Tower Sealing in Set Theory

• Tower Sealing is a principle that ensures precise invariance of universally Baire sets under stationary-tower forcing, defining canonical inner models.
• It marks a critical separation in large-cardinal hierarchies, with Partial Tower Sealing offering a weaker variant that retains key order-preserving properties.
• The study utilizes core model induction and derived model analysis to evaluate forcing extensions, establishing rigorous consistency correlations with large-cardinal axioms.

Tower Sealing is a high-level generic absoluteness principle for universally Baire sets, formulated to capture precise invariance properties of the universally Baire sets and associated canonical inner models under stationary-tower forcing and certain canonical generic embeddings. Its investigation is motivated by the quest for optimal large-cardinal axioms governing the structural robustness of the universally Baire sets, particularly under extensions by stationary-tower and related strong forcings. Tower Sealing and its variants, especially Partial Tower Sealing, mark a critical separation point in the large-cardinal hierarchy beneath the level of supercompactness and have deep ramifications for core model induction, descriptive set theory, and the theory of hod mice.

1. Formal Definitions and Core Principles

Let $\Gamma^\infty$ denote the class of all universally Baire subsets of $\mathbb{R}$, and for any set-generic extension $V[g]$, let $\Gamma^\infty_g$ denote the universally Baire sets in $V[g]$. Let $\delta$ be a Woodin cardinal, and $\mathbb{Q}_{<\delta}$ or $\mathbb{P}_{<\delta}$ denote the countable stationary-tower forcings below $\delta$.

Tower Sealing at $\delta$ asserts the following:

• For all set-generic $g$, $L(\Gamma^\infty_g) \models \mathrm{AD}^+$ and $$\Gamma^\infty_g = \mathbb{R}^{V[g]} \cap L(\Gamma^\infty_g, \mathbb{R}^{V[g]})$$.
• For any $<\delta$-generic $g$ and any $V[g]$-generic $G \subseteq \mathbb{Q}_{<\delta}$, with associated generic ultrapower embedding $j_G : V[g] \to M \subseteq V[g][G]$,

$j_G(\Gamma^\infty_g) = \Gamma^\infty_{g*G}.$

Partial Tower Sealing is a weakening where, instead of demanding exact equality between $j_G(\Gamma^\infty_g)$ and $\Gamma^\infty_{g*G}$, only an order-preserving bijection exists between the definable inner models $L(\Gamma^\infty_{g*G})$ and $L(j_G(\Gamma^\infty_g))$, fixing the $\Gamma^\infty$-sets and preserving classes of indiscernibles. Specifically, for all such $g$, $G$, there is an elementary $$l: L(\Gamma^\infty_{g*G}) \rightarrow L(j_G(\Gamma^\infty_g))$$ with $l \restriction \Gamma^\infty_{g*G} = \mathrm{id}$ and $l$ order-preserving on indiscernibles. If also $\Gamma^\infty_{g*G} = j_G(\Gamma^\infty_g)$, then full Tower Sealing holds (Sargsyan et al., 6 Dec 2025).

The classical Sealing Principle is implied by Partial Tower Sealing. Sealing asserts that after any set-generic extension by $g$, the structure $L(\Gamma^\infty_g, \mathbb{R}^{V[g]})$ is elementary in the analogous structure after further forcing, with the universally Baire sets carried forward canonically.

2. Consistency Strength and Large-Cardinal Context

The exact consistency strength of Tower Sealing, Partial Tower Sealing, and Sealing has been sharply characterized via core model induction techniques and analysis of hod-mice.

• Tower Sealing and Sealing are equiconsistent over the background theory $T$: “ZFC $+$ a proper class of Woodin cardinals $+$ the class of measurable cardinals is stationary” (Sargsyan et al., 2021). In symbols,

$$\operatorname{Con}(T+\text{Sealing}) \;\Longleftrightarrow\; \operatorname{Con}(T+\text{Tower Sealing}).$$

• Partial Tower Sealing is strictly weaker: its consistency is witnessed in generic extensions of hod mice with a single strong cardinal and a proper class of Woodin cardinals, using only the existence of a Woodin limit of Woodin cardinals (Sargsyan et al., 6 Dec 2025).
• Full Tower Sealing is strictly above Partial Tower Sealing in the large-cardinal hierarchy. In particular, attempts to force full Tower Sealing require hypotheses exceeding a Woodin limit of Woodins, plausibly approaching the level of supercompactness.

This delineates the following strict inclusion: $$\operatorname{Con}(\text{ZFC}+\text{Woodin limit of Woodins}) \implies \operatorname{Con}(\text{Partial Tower Sealing}) < \operatorname{Con}(\text{ZFC}+\text{Sealing}) \iff \operatorname{Con}(\text{proper-class Woodins} + \text{supercompact})$$ (Sargsyan et al., 6 Dec 2025).

3. Proof Techniques: Core Model Induction and Derived Models

The verification of Tower Sealing (and its variants) relies on several advanced techniques in inner model theory:

• Core Model Induction (CMI): Used to pass between models of Sealing and models of Tower Sealing and to build canonical fine-structural mice (particularly hybrid and hod mice) with strong and Woodin cardinals. The hybrid fully-backgrounded construction alternates between extender indexing and short-tree strategy addition, culminating at LSA-type hod mouse or traversing the full ordinals (Sargsyan et al., 2021).
• Derived Model Analysis: Canonical derived models, such as $L(\Gamma^\infty_g, \mathbb{R}^{V[g]})$, are constructed via direct limits of hod mice or premice, encoding genericity-iterations. These derived models compute universally Baire sets and support homogeneity and elementary embedding arguments required for (Partial/Full) Tower Sealing (Sargsyan et al., 6 Dec 2025).
• Strategy and Suslin Capturing: Universally Baire sets are shown to be Suslin-co-Suslin, captured by short-tree iteration strategies on countable hulls of $V$; such strategies are generically interpreted across extensions and used as building blocks for the core embeddings.

4. Failure and Strengthening: Full versus Partial Tower Sealing

Full Tower Sealing is shown to fail in generic extensions of “small” hod mice—those with minimal large-cardinal features necessary to support the definitions but insufficient for full invariance. Specifically, one can find small initial segments of these models whose generic ultrapower carries strategies not generated as universally Baire sets in the extension, thereby establishing the failure of $\Gamma^\infty_{g*G} = j_G(\Gamma^\infty_g)$. This phenomenon persists even in minimal lbr-hod pairs with one strong cardinal and a proper class of Woodins (Sargsyan et al., 6 Dec 2025).

A plausible implication is that to achieve Tower Sealing in the sense of full invariance, one requires additional strengths, such as wide-gap extender algebras at a supercompact. Conversely, Partial Tower Sealing can be realized in the lower region of the large-cardinal spectrum and suffices to establish the full Shoenfield-style Sealing, provided certain Hom$^*$-=$\Gamma^\infty$ conditions hold on the universality of homogeneously Suslin sets in the derived model.

5. Equiconsistency, Related Principles, and Hierarchical Placement

Tower Sealing and Sealing are equiconsistent with the theory LSA-over-uB: in all set-generic extensions, there is a model of the Largest Suslin Axiom (LSA) whose Suslin, co-Suslin sets are exactly the universally Baire sets (Sargsyan et al., 2021). Furthermore, the equiconsistency proof reveals:

• No large cardinal strength beyond “proper class Woodins $+$ stationary measurables” is needed for Tower Sealing or Sealing.
• A Woodin limit of Woodins suffices for LSA, which implies (but is not necessary for) Sealing and Tower Sealing.

A summary comparison:

Principle	Consistency Strength	Characterization/Fails in
Full Tower Sealing	> Woodin limit of Woodins, plausibly needs supercompacts	Small hod mice
Partial Tower Sealing	Woodin limit of Woodins suffices	Generic extensions of hod mice as above
Sealing, Tower Sealing	Proper class Woodins + stationary measurables	N/A (they are equiconsistent)

6. Open Questions and Future Directions

The emergent boundaries established by Tower Sealing and its relatives frame central open problems:

• Is full Tower Sealing consistent below supercompactness? It is unknown whether a model of ZFC with only a Woodin limit of Woodins can satisfy full Tower Sealing at every Woodin cardinal, i.e., realize $\Gamma^\infty_{g*G} = j_G(\Gamma^\infty_g)$ universally without invoking supercompacts (Sargsyan et al., 6 Dec 2025).
• Can full Tower Sealing be forced by suitable hod pairs reflecting Woodins? A plausible conjecture is that certain hod pairs with strong cardinals reflecting all Woodins might reconstruct full Tower Sealing within their generic digraphs.
• Structural consequences for inner model theory and descriptive set theory: The presence of Sealing or Tower Sealing marks an obstruction to inner model induction, isolating “third-order” canonical objects and preventing continued core model construction beyond this barrier (Sargsyan et al., 2021, Sargsyan et al., 6 Dec 2025). These principles capture precisely the threshold where new methods or additional large cardinals are required.

7. Significance and Context within Set Theory

Tower Sealing occupies a central role in the structure of generic absoluteness. It sharply delineates, via exact equiconsistency, the region in the large-cardinal hierarchy where universally Baire-based absoluteness is robust under both stationary-tower and related forcings. Its distinction from Partial Tower Sealing illuminates the subtleties of inheriting homogeneity and the persistence of canonical sets under strong generic embeddings. The body of work by Sargsyan and Trang, as well as foundational results by Woodin, forms the backbone for ongoing research into the landscape of hod mice, inner model theory, and the fine structure of large-cardinal-driven absoluteness phenomena (Sargsyan et al., 6 Dec 2025, Sargsyan et al., 2021, Sargsyan et al., 2021).