c} ∈ w(α). But this is absurd. □

Lemma 3. Let w ∈ U_{∞}, with (i : V ⟶ M, g) a constructing pair and lh(w) < i(κ_{w}). Let β < lh(w) be a repeat for w, and v = w ↾ β. Let p = x ⌢ ⟨v, A, H, h⟩ ∈ Q_{v}. Then

p ⌢ ⟨i(v), i(A), i(H), i(H)(κ_{v})⟩ ≤_{i(Q_{v})} i(p)

Proof. It is required to show that ⟨v, A, H, h⟩ can be added in to i(p) = x ⌢ ⟨i(v), i(A), i(H), i(h)⟩. A ∈ F_{v}, so by lemma 2 A ∈ w(β) and therefore v = w ↾ β ∈ i(A). Also A = i(A) ∩ V_{κ_{v}}, H = i(H) ↾ [κ_{v}]^{2}, h = i(h) ↾ κ_{v}. The lemma is proved. □

Now we use the canonical generic sequence constructed in the last section to show that the forcing Q_{w} can preserve large cardinal properties of κ_{w}. It is possible to do this proof in the style of [R], using the Prikry Lemma to define extenders in the generic extension, but the proof we give here is rather shorter and more perspicuous.

Theorem 1. With the same assumptions as in lemma 3 let G be generic for Q_{w↾β}, and do the construction from the last section to produce an α and an H such that H is i_{0α}(Q_{w↾β})-generic over M_{α}, where i_{0α} : V ⟶ M_{α} is the α^{th} iterate of i. Write G = (x⃗, E⃗) where x⃗ is a sequence of measure sequences, and E⃗ is an appropriate sequence of generic collapses; similarly let H = (u⃗, F⃗), and define a new pair G + H = (v⃗, I⃗) where

\begin{aligned}v⃗ & = x⃗ ⌢ ⟨w ↾ β⟩ ⌢ u⃗ ↾ (dom(u⃗) \ \{κ_{w}\}) \\ I⃗ & = E⃗ ⌢ F⃗ \end{aligned}

Then G + H is i_{0α}(Q_{w↾β})-generic over M_{α}, and

∀p p ∈ G ⟺ i_{0α}(p) ∈ G + H

Proof. The genericity is immediate by the geometric condition, so fix p = x ⌢ ⟨w ↾ β, A, H, h⟩ with p ∈ G. By the geometric condition it suffices to show that p ⌢ ⟨i_{0α}(v), i_{0α}(A), i_{0α}(H), i_{0α}(H)(κ_{w})⟩ ∈ G + H, which amounts to showing that the sequences (excepting the first) from u⃗ and the generics from F⃗ are controlled by the top part of this condition. By construction each sequence occurring on u⃗ is a proper subsequence of some i_{0γ}(w ↾ β) for some γ, say i_{0γ}(w) ↾ λ. As A is in all the measures on w ↾ β, i_{0γ}(A) ∈ i_{0γ}(w)(λ) by elementarity, hence i_{0γ}(w) ↾ λ ∈ i_{0γ+1}(A); then i_{0γ}(w) ↾ λ ∈ i_{0α}(A) because i_{γ+1α} has large critical point. That the entries of F⃗ are controlled by the top part is immediate from the observation that i_{02}(H)(κ_{w}, i(κ_{w})) ∈ g. □

Now it is possible to extend the map i_{0α} : V ⟶ M_{α} to a map i^{∗} : V [G] ⟶ M_{α}[G + H], so that certainly the measurability of κ_{w} is preserved. In fact more is true, for observe that if (κ_{w})^{+} < λ < i(κ_{w}) and Pλ ⊂ M then Pλ ⊂ M_{α}, because crit(i_{1α}) = i(κ_{w}) > λ. An easy chain condition argument shows that if this is the case then

(Pλ)_{V [G]} ⊂ M_{α}[G + H]

so that we can preserve any strength that the embedding i may happen to have.

9. The final model.

Main Theorem. If it is consistent that GCH holds and there is a P_{3}κhypermeasurable cardinal, then it is consistent that GCH hold at every successor and fail at every limit.

Proof. The proof of the Main Theorem is now straightforward. We perform the Reverse Easton construction of section 2 to get the initial hypotheses of section 3. Then using the constructing pair (j, F) we build a sequence u with one repeat point β. We force with Q_{u↾β}, and then truncate the model V [G] (in which κ is P_{2}κ-hypermeasurable) at κ. The result is a set model of ZFC in which GCH holds at every successor and fails at every limit, containing many moderately large cardinals.

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