BLOWING UP THE POWER SET OF THE LEAST MEASURABLE

ARTHUR W. APTER AND JAMES CUMMINGS

Abstract. We prove some results related to the problem of blowing up the power set of the least measurable cardinal. Our forcing results improve those of [1] by using the optimal hypothesis.

1. Introduction

In his paper “Forcing the least measurable to violate GCH” [1] the first author proved the consistency of the following situation: κ is the least measurable cardinal, 2^{κ} = κ^{+}, and κ remains the least measurable cardinal after forcing with the Cohen poset for adding κ^{++} subsets of κ. The construction of [1] starts with a model in which GCH holds and κ is κ^{+}-supercompact, and proceeds by an iteration of Prikry forcing and Cohen forcing in the style of Gitik [4, 6].

It is known by work of Gitik [5], Mitchell [11] and Woodin (unpublished) that the exact strength of the failure of GCH at a measurable cardinal is given by the existence of a cardinal λ of Mitchell order λ^{++}. It is clear that the result mentioned in the first paragraph can not be done from a weaker hypothesis than this; our main result is that this hypothesis suffices. Along the way we will prove some other results related to the problem of violating GCH at a measurable cardinal.

Silver gave the first consistency proof for the failure of the GCH at a measurable cardinal; his argument involved starting with a very large (supercompact) cardinal κ and then doing a Reverse Easton iteration in which α^{++} Cohen subsets are added to each inaccessible α ≤ κ. In what follows we refer to Silver’s forcing construction as the standard iteration, and say that the standard iteration succeeds at a measurable cardinal κ if it preserves the measurability of κ.

2000 Mathematics Subject Classification. Primary 03E35, 03E55; Secondary 03E05.

Key words and phrases. Measurable cardinal, GCH, Reverse Easton forcing, inner model.

The first author’s research was partially supported by PSC-CUNY Grants 669408, 667379, and 61449-00-30, and by the Volkswagen-Stiftung (RiP-program at Oberwolfach).

The second author’s research was partially supported by NSF grants DMS-9703945 and DMS-0070549, and by the Volkswagen-Stiftung (RiP-program at Oberwolfach).

Woodin showed that it is possible to violate GCH starting with κ which is (κ + 2)-strong, but his construction (whose ideas we will use in this paper) is more complex than the standard iteration. Gitik’s construction from the optimal hypothesis o(λ) = λ^{++} [5] is even more complex, involving in addition a preliminary Prikry-style forcing.

As part of the proof of the main result, we will show that it is possible to start from a canonical inner model for o(λ) = λ^{++} and to build a model over which the standard iteration succeeds. We also show that no forcing resembling the standard iteration can succeed over this canonical inner model itself. Finally we show that a technical hypothesis on a measurable cardinal κ introduced by Woodin is consistent with κ being the least measurable cardinal. Note on terminology: Our conventions are fairly standard. In particular in a forcing poset p ≤ q means that p is stronger than q. A κ-closed poset is one in which every sequence of length less than κ has a lower bound. A κ-Knaster poset is one in which every κ-sequence of conditions has a subsequence of length κ consisting of pairwise compatible elements. If ℙ_{α} is an iteration of length α and β < α then ℙ_{β} is the subiteration of ℙ_{α} through stage β; if G_{β} is ℙ_{β}-generic then ℙ_{β,α} is the standard factor iteration as computed in V [G_{β}].

2. An inner model argument

As we mentioned in the introduction, Gitik [5] has shown how to start with a canonical inner model for the hypothesis o(κ) = κ^{++} and produce a model in which κ is measurable and 2^{κ} = κ^{++}. The first step in Gitik’s construction is an Easton support iteration of Prikrystyle forcing in which many measurable cardinals are made singular and extension is defined in a special way.

It is natural to ask whether this is really necessary, or whether instead it may be possible to get the desired result by the classical technique of iterating highly closed forcing with Easton support. We show here that this is not the case when forcing over a certain canonical inner model for the hypothesis o(κ) = κ^{++}.

We will use a standard fact from inner model theory. It is shown in Mitchell’s papers on inner models constructed from sequences of measures [11, 9, 10] that if there is an inner model in which o(λ) = λ^{++} for some cardinal λ, then we may construct an inner model V with the following pleasant properties.

(1) V models GCH.

(2) There is a sequence U(α, β) such that

(a) U(α, β) is defined for every β < o^{U}(α), for some ordinalvalued function o^{U}.

(b) U(α, β) is a normal measure on α.

(c) o^{U}(κ) = κ^{++} and κ is the largest point at which o^{U} is

non-zero.

(3) If V [G] is any set-generic extension of V, W is any measure on κ and i : V [G] ⟶ N is the corresponding ultrapower map then i ↾ V arises from some iteration of V with the following properties:

(a) The critical points in the iteration are increasing.

(b) Direct limits are taken at limit stages.

(c) If j_{0γ} : V ⟶ M_{γ} is the map at stage γ in the iteration then M_{γ+1} is the ultrapower by some measure j_{0γ}(U)(α, β) and j_{γγ+1} is the corresponding ultrapower map.

We can now state the main result of this section.

Theorem 2.1. Let V have the properties listed above. Then there is no poset ℙ ∈ V such that

(1) Forcing over V with ℙ adds no ω-sequences of ordinals.

(2) In V^{ℙ} the cardinal κ is measurable and 2^{κ} = κ^{++}.

Proof. We proceed by contradiction. Let G be ℙ-generic, and suppose that in V [G] the cardinal κ is measurable and 2^{κ} > κ^{+}. Let W be any normal measure on κ and let i : V [G] ⟶ N be the associated ultrapower map. By standard facts about measurability we see that i(κ) > (2^{κ})_{V [G]}. By hypothesis (2^{κ})_{V [G]} ≥ κ_{V [G]}^{++}, and clearly κ_{V [G]}^{++} ≥ κ^{++}. We conclude that i(κ) > κ^{++}.

Now let H = i(G), j = i ↾ V and M = ⋃_{α} i(V_{α}). It is routine to check that M is a transitive model of set theory, H is j(ℙ)-generic over M, N = M[H] and j : V ⟶ M is elementary. By standard facts about measurability V [G] ⊧ ^{κ}N ⊆ N.

Let j : V ⟶ M be generated by an iteration as above of length α. Since the critical points are increasing, the first critical point in the iteration must be κ.

Claim. α is finite.

Proof. Suppose not and let x = ⟨κ_{i} : i < ω⟩ be the sequence of the first ω many critical points. Now x ∈ V [G], and so by the closure of N we have x ∈ N. Since N = M[H] and by elementarity the poset j(ℙ) adds no ω-sequences of ordinals, x ∈ M. Since the critical points in the iteration are increasing, x is in the model M_{ω} which appears at stage ω in the iteration.

Since M_{ω} is constructed as a direct limit, x = j_{nω}(y) for some n < ω and y ∈ M_{n}. In particular the n-th point of x must be the image of the n-th point of y. This is impossible because that n-th point is κ_{n}, which can not be in the range of j_{nω} since crit(j_{nω}) = κ_{n}. □

Let α = n for some finite n. Since κ is the largest λ such that o^{U}(λ) ≠ 0, it is easy to see that κ_{i} ≤ j_{0i}(κ) for all i < n. Since M_{n} = M is constructed by a finite iteration using normal measures, we also see that

M_{n} = {j(F)(κ_{0}, …, κ_{n−1}) : F ∈ V, dom(F) = [κ]^{n}}.

Since GCH holds in V, there are only κ^{+} functions from [κ]^{n} to κ. It follows that j(κ) < κ^{++}, which is absurd because j(κ) = i(κ) and i(κ) > κ^{++}. This contradiction finishes the proof. □

3. Arguments of Woodin and Levinski

In this section we outline arguments of Woodin and Levinski, which will be needed for the forcing construction of Section 4.

3.1. Woodin’s argument. Woodin invented a forcing technique for starting with a certain fairly weak embedding hypothesis on κ, and producing a model in which κ is measurable and 2^{κ} = κ^{++}. For more details we refer the reader to Gitik’s paper [5] which shows that Woodin’s hypothesis is consistent relative to o(κ) = κ^{++}, and then explains in detail Woodin’s methods for getting the failure of GCH at a measurable and the failure of SCH at ℵ_{ω} from this hypothesis. In particular, [5, pages 227–229] contains the argument we are about to outline below.

The hypothesis we make on κ is that GCH holds and there is an embedding j : V ⟶ M such that crit(j) = κ, ^{κ}M ⊆ M and κ_{M}^{++} = κ^{++}. We note that the existence of such an embedding is immediate if κ is (κ + 2)-strong.

We will begin by doing a Reverse Easton iteration ℙ of length κ, in which α^{++} subsets are added to each α < κ. Let G be generic for ℙ. Now let ℚ = Add(κ, κ^{++})_{V [G]} and let g be ℚ-generic over V [G].

We will factor j through the ultrapower by the normal measure U = {X : κ ∈ j(X)}. Let i : V ⟶ M_{0} = Ult(V, U) be the canonical ultrapower map, and let k : M_{0} ⟶ M be the map given by k : [F]_{U} ⟼ j(F)(κ). It is easy to see that k is elementary, k ∘ i = j and crit(k) = κ_{M_{0}}^{++}. We let λ = κ_{M_{0}}^{++} and note that by GCH and elementarity λ < i(κ) < κ^{++}.

Now let g_{0} be the Add(κ, λ)-generic obtained by restricting g, and observe that 2^{κ} = κ^{+} in V [G ∗ g_{0}]. Let ℝ = Add(κ^{+}, κ^{++})_{V [G∗g_{0}]} and let h be ℝ-generic over V [G ∗ g]. Our final model will be V [G ∗ g ∗ h]. We note for the record that V [G ∗ g] is an extension of V [G ∗ g_{0}] by κ^{+}-c..c forcing, and so by Easton’s lemma forcing with ℝ over V [G ∗ g] adds no κ-sequences of ordinals.

The iteration i(ℙ) in M_{0} may be factored as ℙ ∗ ℚ̇_{0} ∗ 𝕊̇_{0} where ℚ_{0} = Add(κ, λ)_{M_{0}^{ℙ}}. It is easy to see that G ∗ g_{0} is ℚ ∗ ℚ̇_{0}-generic over M_{0} and that V [G ∗ g_{0}] ⊧ ^{κ}M_{0}[G ∗ g_{0}] ⊆ M_{0}[G ∗ g_{0}] We may use the standard method to build H_{0} ∈ V [G ∗ g_{0}] which is 𝕊_{0}-generic over M_{0}[G ∗ g_{0}].

It is routine to lift k : M_{0} ⟶ M to a new map k^{+} : M_{0}[G ∗ g_{0}] ⟶ M[G ∗ g], and to see that

M[G∗g] = {k^{+}(F)(a) : F ∈ M_{0}[G∗g_{0}], a ∈ [κ^{++}]^{<ω}, dom(F) = [λ]^{<ω}}.

Since 𝕊_{0} is λ^{+}-closed in M[G ∗ g_{0}] we may transfer H_{0} along k^{+}. The result is a filter H which is generic over M[G ∗ g] for 𝕊, where 𝕊̇ is the last term in the factorisation j(ℙ) = ℙ ∗ ℚ̇ ∗ 𝕊̇. We may now lift k^{+} to k^{++} : M_{0}[G ∗ g_{0} ∗ H_{0}] ⟶ M[G ∗ g ∗ H], and may also lift j to j^{+} : V [G] ⟶ M[G ∗ g ∗ H].

Since H_{0} ∈ V [G ∗ g_{0}], we see that V [G ∗ g_{0}] ⊧ ^{κ}M_{0}[G ∗ g_{0} ∗ H_{0}] ⊆ M_{0}[G ∗ g_{0} ∗ H_{0}]. We may lift i to get i^{+} : V [G] ⟶ M_{0}[G ∗ g_{0} ∗ H_{0}], and may use the closure of M_{0}[G ∗ g_{0} ∗ H_{0}] to argue that in V [G ∗ g_{0}] the forcing i^{+}(ℚ) is equivalent to ℝ. We may therefore find h^{∗} ∈ V [G ∗ g ∗ h] which is V [G ∗ g_{0}]-generic for i^{+}(ℚ). Note that a fortiori h^{∗} is M_{0}[G ∗ g_{0} ∗ H_{0}]-generic for i^{+}(ℚ).

We now transfer h^{∗} along k^{++} to get h^{∗∗} which is M[G∗g∗H]-generic for j^{+}(ℚ). Unfortunately h^{∗∗} is probably not compatible with g and j^{+}, but we may alter h^{∗∗} to obtain h^{∗∗∗} which is still M[G ∗ g ∗ H]-generic for j^{+}(ℚ) and is also such that j^{+}“g ⊆ h^{∗∗∗}.

We may now lift j^{+} to j^{++} : V [G ∗ g] ⟶ M[G ∗ g ∗ H ∗ h^{∗∗∗}]. We are still not quite done, but since ℝ is sufficiently distributive we may transfer h along j^{++} to obtain h^{†} which is j^{++}(ℝ)-generic over M[G ∗ g ∗ H ∗ h^{∗∗∗}]. We now lift to get j^{+++} : V [G ∗ g ∗ h] ⟶ M[G∗g∗H∗h^{∗∗∗}∗h^{†}]. This witnesses that κ is measurable in V [G∗g∗h], and we are done.

3.2. Levinski’s argument. Levinski [8] showed that the situation in which κ is measurable and GCH holds at κ while failing unboundedly often below is consistent relative merely to the existence of a measurable cardinal. We outline a version of the argument here.

Suppose that κ is measurable and GCH holds. Let ℙ_{κ} be a Reverse Easton iteration in which α^{++} Cohen subsets are added to each inaccessible α < κ. Let G be ℙ_{κ}-generic, let ℚ = Add(κ, κ^{+})_{V [G]} and let g be ℚ-generic over V [G].

Let U be a normal measure on κ and let i : V ⟶ M = Ult(V, U) be the associated ultrapower map. Let λ = κ_{M}^{++} and observe that κ^{+} < λ < κ^{++}. We may therefore find g^{∗} ∈ V [G∗g] which is Add(κ, λ)- generic over V [G] and is such that V [G ∗ g] = V [G ∗ g^{∗}].

Arguing as in the last subsection we may build H ∈ V [G ∗ g] which is generic over M[G ∗ g^{∗}] for i(ℙ_{κ})_{κ+1,i(κ)}, and may then extend to get a map i^{+} : V [G] ⟶ M[G ∗ g^{∗} ∗ H].

Since Add(κ, κ^{+}) has κ^{+} maximal antichains and |j(κ^{+})| = κ^{+}, we may also build h ∈ V [G ∗ g] which is i^{+}(ℚ)-generic over M[G ∗ g^{∗} ∗ H]. The construction may be done in a way such that i^{+}“g ⊆ h.

We may now lift again to get i^{++} : V [G ∗ g] ⟶ M[G ∗ g^{∗} ∗ H ∗ h]. This shows that κ is measurable in V [G ∗ g].

4. A forcing argument

In this section we show how to begin with a canonical model with o(κ) = κ^{++} of the sort discussed in Section 2 and to create by forcing a model in which 2^{κ} = κ^{+}, κ is the least measurable and κ remains the least measurable after adding κ^{++} Cohen subsets of κ. The forcing construction proceeds in a number of stages, some of which are quite standard and will not be described in any detail.

Stage One: We do the iterated Prikry-style forcing described in Gitik’s paper [5]. After this forcing we have a universe in which GCH holds and there is an embedding j : V ⟶ M such that crit(j) = κ, κM ⊆ κ and j(κ) > κ^{++}.

We claim that without loss of generality we may assume that κ is not measurable in M. If κ is measurable in M then there is U ∈ M which is a measure of Mitchell order zero, that is to say κ is not measurable in Ult(M, U). We may now replace M by Ult(M, U) and j by the composite embedding i_{U}^{M} ∘ j, where i_{U}^{M} is the canonical ultrapower embedding from M to Ult(M, U).

Now let E be the (κ, κ^{++} + 1)-extender approximating j. Let j_{E} : V ⟶ Ult(V, E) be the ultrapower of V by E and k : Ult(V, E) ⟶ M the standard map such that k ∘ j_{E} = j. We know that j_{E}(κ) > κ^{++} and also crit(k) > κ^{++}, so that in particular k(κ) = κ and κ is not measurable in Ult(V, E). Standard arguments also show that Ult(V, E) is closed under κ-sequences.

Replacing j by j_{E}, we may assume that in addition to the properties of j listed above, j arises as the ultrapower by a (κ, κ^{++} + 1)-extender.

Stage Two: We force with Woodin’s “fast function” forcing. The conditions are partial functions p from κ to κ such that

• The domain of p consists of inaccessible cardinals γ < κ which are closed under p. That is to say if α and β are in dom(p) and α < β then p(α) < β.

• For every β ∈ dom(p), the cardinality of dom(p)∩β is less than β.

The ordering is inclusion. We refer the reader to Hamkins’ paper [7] for a detailed discussion of this forcing. We note that by a standard Δ-system argument, since κ is Mahlo the fast function forcing has the κ-Knaster property.

Let ℙ be the fast function forcing, let G be ℙ-generic, and let f : κ ⟶ κ be the fast function added by G. Arguing as in Theorem 1.6 of [7] we may build H ⊆ j(ℙ) such that

(1) H is j(ℙ)-generic over M.

(2) j“G ⊆ H.

(3) The condition {(κ, κ^{++})} is in H. The arguments given in [7] show that G is ℙ-generic over M, and the models M[G] and M[H] agree up to at least the rank of the first Minaccessible greater than κ. Also, by Theorem 1.3 of [7], GCH holds in V [G].

We may lift j to get a new map j^{+} : V [G] ⟶ M[H] such that j^{+}(f)(κ) = κ^{++}. It is a standard fact that j^{+} arises as the ultrapower by a (κ, κ^{++} + 1) extender, and since κ^{++} = j^{+}(f)(κ) we see that j^{+} is actually the ultrapower by a (κ, κ^{++})-extender.

We claim that κ is not measurable in M[H]. It follows by the factorisation argument of Lemma 1.1 in [7] that it is enough to show that κ is not measurable in M[G]. This is immediate because ℙ is κ-Knaster in M, and by a well-known argument (see for example [2]) a κ-Knaster forcing can not create measurability at κ.

We also claim that M[H] is closed under κ-sequences inside V [G]. Since these are models of ZFC it is enough to check that every κsequence of ordinals from V [G] is in M[H]. This is immediate because V ⊧ ^{κ}M ⊆ M, and since G is generic for κ-c.c forcing it follows that V [G] ⊧ ^{κ}ON ⊆ M[G].

To simplify notation we relabel so that V [G] becomes V, j^{+} becomes j and M[H] becomes M. We are now in the following situation: GCH holds and there is j : V ⟶ M such that crit(j) = κ, ^{κ}M ⊆ M, κ is not measurable in M, j is the ultrapower by a (κ, κ^{++})-extender and there is f : κ ⟶ κ such that j(f)(κ) = κ^{++}.

Stage Three: We now do a Reverse Easton iteration ℙ_{κ+1}, in which we force at α ∈ dom(f) with the Levy collapse Coll(α^{+}, < f(α)) and then force at κ with Coll(κ^{+}, < κ^{++}). We recall from the previous stage that if α ∈ dom(f) then α is inaccessible and f“α ⊆ α.

Let G be ℙ_{κ}-generic over V, and let g be generic over V [G] for Coll(κ^{+}, < κ^{++}). We note that g does not collapse cardinals, since V [G] models GCH and Coll(κ^{+}, < κ^{++}) is equivalent to Add(κ^{+}, κ^{++}).

We factor j : V ⟶ M through the ultrapower by the normal measure U induced by j. Let M_{0} = Ult(V, U). As usual we have an embedding i : V ⟶ M_{0} and a map k : M_{0} ⟶ M such that k ∘ i = j. Since crit(k) > κ and κ ∈ dom(j(f)), we see that κ ∈ dom(i(f)).

Let μ = i(f)(κ). Consider the iteration i(ℙ_{κ}) and let λ be the least point greater than κ in the support of this iteration. λ is an M_{0}- inaccessible closure point of i(f), so in particular λ > i(f)(κ) = μ. We note also that at stage κ the iteration i(ℙ_{κ}) adds a generic for Coll(κ^{+}, < μ).

It is easy to see that M_{0}[G] is closed under κ-sequences in V [G]. Working in M_{0}[G], let ℝ be the term forcing consisting of Coll(κ^{+}, < μ)-names for elements of the factor forcing i(ℙ_{κ})_{κ+1,i(κ)}. Standard arguments tell us that in M_{0}[G] the poset ℝ is λ-closed and has i(κ) antichains. We may therefore build H^{∗} ∈ V [G] which is ℝ-generic over M_{0}[G].

Since M is the ultrapower of V by a (κ, κ^{++})-extender it follows that M = {j(f)(a) : f ∈ V, dom(f) = [κ]^{<ω}, a ∈ [κ^{++}]^{<ω}}. Since j can be factored as k composed with i, this implies that M = {k(F)(a) : F ∈ M_{0}, a ∈ [κ^{++}]^{<ω}}. Since k(μ) = j(f)(κ) = κ^{++} we may consider the domain of the functions F in this representation of M to be [μ]^{<ω}. It is now routine to lift k to a map k^{+} : M_{0}[G] ⟶ M[G], and also to see that

M[G] = {k^{+}(F)(a) : F ∈ M_{0}[G], a ∈ [κ^{++}]^{<ω}, dom(F) = [μ]^{<ω}}.

Since ℝ is λ-closed in M_{0}[G] we may transfer H^{∗} along k^{+} to get H^{∗∗}, which is generic over M[G] for the term forcing consisting of Coll(κ^{+}, < κ^{++})-terms for elements of j(ℙ_{κ})_{κ+1,j(κ)}. We may now use the generic g to realise H^{∗∗}, and obtain H which is generic over V [G∗g] for j(ℙ_{κ})_{κ+1,j(κ)}.

It is routine to lift j to j^{+} : V [G] ⟶ M[G ∗ g ∗ H], and to see that

M[G∗g ∗H] = {j^{+}(f)(a) : f ∈ V [G∗g], a ∈ [κ^{++}]^{<ω}, dom(f) = [κ]^{<ω}}.

Since g is generic for κ^{+}-closed forcing we may transfer it along j^{+} to get a generic g^{+}, and then lift again to get j^{++} : V [G ∗ g] ⟶ M[G ∗ g ∗ H ∗ g^{+}]. It is easy to see that j^{++} is the ultrapower by a (κ, κ^{++})-extender.

Since V [G] ⊧ ^{κ}ON ⊆ M[G] and g is generic for κ^{+}-closed forcing, it is easy to see that V [G ∗ g] ⊧ ^{κ}M[G ∗ g ∗ H ∗ g^{+}] ⊆ M[G ∗ g ∗ H ∗ g^{+}]. We claim that κ is not measurable in M[G ∗ g ∗ H ∗ g^{+}]. By closure it will suffice to see that κ is not measurable in M[G ∗ g], and since ℙ_{κ} is κ-Knaster we know that κ is not measurable in M[G]. The following easy lemma is therefore sufficient.

Lemma 4.1. Let 2^{κ} = κ^{+} and let 𝕊 be κ^{+}-closed. If κ is not measurable in V, then κ is not measurable in V^{𝕊}.

Proof. Suppose for a contradiction that κ is measurable in V^{𝕊}, and let U̇ name a measure. Enumerate the power set of κ as ⟨X_{α} : α < κ^{+}⟩ and build a decreasing sequence ⟨p_{α} : α < κ^{+}⟩ where p_{α} decides whether X_{α} ∈ U̇. Let U_{0} = {X_{α} : ∃α < κ^{+} p_{α} ⊩ X_{α} ∈ U̇} and check that U_{0} is a measure on κ. □

Relabelling our models and embeddings as we did at the end of the last stage, we are in the following situation: GCH holds and there is j : V ⟶ M such that crit(j) = κ, ^{κ}M ⊆ M, κ is not measurable in M, j is the ultrapower by a (κ, κ^{++})-extender and κ^{++} = κ_{M}^{++}.

Stage Four: We now do a Reverse Easton iteration ℙ_{κ} of length κ, adding a non-reflecting stationary set of ordinals of cofinality ω to each V -measurable α < κ. See [2] for a detailed discussion of this kind of iteration.

Let G be ℙ_{κ}-generic. Since κ is not measurable in M the forcing j(ℙ_{κ}) does not act at κ, and so in M[G] the factor forcing j(ℙ_{κ})_{κ,j(κ)} is highly strategically closed. By the usual arguments M[G] is closed under κ-sequences in V [G], and we may build H ∈ V [G] which is j(ℙ_{κ})_{κ,j(κ)}-generic over M[G].

We may then lift j to get j^{+} : V [G] ⟶ M[G ∗ H]. By arguments similar to those we have used in previous stages, M[G ∗ H] is closed under κ-sequences inside V [G] and κ is not measurable in M[G ∗ H]. Also j^{+} is the ultrapower by a (κ, κ^{++})-extender, all the models we have mentioned agree on the value of κ^{++}, and GCH holds in V [G].

We claim that κ is the least measurable cardinal in V [G]. We start by observing that all V -measurable cardinals below κ are no longer measurable. Suppose that λ < κ and λ is measurable in V [G] but not in V; then λ is not in the support of ℙ_{κ}, so λ is measurable in V [G_{λ}]. Now λ must be Mahlo in V and so ℙ_{λ} is λ-Knaster, but this is impossible since λ-Knaster forcing cannot create measurability at λ. For more details, we refer the reader to Lemma 3 of [2].

Relabelling once again, we are now in the following situation: GCH holds and there is j : V ⟶ M such that crit(j) = κ, ^{κ}M ⊆ M, κ is not measurable in M and is the least measurable in V, j is the ultrapower by a (κ, κ^{++})-extender and κ^{++} = κ_{M}^{++}.

Stage Five: We will now force with a certain term forcing. Let ℙ_{κ} be the Reverse Easton iteration in which α^{++} Cohen subsets are added to each inaccessible less than κ. Let ℚ̇ be a term for Add(κ, κ^{++})_{V^{ℙκ}} and ℚ̇_{0} be a term for Add(κ, κ^{+})_{V^{ℙκ}}. Let ℝ̇ be a term for Add(κ^{+}, κ^{++})_{V^{ℙκ∗ℚ̇_{0}}}. Finally we define ℝ^{∗} to be the term forcing consisting of ℙ_{κ} ∗ ℚ̇_{0}-terms for elements of ℝ.

Using the arguments of [3] it is possible to show that ℝ^{∗} is equivalent in V to Add(κ^{+}, κ^{++})_{V}. In particular ℝ^{∗} does not collapse cardinals or add any κ-sequences of ordinals. It follows that if in V^{ℝ^{∗}} we compute the Reverse Easton iteration to add α^{++} subsets to each inaccessible α ≤ κ, we end up with a forcing which is isomorphic to ℙ_{κ} ∗ ℚ̇. Let G^{∗} be ℝ^{∗}-generic over V.

We now claim that forcing with ℙ_{κ} ∗ ℚ̇ over V [G^{∗}] preserves the measurability of κ. To see this let G ∗ g be ℙ_{κ} ∗ ℚ̇-generic over V [G^{∗}]. We note that a fortiori G ∗ g is ℙ_{κ} ∗ ℚ̇-generic over V. We set g_{0} to be the restriction of g to ℚ_{0}, so that G ∗ g_{0} is ℙ_{κ} ∗ ℚ̇_{0}-generic over V.

We use G ∗ g_{0} to realise the term generic G^{∗}, thereby obtaining h which is ℝ-generic over V [G ∗ g_{0}]. V [G ∗ g] is a κ^{+}-c.c extension of V [G ∗ g_{0}] and ℝ is κ^{+}-closed in V [G ∗ g_{0}], so by Easton’s lemma h is ℝ-generic over V [G ∗ g].

By Woodin’s argument described in Section 3 κ is measurable in V [G ∗ g ∗ h]. By Easton’s lemma and the fact that ℙ_{κ} ∗ ℚ̇ is κ^{+}-c.c. we know that the power set of κ in V [G ∗ g × G^{∗}] is equal to the power set of κ in V [G ∗ g], which in turn is equal to the power set of κ in V [G ∗ g ∗ h]. So κ is measurable in V [G ∗ g × G^{∗}].

We can now prove the main result of this paper.

Theorem 4.2. If it is consistent that there exists a cardinal λ of Mitchell order λ^{++}, then it is consistent that if κ is the least measurable cardinal then 2^{κ} = κ^{+} and κ remains the least measurable cardinal after adding κ^{++} Cohen subsets of κ.

Proof. Consider the intermediate model V [G ∗ g_{0} × G^{∗}] from Stage Five. By Levinski’s argument from Section 3 κ is measurable in this model, and familiar arguments give us that in this model κ is the least measurable and 2^{κ} = κ^{+}.

It is also easy to see that V [G ∗ g_{0} × G^{∗}] has the same < κ-sequences of ordinals as V [G], so that Add(κ, κ^{++})_{V [G]} = Add(κ, κ^{++})_{V [G∗g_{0}×G^{∗}]}. Clearly V [G ∗ g × G^{∗}] can be viewed as an extension of V [G ∗ g_{0} × G^{∗}] by this poset.

We have shown that the model V [G ∗ g_{0} × G^{∗}] is as required. □

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Department of Mathematics, Baruch College of CUNY, New York NY 10010

E-mail address: awabb@cunyvm.cuny.edu
URL: http://math.baruch.cuny.edu/ apter

Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh PA 15213

E-mail address: jcumming@andrew.cmu.edu
URL: http://www.math.cmu.edu/users/jcumming/