1 The consistency of $\sf {Sealing}$

The following is our main theorem. We say that $\phi $ and $\psi $ are equiconsistent over theory T if there is a model of $T\cup \{\phi \}$ if and only if there is a model of $T\cup \{\psi \}$ .

The following variation of $\sf {Sealing}$ , called $\sf {Tower \ Sealing}$ , is a stronger statement than the version isolated by Woodin [Reference Neeman11]. $\sf {Tower \ Sealing}$ turns out to be equiconsistent with $\sf {Sealing}$ and $\sf {LSA}$ - $\sf {over}$ - $\sf {uB}$ over the base theory of Theorem 1.1.

Woodin has observed that assuming a proper class of Woodin cardinals which are limits of strong cardinals, $\sf {Tower \ Sealing}$ implies $\sf {Sealing}$ . The converse is not known. However, we can show the following theorem.

Woodin, in [Reference Neeman11], shows $\sf {Sealing}$ and a weak form of $\sf {Tower \ Sealing}$ are consistent relative to the existence of a supercompact cardinal and a proper class of Woodin cardinals. As explained above, a corollary of the theorems above, and recent work in [Reference Sargsyan and Trang16, Reference Hugh Woodin26], is the following theorem, which significantly improves Woodin’s theorem. Throughout this paper, we let $\sf {WLW}$ denote the theory “ $\sf {ZFC}$ + there is a Woodin cardinal which is a limit of Woodin cardinals.”

In fact, the proof Theorem 1.4 shows that in a generic extension of an excellent hybrid mouse (cf. Definition 3.1), both $\sf {Sealing}$ and $\sf {Tower \ Sealing}$ hold. As mentioned in the introduction, the consistency of the existence of an excellent hybrid mouse follows from Con $(\sf {WLW})$ .

2 Self-iterability and $\sf {Sealing}$

Next, we state another theorem about the consistency of $\sf {Sealing}$ that implies $\sf {Sealing}$ and $\sf {LSA}$ - $\sf {over}$ - $\sf {uB}$ are not equivalent. The Unique Branch Hypothesis ( $\textsf {UBH}$ ) is the statement that every normal iteration tree ${\mathcal {T}}$ on V has at most one cofinal well-founded branch. The Generic Unique Branch Hypothesis ( $\sf {gUBH}$ ) says that $\sf {UBH}$ holds in all set generic extensions. The notions of a pre-iterable structure and a generically universally Baire (guB) strategy are discussed in detail in [Reference Sargsyan and Trang18, Section 1].

Suppose P is a transitive model of set theory. We let ${\sf {ile}}(P)$ be the set of inaccessible-length extenders of P. More precisely $\sf {ile}(P)$ consists of extenders $E\in P$ such that $P\vDash ``lh(E)$ is inaccessible and $V_{lh(E)}=V_{lh(E)}^{Ult(V, E)}$ ,” where $lh(E)$ is the length of the extender E.

The notions in Definition 2.1 abstract some of the essential properties of universally Baire strategies of fine-structural extender models that we need for the proof of the following theorem.

Recall from [Reference Hugh Woodin26] the statement of Hod Pair Capturing $(\sf {HPC})$ : For any Suslin, co-Suslin set $A\subseteq \mathbb {R}$ , there is a least-branch (lbr) hod pair $({\mathcal {P} },\Sigma )$ such that A is definable from parameters over $(HC,\in , \Sigma )$ . No Long Extender $(\sf {NLE})$ is the statement: there is no countable, $\omega _1+1$ -iterable pure extender premouse M such that there is a long extender on the M-sequence. The notion of least-branch hod mice (lbr hod mice) is defined precisely in [Reference Hugh Woodin26, Section 5].

Remark 2.6, Theorem 2.5, and the fact that self-iterability and $\sf {gHPC}$ hold in any generic extension of an lbr hod mouse with a proper class of Woodin cardinals give us the following.

Corollary 2.7 is surprising and unexpected in light of the above results. For example, generic absoluteness for $L({\mathbb {R}})$ , namely, for all successive generics g and h, there is an elementary embedding $j:L({\mathbb {R}}_g)\rightarrow L({\mathbb {R}}_{g*h})$ , is equivalent to the existence and the universal Baireness of the next canonical set beyond $L({\mathbb {R}})$ , namely ${\mathbb {R}}^{\#}$ .Footnote ⁷ While one cannot hope that $\sf {Sealing}$ would imply both the existence and the universal Baireness of the next canonical set of reals beyond $\Gamma ^{\infty }$ ,Footnote ⁸ one could still hope that the cause of $\sf {Sealing}$ is the existence of some nice set of reals just like the cause of the generic absoluteness of $L({\mathbb {R}})$ is the universal Baireness of ${\mathbb {R}}^{\#}$ .Footnote ⁹ Because the next nice set beyond $\Gamma ^{\infty }$ cannot be universally Baire, the best we can hope for is that the next set beyond $\Gamma ^{\infty }$ creates an $\sf {LSA}$ model over $\Gamma ^{\infty }$ . In fact, this discussion was the original motivation for isolating $\sf {LSA}$ - $\sf {over}$ - $\sf {UB}$ . Contrary to our expectations, what causes $\sf {Sealing}$ to hold may not be coded into a set of reals as demonstrated by Corollary 2.7.

3 Outline of proofs

We first give a brief outline of the proofs of Theorems 1.1 and 1.3. The crucial notion involved in these proofs is that of an excellent hybrid premouse, which is a kind of short-tree strategy hybrid premice, where the short-tree strategy is that of an lsa hod premouse. The notions of lsa hod premice and their short-tree strategies are crucial for the analysis of models of $\sf {LSA}$ and are treated in detail in [Reference Steel21]. Roughly an lsa hod premouse ${\mathcal {P} }$ is a hod premouse with the largest Woodin cardinal $\delta ^{\mathcal {P} }$ , ${\mathcal {P} } = ({\mathcal {P} }|\delta ^{\mathcal {P} })^{\sharp }$ , and letting $\kappa $ be the least $<\delta ^{\mathcal {P} }$ -strong cardinal in ${\mathcal {P} }$ , then $\kappa $ is a limit of Woodin cardinals. If $\Sigma $ is an iteration strategy of ${\mathcal {P} }$ , then the short part $\Sigma ^{sh}$ of $\Sigma $ is the part of $\Sigma $ that acts on short trees according to $\Sigma $ , so $\Sigma ^{sh}({\mathcal {T}})$ is defined and is $\Sigma ({\mathcal {T}})$ if and only if ${\mathcal {T}}$ is short.Footnote ¹⁰ We say that $({\mathcal {P} },\Lambda )$ is a short-tree strategy (sts) hod pair if ${\mathcal {P} }$ is an lsa hod premouse and $\Lambda $ is a short-tree strategy of ${\mathcal {P} }$ . If ${\mathcal { Q}}$ is a $\Lambda $ -premouse above ${\mathcal {P} }$ , then we say that ${\mathcal { Q}}$ is an sts premouse based on ${\mathcal {P} }$ .Footnote ¹¹

Suppose $({\mathcal {P} }, \Lambda )$ is an sts hod pair, where $\Lambda $ has hull condensation and strong branch condensation (see [Reference Steel21] for a detailed discussion of these notions). $\Lambda $ has branch condensation for pullbacks if whenever $\pi :{\mathcal { Q}}\rightarrow {\mathcal {P} }$ is elementary, the $\pi $ -pullback of $\Lambda $ has branch condensation. For a more detailed discussion of these topics, see [Reference Sargsyan15, Reference Steel20, Reference Steel21].

We say that $[\nu , \delta ]$ is a window if there are no Woodin cardinals in the interval $(\nu , \delta )$ . Given a window w, we let $\nu ^w$ and $\delta ^w$ be such that $w=[\nu ^w, \delta ^w]$ . We say that window w is above $\kappa $ if $\nu ^w\geq \kappa $ . Suppose $\kappa $ is a cardinal. We say window-based self-iterability $(\sf {WBSI})$ holds at $\kappa $ if for any window w that is above $\kappa $ and for any successor cardinal $\eta \in (\nu ^w, \delta ^w)$ , setting $Q= H_{\eta ^+}$ , Q has an $Ord$ -iteration strategy $\Sigma $ which acts on iterations that only use extenders with critical points $>\nu ^w$ .

For one direction of Theorems 1.1 and 1.3, assume the existence of an excellent hybrid premouse ${\mathcal {P} }$ . Let $\delta = \delta ^{\mathcal {P} }$ , $\kappa $ be the least $<\delta $ -strong cardinal in ${\mathcal {P} }_0$ , and $g\subset Coll(\omega ,\delta )$ be ${\mathcal {P} }$ -generic. Then one can show in ${\mathcal {P} }[g]$ , $\sf {Sealing}$ , $\sf {Tower \ Sealing}$ , and $\sf {LSA}$ - $\sf {over}$ - $\sf {uB}$ hold. We give a sketch of $\sf {Sealing}$ holds; the reader can consult [Reference Steel20, Section 3] for a detailed proof. Let $\Lambda $ be the short-tree strategy of ${\mathcal {P} }_0$ coded by $S^{\mathcal {P} }$ . The smallness assumptions in clause (1) of Definition 3.1 amongst other things imply that all local non-Woodin cardinals of a hod premouse (or hybrid premouse) are witnessed by ${\mathcal { Q}}$ -structures which are initial segments of ${\mathcal {P} }$ and are tame. Clause (3) prevents us from constructing a “smaller” excellent hybrid premouse by performing certain fully backgrounded sts constructions over $({\mathcal {P} }_0|\tau )^{\sharp }$ inside ${\mathcal {P} }$ . Clauses (2) and (4) (combined with clause (1)) allow us to extend $\Lambda $ and the window-based strategy for ${\mathcal {P} }|\eta $ , where $\eta \in [\nu ^w, \delta ^w]$ and $[\nu ^w, \delta ^w]$ is a window above $\delta $ , to any set generic extension over ${\mathcal {P} }$ (cf. [Reference Steel20, Section 2]. Let h be a generic filter over ${\mathcal {P} }[g]$ and $\Lambda ^h$ be the interpretation of $\Lambda $ in ${\mathcal {P} }[g*h]$ . We then can “capture” $\Gamma ^{\infty }_{g*h}$ in the following sense: letting $\Delta ^h$ be the collection of $A\subseteq \mathbb {R}^{{\mathcal {P} }[g*h]}$ in ${\mathcal {P} }[g*h]$ such that for some countable ${\mathcal {T}}$ according to $\Lambda ^h$ such that the iteration embedding restricted to ${\mathcal {P} }_0^b = ({\mathcal {P} }_0|(\kappa ^+)^{{\mathcal {P} }})$ exists, then

(1) $$ \begin{align} \Delta^h = \Gamma^{\infty}_{g*h}. \end{align} $$

[Reference Steel20, Lemma 3.3] shows that $L(\Delta ^h, \mathbb {R}^{{\mathcal {P} }[g*h]})$ can be realized as the derived model of an iterate of ${\mathcal {P} }_0$ . The proof of [Reference Steel20, Lemma 3.3] and various lemmas in [Reference Steel20, Section 2] uses substantially the fact that window-based iterability in ${\mathcal {P} }$ and its generic extensions implies various backgrounded constructions in ${\mathcal {P} }$ and in its generic extensions converge. Equation (1) then implies clause (1) of $\sf {Sealing}$ . A standard argument (cf. [Reference Steel20, Lemma 3.4]) shows that for any generic filter k over ${\mathcal {P} }[g*h]$ , there is an elementary embedding $j: L(\Delta ^h, \mathbb {R}^{{\mathcal {P} }[g*h]})\rightarrow L(\Delta ^{h*k}, \mathbb {R}^{{\mathcal {P} }[g*h*k]})$ such that for any $A\in \Delta ^h$ , $j(A) = A^{k}$ . This implies clause (2) of $\sf {Sealing}$ .

The proof of Theorem 2.2’s main idea is similar to the sketch above. The self-iterability hypothesis of the theorem for the most part allows us to simulate the proof sketched above. We can find a transitive model R Footnote ¹³ with sufficient iterability such that $H_{\kappa ^+}\subset R$ and there is an iterate S of R such that for some $\lambda $ , a limit of Woodin cardinals of S, for some S-generic $g\subseteq Coll(\omega , <\lambda )$ , letting $\mathbb {R}^*_g = \bigcup _{\alpha <\lambda } \mathbb {R}^{S[g \cap Coll(\omega , \alpha )]}$ , and $Hom^*_g$ be the collection of sets $A\in S(\mathbb {R}^*_g)\cap {\wp }(\mathbb {R})$ such that for some $\alpha <\lambda $ , there is a pair $(T, U)\in S[g\cap Coll(\omega , \alpha )]$ such that $S[g\cap Coll(\omega , \alpha )]\vDash ``(T, U)$ are $<\lambda $ -absolutely complementing trees” and $p[T]^{V({\mathbb {R}}^*)}=A$ , then we have the following:

• • $\mathbb {R}^*_g = \mathbb {R}^V$ .

• • $Hom^*_g = \Gamma ^{\infty }$ .

The fact that $\kappa $ is strong is used substantially in the proof of $Hom^*_g = \Gamma ^{\infty }$ above. $\sf {Sealing}$ holds in $V^{Coll(\omega ,\kappa ^+)}$ then follows by a standard argument. See [Reference Sargsyan and Trang18].

Now we sketch the proof of the other direction of Theorems 1.1 and 1.3. In particular, assuming $\sf {Sealing}$ (or $\sf {Tower \ Sealing}$ , or $\sf {LSA}$ - $\sf {over}$ - $\sf {uB}$ ) and the large cardinal base theory of the theorems, we want to show that excellent hybrid premice exist. The proof of this occupies Sections 4–10 of [Reference Steel20] and involves many details and ideas. Many of these ideas, especially the notion of condensing sets and various hybrid (sts) backgrounded constructions, feature prominently in the HOD analysis of models of $\sf {LSA}$ as well as the core model induction proof that $\sf {LSA}$ is consistent relative to $\sf {PFA}$ (cf. [Reference Steel21]). The main thing we need to prove is: assuming no excellent hybrid premice exist, then certain hybrid (sts) fully backgrounded constructions converge (see [Reference Steel20, Sections 8–10]) to a model ${\mathcal {P} }$ , which turns out to be an excellent hybrid premouse. There are many consequences that we need from the hypothesis to guarantee the convergence of such constructions, but one main set of consequences is described in [Reference Steel20, Section 5]: there is a stationary class S of measurable cardinals $\mu $ which is a limit of Woodin cardinals, there is a $\nu < \textrm {min}(S)$ , and there are sequences $(Y_{\mu }: \mu \in S)$ , $(A_{\mu }: \mu \in S)$ such that the following hold for $\mu \in S$ :

1. (i) letting $h\subset Coll(\omega , <\mu )$ be V-generic, then $Hom^*_h = \Gamma ^{\infty }_h$ ,

2. (ii) $A_{\mu }\in Y_{\mu }$ and $A_{\mu }$ codes $H_{\mu }$ , and the hod limit of $L(Hom^*_h, \mathbb {R}^*_h)$ ,Footnote ¹⁴

3. (iii) whenever $Y_{\mu } \subseteq X\prec H_{\mu ^+}$ and X has size $<\mu $ , is closed under $\nu $ -sequences, X captures Lp $^{cuB,\Psi _{\mu }}(A_{\mu })$ in the sense that letting $\pi _X: M_X\rightarrow X$ be the uncollapse map, then $\pi _X^{-1}(Lp^{cuB,\Psi _{\mu }}(A_{\mu })) = Lp^{cuB,\Lambda }(\pi _X^{-1}(A_{\mu }))$ , where $\Lambda $ is the $\pi _X$ -pullback strategy of $\Psi _{\mu }$ .Footnote ¹⁵

$\sf {Sealing}$ (or $\sf {Tower \ Sealing}$ , or $\sf {LSA}$ - $\sf {over}$ - $\sf {uB}$ ) and the fact that $\mu $ is measurable can be used to show

$$ \begin{align*} \text{cof}(o(Lp^{cuB,\Psi_{\mu}}(A_{\mu}))) < \mu. \end{align*} $$

The significance of $\nu $ is that it is an ordinal that bounds cof $(o(Lp^{cuB,\Psi _{\mu }}(A_{\mu })))$ for all $\mu \in S$ and $\nu $ is obtained from the usual Fodor’s argument applied to the (class) function $\mu \mapsto \textrm {cof}(o(Lp^{cuB,\Psi _{\mu }}(A_{\mu })))$ ; this in turn allows us to “capture” $Lp^{cuB,\Psi _{\mu }}(A_{\mu })$ by a club of X as described in (iii) above. The reader can consult [Reference Steel20] for details on how these facts are put together into a proof of convergence of various hybrid fully backgrounded constructions.

4 Variations of $\sf {Sealing}$ and open problems

The rather mild assumption used in Theorems 1.1 and 1.3 that the class of measurable cardinals is stationary is probably not needed. As mentioned above, it is mainly used in [Reference Steel20, Section 5] to show that a certain (definable) regressive class function is constant on a stationary class of measurable cardinals. The existence of the ordinal $\nu $ above follows from this. We suspect that refinements of our techniques will eliminate this assumption. We conjecture the following.

As discussed in [Reference Steel20], the core model induction becomes very difficult past $\sf {Sealing}$ . A good test problem for practitioners of the core model induction is the following problem.

We know from the results above that $\sf {WLW}$ is stronger than $\sf {Sealing}$ and is roughly the strongest natural theory at the limit of traditional methods for proving iterability. We believe it is plausible to develop core model induction methods for obtaining canonical models of $\sf {WLW}$ from just $\sf {PFA}$ .Footnote ¹⁶ The paper [Reference Sargsyan and Trang19] is the first step towards this goal; in the paper, we have constructed from $\sf {PFA}$ hod mice that are stronger than $\mathcal {P}$ in Definition 3.1.

The following are variations of the theories discussed above.

The proofs of Theorems 1.1 and 1.3 show the following. Let $T=``$ there exists a proper class of Woodin cardinals and the class of measurable cardinals is stationary.” Then the following theories are equiconsistent:

1. 1. $\sf {Sealing}+T$ .

2. 2. $\sf {Sealing}^++T$ .

3. 3. $\sf {Sealing}^- + T$ .

4. 4. $\sf {Tower \ Sealing} + T$ .

5. 5. $\sf {LSA}$ - $\sf {over}$ - $\sf {uB}+T$ .

6. 6. $\sf {LSA}$ - $\sf {over}$ - $\sf {uB}^-+T$ .

The following conjecture, if true, would be an ultimate analog of the main result of [Reference Steel22], which shows, assuming a proper class of measurable cardinals, the statement “ $\sf {AD}$ holds in $L(\mathbb {R})^{V^{\mathbb {P}}}$ for every poset $\mathbb {P}$ ” is equivalent to “for every poset $\mathbb {P}$ , $L(\mathbb {R})^{V^{\mathbb {P}}}\vDash $ there is no uncountable sequence of distinct reals.”

We end this paper by the following question. In every known model of ${\sf {MM}} + $ there is a proper class of Woodin cardinals, $(*)^{++}$ fails.