# from iterability

## Abstract

We show that if has a proper class of Woodin cardinals, a strong cardinal, and a generically universally Baire iteration strategy (as defined in the paper) then holds after collapsing the successor of the least strong cardinal to be countable. This result is complementary to other work by the authors where it is shown that holds in a generic extension of a certain minimal universe. The current theorem is more general in that no minimality assumption is needed. A corollary of the main theorem is that is consistent relative to the existence of a Woodin cardinal which is a limit of Woodin cardinals. This improves significantly on the first consistency of obtained by W.H. Woodin.

The ( is a determinacy axiom isolated by Woodin. It asserts that the largest Suslin cardinal is inaccessible for ordinal definable bijections. Let ) be the statement that in all (set) generic extensions there is a model of whose Suslin, co-Suslin sets are the universally Baire sets. The other main result of the paper shows that assuming has a proper class of inaccessible cardinals which are limit of Woodin cardinals, a strong cardinal, and a generically universally Baire iteration strategy, in the universe where , is for the collapse of the successor of the least strong cardinal to be countable, the theory -generic - - fails; this implies that is not equivalent to (over the base theory of This is interesting and somewhat unexpected, in light of other work by the authors. Compare this result with SteelтАЩs well-known theorem that ).тАЬ holds in all generic extensionsтАЭ is equivalent to тАЬthe theory of is sealedтАЭ in the presence of a proper class of measurable cardinals.

We identify elements of the Baire space with reals. Throughout the paper, by a тАЬset of reals we mean тАЭ, A set of reals . is * Baire -universally* if there are trees on for some such that and whenever is a in -generic, , We write . for this is the canonical interpretation of ; in .тБаFootnote^{1} is *universally Baire* if is Baire for all -universally Let . be the set of universally Baire sets. Given a generic we let , and The next definition is due to Woodin. .

is a form of Shoenfield-type generic absoluteness for the theory of universally Baire sets. In this paper, we will avoid motivational discussion as Reference ST19 has a lengthy introduction to the subject. We should say, however, that is an important hypothesis in set theory and particularly in inner model theory for several reasons. If a large cardinal theory implies then the Inner Model Program for building canonical inner models of cannot succeed (at least with the criteria for defining тАЬcanonical inner modelsтАЭ as is done to date), cf Reference ST19, Sealing Dichotomy. signifies a place beyond which new methodologies are needed in order to advance the Core Model Induction techniques. In particular, to obtain consistency strength beyond from strong theories such as the Proper Forcing Axiom, one needs to construct canonical subsets of (third-order objects), instead of elements of like what has been done before (see Reference ST19, Section 1 for a more detailed discussion). The consistency of was first demonstrated by Woodin, who showed that if there is a proper class of Woodin cardinals and a supercompact cardinal then holds after collapsing to be countable. WoodinтАЩs proof can be found in Reference Lar04.

One of the main corollaries of the Theorem 0.4 is that the set theoretic strength of is below a Woodin cardinal that is a limit of Woodin cardinals; this improves significantly the aforementioned result of Woodin. Another proof of this fact was presented in Reference ST19, where the authors establish an actual equiconsistency for One advantage of the proof in this paper is that no smallness assumption is made (unlike .Reference ST19). Another, perhaps more important, advantage of the current proof over the one presented in Reference ST19 is that this proof is more accessible. Our proof of is based on iterability and uses recent ideas from descriptive inner model theory. However, in this paper, our aim is to present the proof of our main theorem, Theorem 0.4, without using any fine structure theory or heavy machinery from inner model theory, so that the paper is accessible to the widest possible audience. We will only assume general knowledge of iterations, iteration strategies and WoodinтАЩs extender algebra, all of which are topics that can be presented without any fine structure theory. For instance, the reader can consult Reference Far11 or Reference MS94. The fact that the hypothesis of Theorem 0.4 is weaker than a Woodin cardinal that is a limit of Woodin cardinals follows from a very recent work of Steel (Reference Ste16b) and the first author (Reference Sar) (but also see Reference Sar20), and this fact will not be proven here, as it is well beyond the scope of this paper.

Given a transitive model of set theory and a -cardinal we let , We say . is a extender over -short if there is a embedding -elementary such that

- (1)
is transitive,

- (2)
and ,

- (3)
and ,

- (4)
and .

is called the *critical point of * and the *length of *. We write and . is then called the *ultrapower of by * and is uniquely determined by and We write . Given a set . and an extender we say , *coheres* if For more on short extenders, the reader can consult .Reference MS94 or Reference Far11.

We can also define the notion of *a long extender*, though we will not need the precise definition in this paper. Roughly speaking, given an elementary embedding with critical point an ordinal , and letting , be least such that we can define an extender , of length from This is a function . given by: If . then , is a long extender. For more details on long extenders, see Reference Woo10b.

Suppose is a transitive model of set theory. We let be the set of *inaccessible-length extenders* of More precisely . consists of short extenders such that is inaccessible and .тАЭ

When we talk about iterability for we mean iterability with respect to extenders in , (and its images). Thus, the relevant iterations are those that are built by using extenders in and its images.

Recall from Reference MS94 that an iteration is *normal* if the extenders used in it have increasing lengths and each extender used along is applied to the least possible model, i.e. is applied to the first model where the ultrapower makes sense. Following Jensen, we will say that is a *smooth iteration* (of its base model) if it can be represented as a *stack* of normal iterations. More precisely, where is a normal iteration of the base model of and for , is a normal iteration of the last model of if is a successor ordinal and on the direct limit of under the iteration embeddings if is limit. We say that a pre-iterable structure is *smoothly iterable* if player II has a wining strategy in the iteration game of arbitrary length that produces smooth iterations. Recall that in iteration games, player I picks the extenders while player II plays branches at limit steps. We say that is an iteration strategy for if it is a strategy for in the iteration game that produces arbitrary length smooth iterations of .

Finally we state self-iterability. The Unique Branch Hypothesis ( is the statement that every normal iteration tree ) on has at most one cofinal well-founded branch. The Generic Unique Branch Hypothesis ( says that ) holds in all set generic extensions. The notion of generically universally Baire (guB) strategy appears in the next section as Definition 1.5.

Notice that because of clause 1, the iteration strategy in clause 2 is unique.

As mentioned above, a corollary of Theorem 0.4, via a non-trivial amount of work in Reference Ste16b and Reference Sar (but also see Reference Sar20),тБаFootnote^{2} is

^{2}

The existence of an lbr hod premouse as in Reference Sar20, Theorem 1.2 follows from the existence of a Woodin limit of Woodin cardinals by Reference Sar20, Step 4. Then letting be as in Reference Sar20, Theorem 1.2, satisfies the hypothesis of Theorem 0.4.

The main idea behind the proof of Theorem 0.4 originates in Reference ST19. The most relevant portion of that paper is Reference ST19, Theorem 3.1. We should note that the hypothesis of Theorem 0.4 cannot be weakened to just for plus-2 iterations as this form of holds in a minimal mouse with a strong cardinal, a class of Woodin cardinals and a stationary class of measurable cardinals,тБаFootnote^{3} but this theory is weaker than as shown by Reference ST19, Theorem 3.1.

The was introduced by Woodin in Reference Woo10a, Remark 9.28. The terminology is due to the first author. Here is the definition. In the following, we say that a cardinal is * -inaccessible* if for every there is no surjection that is definable from ordinal parameters.

In the hierarchy of determinacy axioms, which one may appropriately call the ,тБаFootnote^{4} is an anomaly as it belongs to the successor stage of the but does not conform to the general norms of the successor stages of the Prior to .Reference ST, was not known to be consistent. Reference ST shows that it is consistent relative to a Woodin cardinal that is a limit of Woodin cardinals. Nowadays, the axiom plays a key role in many aspects of inner model theory, and features prominently in WoodinтАЩs framework (see Reference Woo17, Definition 7.14 and Axiom I and Axiom II on page 97 of Reference Woo17).тБаFootnote^{5}

^{4}

Solovay defined what is now called the (see Reference Woo10a, Definition 9.23). It is a closed sequence of ordinals with the largest element where , is the least ordinal that is not a surjective image of the reals. One then obtains a hierarchy of axioms by requiring that the has complex patterns. is an axiom in this hierarchy. The reader may consult Reference Sar13 or Reference Woo10a, Remark 9.28.

^{5}

The requirement in these axioms that there is a strong cardinal which is a limit of Woodin cardinals is only possible if .

Reference ST19 shows that is equiconsistent with over the theory is a proper class of Woodin cardinals and the class of measurable cardinals is stationaryтАЭ. In this paper, we show that in general, one cannot replace тАЬequiconsistentтАЭ with тАЬequivalentтАЭ. Recall from тАЬthereReference Ste16b the statement of *Hod Pair Capturing* for any Suslin co-Suslin set : there is a least-branch (lbr) hod pair , such that is definable from parameters over .*No Long Extender* is the statement: there is no countable, pure extender premouse -iterable such that there is a long extender on the The notion of least-branch hod mice (lbr hod mice) is defined precisely in -sequence.Reference Ste16b, Section 5.

Remark 0.10(1), Theorem 0.9, and the fact that self-iterability and hold in any generic extension of an lbr hod mouse with a proper class of Woodin cardinals give us the following.

Corollary 0.11 is surprising. For example, generic absoluteness for namely that for all successive generics , and there is an elementary embedding is equivalent to the existence and the universally Bairness of the next canonical set beyond , namely , .тБаFootnote^{6} While one cannot hope that would imply both the existence and the universal Bairness of the next canonical set of reals beyond ,тБаFootnote^{7} one could still hope that the cause of is the existence of some nice set of reals just like the cause of the generic absoluteness of is the universally Bairness of .тБаFootnote^{8} Because the next nice set beyond cannot be universally Baire, the best we can hope for is that the next set beyond creates an model over In fact, this discussion was the original motivation for isolating . Contrary to our expectations, what causes . may not be coded into a set of reals as demonstrated by Corollary 0.11.

^{6}

This fact is due to Steel and Woodin. For example, see genericity iterations in Reference Ste10.

Throughout this paper, except in Section 1, we assume the hypothesis of Theorem 0.4. Throughout this paper, except in Section 1, will stand for the least strong cardinal. In this paper, especially in Section 2, we will make heavy use of NeemanтАЩs тАЬrealizable maps are genericтАЭ result that appears as Reference Nee02, Corollary 4.9.2. Sections 4 and 5 make heavy use of the results of Section 2 to show that for -generic where , is as in Theorem 0.4, for generic one can realize , as the derived model of an iterate of a countable substructure of for some large (Lemma 5.1). This is then used to prove Theorem 0.4 in Section 6. The last section proves Theorem 0.9.

## 1. Generically universally Baire iteration strategies

In this paper we will need three properties of iteration strategies, namely *Skolem-hull condensation*, *pullback condensation* and *generically universal Bairness*. We now define these notions.

We say is an *iterable pair* if is a pre-iterable structure and is a strategy for it. Suppose is an iterable pair. If is a smooth iteration of according to with last model then we write for the strategy of induced by Namely, . When . is independent of we will drop it from our notation. Given a -cardinal we write , for the fragment of that acts on smooth iterations based on Here recall that . .

Continuing with suppose , is elementary. Given a smooth iteration of we can define the copy on which may or may not have well-founded models. The construction of was introduced in Reference MS94 on page 17. Suppose now that is such that is according to and is of limit length. Let It follows from the construction of . that yields a well-founded branch of .

We then say is the of -pullback if for any smooth iteration on that is according to , is according to It is customary to let . be .

The following theorems are easy consequences of ( and are probably not due to the authors. ),

Suppose is an iterable pair. Given a strong limit cardinal and set ,

Given a structure in a language extending the language of set theory with a transitive universe, and an we let , be the transitive collapse of and be the inverse of the transitive collapse. In general, the preimages of objects in will be denoted by using as a subscript, e.g. Suppose in addition . where is a pre-iterable structure and is an iteration strategy of We will then write . to mean that and the strategy of that we are interested in is We set . .

Motivated by the definition of universally Baire sets that involves club of generically correct hulls, we make the following definition.

In Definition 1.5, we could demand that there is a club of with the desired properties. However that would be equivalent to our definition as we can let above code the desired club. In the next section our goal is to prove some basic facts about -strategies.

## 2. Generic interpretability of guB strategies

As we said in the introduction, from this point on we work under the hypothesis of Theorem 0.4. However, we will not use the existence of a strong cardinal until Section 5.

Let be the guB-strategy of and fix a generic prescription for (see Definition 1.5). We will omit from our notation and just write instead of Given a cardinal . we will write for the fragment of that acts on iterations based on Often we will treat . as a strategy for rather than a strategy for Similarly, given an interval . we will write for the fragment of on iterations based on above To make the notation simpler, often we will not specify the domain of . that we have in mind (as in Lemma 2.1).

Let be a Woodin cardinal of We first prove that . has canonical extensions in generic extensions of As a first step, we prove the following useful capturing result. .