PRESERVING MEASURABILITY WITH COHEN ITERATIONS

We describe a weak version of Laver indestructibility for a μ-tall cardinal κ, μ > κ+, where “weaker” means that the indestructibility refers only to the Cohen forcing at κ of a certain length. A special case of this construction is: if μ is equal to κ+n for some 1 < n < ω, then one can get a model V∗ where κ is measurable, and its measurability is indestructible by Add(κ, α) for any 0 ≤ α ≤ κ+n (Theorem 3.3).


Introduction
Assume κ is supercompact.In [7], Laver defined an iteration P of length κ such that in V[P], 1 κ is still supercompact and every further κ-directed closed forcing preserves the supercompactness of κ (P is often called the Laver preparation).We also say that κ is Laver-indestructible in V [P].The proof of this indestructibility result is essentially based on two useful properties of a supercompact cardinal κ in V: (i) for every μ ≥ κ, one can choose an elementary embedding j ∶ V → M with critical point κ such that M is closed under μ-sequences existing in V; this closure is then used to find a master condition in M and proceed with a lifting argument which ensures that supercompactness is preserved, 2 (ii) there is a single function f ∶ κ → V κ such that for every x ∈ V, one can choose an embedding j in (i) so that j(f )(κ) = x (this f is often called the Laver function).
A typical example of a κ-directed closed forcing is the Cohen forcing at κ, which we will denote by Add(κ, α), 3 where α is any ordinal larger than 0. The fact that over V[P], Add(κ, α) preserves the measurability of κ is very useful when one wishes to use some 1 V[P] indicates a P-generic extension of V whenever it is not important to distinguish specific P-generic filters.For instance the statement "φ holds in V[P]" means that φ holds in V[G] for every P-generic filter G. 2 Assume j ∶ V → M is an elementary embedding, P is a forcing notion, G is P-generic over V, and H is j(P)-generic over M. Then a sufficient condition for j to lift, i.e. a sufficient condition for the existence of With supercompactness, we can often argue that j"G is a condition in M (a master condition), and H can then be built below this master condition.For more details, see [3]. 3 Formally speaking, conditions in Add(κ, α) are partial functions of size < κ from κ × α to 2. The ordering is by reverse inclusion.
large cardinal properties of κ in V[P][Add(κ, α)] (see for instance [4] where a model with the tree property at κ ++ , κ strong limit singular with cofinality ω, is constructed starting with a supercompact κ).
A natural question is whether a "Laver-like" indestructibility is available also for smaller large cardinals.As it turns out, it is the property (i) above which is more important: it is known that for instance a strong cardinal4 κ has the analogue of the Laver function, but it is not known whether it can be made indestructible under κ-directed closed forcings. 5n this short paper we use the idea of Woodin (as described in [2]) to argue that it is possible to have a limited indestructibility of a μ-tall cardinal6 κ, κ + < μ regular, in the sense that we can successively extend V ⊆ V 1 ⊆ V * so that forcing with Add(κ, μ) over V * yields the measurability of κ.See Section 2.
Remark 1.1.We assume that the reader is familiar with the lifting arguments.The general reference is [3]; the more specific constructions used in the present paper are also given in [2].

Tall cardinals
In this section, we assume GCH.Let κ be μ-tall cardinal for some regular κ + < μ.Let j ∶ V → M be a μ-tall embedding with the extender representation: In particular, M is closed under κ-sequences in V and μ < j(κ) < μ + .Let U be the normal measure derived from j, and let i ∶ V → N be the ultrapower embedding generated by U. Let k ∶ N → M be elementary so that j = k ∘ i.Note that κ is the critical point of j, i and j, i have support κ, i.e. every element of M and N is of the form j(f )(α), or i(f )(κ) respectively, for some f with domain κ.In contrast, the critical point of k is (κ ++ ) N and k has support which we denote ν, where (κ ++ ) N < ν < i(κ), i.e. every element of M can be written as k(f )(α) for some f in N with domain ν. 9 Let P denote the forcing Add(κ, μ) in V, Q = i(P), and let g be a Q-generic filter over V. Then the following hold: Theorem 2.1.GCH.Forcing with Q preserves cofinalities and the following hold in V[g]: Proof.We show that Q is κ + -closed and κ ++ -cc in V. Closure is obvious by the fact that N is closed under κ-sequences in V. Regarding the chain condition, notice that every element of Q can be identified with the equivalence class of some function ; it suffices to check that the ordering ≤ on these f 's is κ ++ -cc.Let A be a maximal antichain in this ordering; take an elementary substructure M in some large enough H(θ) of V which contains all relevant data, has size κ + and is closed under κ-sequences.Then it is not hard to check that A ∩ M is maximal in the ordering (and so A ⊆ M ), and therefore has size at most κ + .
(i) and (ii).These follow by κ + -distributivity of Q in V and the fact that j, i have support κ: the pointwise image of g generates a generic for j(Q) and i(Q), respectively.
(iii).i(Q) is i(κ + )-closed in N, and since ν < i(κ + ), we use the distributivity of i(Q) and the fact that k has support ν to argue that the pointwise image of i 1 (g) generates a generic filter which is equal to j 1 (g) by commutativity of j, i, k. (iv).
There are therefore mutually generic over N by Easton's lemma.Remark 2.2.It would be tempting to expect that j 1 is still H(μ)-hypermeasurable if the original j was: however g is not included in M[j 1 (g)] and j 1 is therefore just μ-tall.There are some delicate issues involved if one wishes to preserve the H(μ)-hypermeasurability of κ in Theorem 2.1.A natural strategy is to prepare below κ by a reverse Easton iteration.This approach is taken in [2] where it is also shown that if μ = κ ++ , then Q is isomorphic to Add(κ + , κ ++ ) and thus the preparation can be implemented by iterating Add(α + , α ++ ) at all inaccessible α ≤ κ.In [5], this representation is shown for μ = κ +n for 2 ≤ n < ω, i.e. i(Add(κ, κ +n )) is isomorphic to Add(κ + , κ +n ).It seems it is possible to continue up to the first cardinal above κ with cofinality κ, but it is unclear whether it can be extended further.
Remark 2.3.The loss of the H(μ)-hypermeasurability of j 1 may prevent the use of this method in more complicated situations (such as a subsequent definition of Radin forcing to achieve results of a more global character).

Let us work in the model
to denote the resulting models and embeddings in Theorem 2.1.Using a fast-function forcing of Woodin, we can assume that there is f ∶ κ → κ in V such that j(f )(κ) = μ.Let us denote f (α) by μ α ; let C(f ) denote the closed unbounded set of the closure points of f : if α ∈ C(f ), then for all β < α, f (β) < α.

Theorem 2.4. There is a forcing iteration
Proof.Define R κ to be the following Easton-supported iteration: α denotes the forcing Add(α, μ α ).The proof uses the usual surgery argument (see [3]) with Fact 2.5 which allows us to use the generic filter g added in (for the proof, see Fact 2 in [2]). 10Fact 2.5.Let S be a κ-cc forcing notion of cardinality κ, κ <κ = κ.Then for any μ, the term forcing Q μ = Add(κ, μ) V [S] /S is isomorphic to Add(κ, μ).

Now we proceed with the proof of Theorem
Using the elementarity of i 1 , Fact 2.5 applied with S = i 1 (R κ ) and i 1 (Add(κ, μ)) shows that g -which is present in V 1 -yields a generic filter g ′ for the forcing i which is then modified by the standard surgery argument to allow for lifting [2]); i.e. if we denote the lifting of j 1 by j 2 , then witnesses the measurability, and in fact μ-tallness, of κ.

Hypermeasurable cardinals
It seems natural to extend Theorem 2.4 and have that the measurability of κ ensured by Add(κ, α) for any ordinal α, 0 < α ≤ μ.We will show that this can be achieved with some additional assumptions on μ.For concreteness, we will focus on the example where μ = κ +n for some 1 < n < ω.
First, in Theorem 3.1, we provide a standard construction which actually forces κ to stop being measurable in V * ; the measurability of κ is then resurrected by Add(κ, α) for any κ + ≤ α ≤ κ +n .Theorem 3.1.(GCH) Let 1 < n < ω be fixed and assume κ is H(κ +n )-hypermeasurable.Then there is an iteration

and for some reverse Easton iteration
In V * , the measurability -in fact the hypermeasurability -of κ is resurrected by Cohen forcing Add(κ, α) for any κ + ≤ α ≤ κ +n .
10 Recall that Q μ -mentioned in Fact 2.5 -is the term forcing defined as follows: the elements of Q μ are names τ such that τ is an S-name and it is forced by 11 For simplicity, we use the notation j 1 , i 1 , k 1 to denote the partial liftings of the embeddings j 1 , i 1 , k 1 .
Proof.Let j be an extender embedding witnessing the H(κ +n )-hyper-measurability of κ, and let i be a normal embedding generated by the normal measure U derived from j. Recall Lemma 3.2 from [5] which implies that if i ∶ V → N is an embedding generated by a normal measure on κ, then (3.2) Add(i(κ), i(κ) +n ) N ≅ Add(κ + , κ +n ).
Define P 1 is an Easton-supported iteration Let j 1 and i 1 be the liftings of j and i.
In V 1 define P κ as an Easton supported iteration: by the application of the gap-forcing theorem of [6]: a hypothetical embedding k with critical point κ found in V * could be written as an embedding from V 1 [P κ ] to some N[j(P κ )], with N ⊆ V 1 ; in particular a generic filter for j(P κ ) would need to add a non-trivial generic filter at stage κ which cannot be found in The rest of the Theorem follows from the following Claim: Proof.It suffices to show the Claim for α's which are cardinals.So assume κ +m = |α| for some 1 ≤ m ≤ n.Choose in V 1 an embedding j m ∶ V 1 → M m which witnesses the H(κ +m )-hypermeasurability of κ with κ +m < j m (κ) < κ +m+1 (this is possible since , and therefore the generic for Add(κ, κ +m ) V 1 [P κ ] provides a generic for Add(κ, κ +n ) M m [P κ ] .The argument is then finished as in Theorem 2.4, using the fact that the generic g for i 1 (Add(κ, κ +n )) is also generic for i 1 (Add(κ, κ +m )).
This concludes the proof of Theorem 3.1.Note that the method in the proof of Theorem 3.1 does not work for the case of α smaller than κ + : every elementary embedding k ∶ V 1 → M with critical point κ sends κ above κ + and therefore κ equivalent to the Cohen forcing at κ of length at least κ + .It follows that to lift the embedding, we need to force over V 1 [P κ ] with a Cohen forcing at κ of length at least κ + .If α < κ + , this condition is not satisfied.We remedy this by a more complicated construction in Theorem 3.3.Theorem 3.3.With the assumptions and the notation as in Theorem 3.1, one can define P κ so that κ is measurable in V * , and its measurability -in fact hypermeasurability -is indestructible by Add(κ, α) for any 0 < α ≤ κ +n .
Then one can argue that κ is still measurable in V * : while lifting the embedding j 1 , it suffices to work below a condition in j 1 (P κ ) which chooses the trivial forcing {1} at stage κ.
To argue that for any 0 < α ≤ κ +n , κ is still measurable in V * [Add(κ, α)], work below a condition in j 1 (P κ ) which chooses the right forcing at stage κ.

Open questions
Q1.Is it possible to generalise Theorem 2.4 so that μ is still H(μ)-hypermeasurable if the original embedding j was H(μ)-hypermeasurable ?This would require some sort of preparation below κ in the model V 1 (analogously to the methods in Theorem 3.1).
A related question is this: Q2.Is it possible to characterise the forcings i(Add(κ, μ)), where i ∶ V → N is a normal measure ultrapower as in Theorem 2.1 ?We know that this forcing does not collapse (it is κ + -closed and κ ++ -cc in V), but does it have a uniform representation?In particular, is it isomorphic to Add(κ + , μ) of V?