Partial Tower Sealing in Inner-Model Theory

• Partial Tower Sealing is a weak variant of Tower Sealing that permits a strict inclusion between derived pointclasses while retaining key structural and absoluteness features.
• It employs stationary tower forcing in hod mice models with strong and Woodin cardinals to achieve a form of generic absoluteness for universally Baire sets.
• The approach is consistent below ZFC with a Woodin limit of Woodin cardinals, marking a significant reduction in large cardinal strength compared to full Tower Sealing.

Partial Tower Sealing is a weak form of the Tower Sealing phenomenon in descriptive inner-model theory, primarily formulated within generic extensions of hod mice models that include a strong cardinal and a proper class of Woodin cardinals. While full Tower Sealing achieves a complete “sealing off” of universally Baire sets under stationary tower forcing, the partial version retains most structural and absoluteness features but permits a strict inclusion between certain derived pointclasses. Partial Tower Sealing is consistent below ZFC plus the existence of a Woodin limit of Woodin cardinals, and crucially, it implies the more widely studied sealing/generic absoluteness for universally Baire sets without necessitating large cardinal axioms as strong as a supercompact (Sargsyan et al., 6 Dec 2025).

1. Historical Context: Tower Sealing and Stationary Tower Forcing

Tower Sealing originates in Woodin's analysis of stationary tower forcing, an extension of the classical notion of forcing, constructed from stationary sets in $P_{\omega_1}(\infty)$. With a proper class of Woodin cardinals, one defines the countable stationary tower forcing $\mathbb{Q}_{<\delta}$ for a Woodin cardinal $\delta$. A $V$-generic filter $G \subseteq \mathbb{Q}_{<\delta}$ yields an elementary embedding $j_G: V \to M \subseteq V[G]$. Tower Sealing at $\delta$ posits that generically added sets of reals—specifically, universally Baire sets—are precisely “sealed off” at the image $j_G(\Gamma_g^\infty)$, with an elementary map between the models of $L(\Gamma_{g*G}^\infty)$ and $L(j_G(\Gamma_g^\infty))$ fixing all sets and matching indiscernibles. In classical inner-model theory, such behavior facilitates deep generic absoluteness results for sets of reals (universally Baire sets), often requiring a supercompact cardinal. Hod mice—hybrid premice with iteration strategies and extender sequences—are constructed to recover many of Woodin’s phenomena using only strong and Woodin cardinals.

2. Formal Definitions

Partial Tower Sealing is formalized in the language of stationary tower forcing and pointclass-based inner models:

• Stationary Tower Forcing: For a fixed Woodin cardinal $\delta$, define $\mathbb{Q}_{<\delta}$ as the countable stationary tower forcing of length $\delta$ [Woodin:Stationary Tower].
• Tower Sealing at $\delta$: Given $g \subseteq \mathrm{Coll}(\omega, <\delta)$ generic over $V$, let

$$\Gamma_g^\infty = \{A \subseteq \mathbb{R} \mid A \text{ is universally Baire in } V[g]\}$$

For any further $V[g]$-generic $G \subseteq \mathbb{Q}_{<\delta}$ and the corresponding $j_G$, Tower Sealing requires that both \begin{align*} L(\Gamma_g^\infty) \cap \mathbb{R} &= \Gamma_g^\infty \ L(\Gamma_g^\infty) &\models \mathrm{AD^+} \end{align*} and that there exists an elementary embedding

$$\ell: L(\Gamma_{g*G}^\infty) \to L(j_G(\Gamma_g^\infty))$$

acting as the identity on $\Gamma_{g*G}^\infty$ and preserving indiscernibles.

• Full Tower Sealing: The condition strengthens to $\Gamma_{g*G}^\infty = j_G(\Gamma_g^\infty)$.
• Partial Tower Sealing: All conditions above are retained except only $$\Gamma_{g*G}^\infty \subsetneqq j_G(\Gamma_g^\infty)$$ is required (strict inclusion allowed).
• Sealing: The weakest variant, Shoenfield-type absoluteness, asserts for $<\delta$-generic filters $g, h$, the existence of an elementary embedding

$$j: L(\Gamma_g^\infty, \mathbb{R}_g) \to L(\Gamma_{g*h}^\infty, \mathbb{R}_{g*h})$$

with $j(A) = A^h$ for all $A \in \Gamma_g^\infty$.

3. Main Theorems and Model-Theoretic Statements

Several principal results delineate the implications and boundaries of Partial Tower Sealing:

• Partial Tower Sealing ⟹ Sealing: If $\delta$ is a Woodin limit of Woodin cardinals, and for every $<\delta$-generic $g$ and further $\mathbb{Q}_{<\delta}$-generic $G$ over $V[g]$ one has $\Gamma_{g*G}^\infty = \mathrm{Hom}^*_{g*G}$ (the homogeneously Suslin pointclass in the "derived model" at $g*G$), then Partial Tower Sealing at $\delta$ implies Sealing at $\delta$.
• Partial Tower Sealing in Hod Mice: Let $(\mathcal{M},\Psi)$ be a layered (or lbr) hod-pair with a strong cardinal $\kappa$ and a proper class of Woodin cardinals. If $g \subseteq \mathrm{Coll}(\omega, \kappa^+)$ is generic, Partial Tower Sealing holds at every Woodin cardinal $\delta > \kappa$ in $V[g]$.
• Consistency Strength: Partial Tower Sealing is consistent relative to $$\mathrm{ZFCC}+\text{“there is a Woodin limit of Woodin cardinals”}$$ (denoted $\mathsf{WLW}$). This places its strength strictly below full Tower Sealing, which typically demands much larger cardinal axioms.
• Failure of Full Tower Sealing: In the same generic extension of a hod mouse as above, full Tower Sealing fails at every Woodin cardinal $\delta$. That is, there exist $G, j_G$ and $A \in j_G(\Gamma_g^\infty)$ so that $A \notin \Gamma_{g*G}^\infty$.

4. Proof Techniques and Methodological Framework

The validation of Partial Tower Sealing exploits the derived-model theorem and specific genericity iterations in hod mice:

• Derived Model Representation: The derived-model theorem by Steel (2008) and Sargsyan–Trang (2021) provides a framework to realize $L(\Gamma_{g*G}^\infty, \mathbb{R}_{g*G})$ as the derived model of a direct-limit hod mouse, constructed through genericity iterations and diagonalizations on cofinal enumerations of the relevant pointclasses and reals.
• Sealing Embedding Construction: The central sealing map $\ell$ arises from a dual system of direct-limiting hod mice—one over $\Gamma_{g*G}^\infty$ and reals, and one over $j_G(\Gamma_g^\infty)$. Canonical maps between initial segments yield an isomorphism of the derived models, fixing reals and the pointclass sets pointwise.
• Implication Partial ⟹ Sealing: By combining derived model isomorphisms across two successive stationary tower extensions ($$L(\Gamma^\infty) \to L(\Gamma_g^\infty) \to L(\Gamma_{g*G}^\infty)$$), and exploiting homogeneity and the Shoenfield factorization, one ultimately splices together a generic absoluteness embedding.
• Demonstrating Failure of Full Tower Sealing: The divergence $$\Gamma_{g*G}^\infty \subsetneqq j_G(\Gamma_g^\infty)$$ is established by locating a strategy predicate for a fragment of the hod mouse present in the target but not the sealed-off image. The least-disagreement comparison method against the background hod pair shows the strict non-identity of the two pointclasses.

5. Relations to Hod Mice and Large Cardinals

Partial Tower Sealing demonstrates that advanced generic absoluteness can be obtained in inner models constructed from hod mice possessing strong and Woodin cardinals. Unlike classical full Tower Sealing, whose consistency traditionally demands a supercompact cardinal, the partial version is realized below a Woodin limit of Woodins. This marks an important shift in the hierarchy of model-theoretic consistency strengths required for robust generic absoluteness in the context of sets of reals and pointclass-based constructions.

In this framework, universally Baire sets, derived models, and their associated “sharps” for the relevant pointclasses are analyzed using the stationary tower and direct-limit mice machinery. The existence of a proper class of Woodin cardinals (but not necessarily a supercompact) suffices for most consequences, with failure of full Tower Sealing precisely manifesting the limits of current inner-model technology.

6. Mathematical and Foundational Significance

Partial Tower Sealing occupies a nuanced position in foundational set theory and inner-model theory:

• Consistency Strength: It is strictly weaker than the consistency strength of full Tower Sealing, as it is established below $\mathsf{WLW}$.
• Implications for Universally Baire Sets: The phenomenon implies Sealing, yielding Shoenfield-type generic absoluteness for the theory of universally Baire sets without supercompacts.
• Inner-Model Constructions: Hod mice constructions suffice for most known consequences of Tower Sealing, providing new avenues for inner-model theorists.
• Limitations and Open Problems: The strict failure of full Tower Sealing signals that further large cardinal machinery beyond current hod mouse or extender model methods may be required for the strongest generic absoluteness, possibly necessitating or implying a supercompact cardinal.
• Research Directions: Open questions involve whether full Tower Sealing can be derived from large-cardinal assumptions strictly weaker than a supercompact, or whether its existence necessarily entails a supercompact.

7. Reference Framework

Foundational works underlying Partial Tower Sealing include:

Reference Author(s)	Contribution	Publication Detail
P. Larson	Stationary Tower	AMS Univ. Lecture Series 32
H. Woodin	Suitable Extender Models	J. Math. Logic 10 (2010)
J. Steel	Derived Model Theorem	Logic Colloquium 2006, LNL 32
G. Sargsyan, N. Trang	Sealing from Iterability	Trans. AMS B 8 (2021)
G. Sargsyan, N. Trang	Partial Tower Sealing	(Sargsyan et al., 6 Dec 2025)

These works are central in developing the concepts and technical machinery supporting Tower Sealing, hod mice constructions, generic absoluteness, and the inner-model theory of sets of reals.