Christoph Weiß's paper 'The combinatorial essence of supercompactness' introduced the two-cardinal combinatorial principles TP, SP, ITP and ISP, which characterize strong compactness and supercompactness at inaccessible cardinals yet remain meaningful at successors. Its Conclusion (Section 6) states a single motivating open problem: 'the quest for an inner model for a supercompact cardinal' — construct a canonical (fine-structural) inner model of ZFC containing a supercompact cardinal. Two related calibration questions are suggested by the paper but are not posed in it as open problems: the exact consistency strength of the principles ITP(κ,λ) and ISP(κ,λ) at successor cardinals (the abstract calls the forcing upper bounds — an ineffable cardinal for ISP(ω₂,ω₂); a supercompact cardinal for ISP(ω₂,λ) for all λ — 'supposedly optimal', and the inner-model lower bounds — an inner model with a proper class of strong and Woodin cardinals, via failure of weak square — 'the best currently known'), and the exact consistency strength of the Proper Forcing Axiom (PFA), which implies ISP(ω₂) (the Conclusion describes the Viale–Weiß iteration theorems as 'strong heuristic evidence on the lower bound of the consistency strength of PFA'). As of August 2026 the inner model problem is open: the inner-model frontier has reached fine structure for mice at the level of κ⁺-supercompactness and no canonical inner model with a supercompact cardinal has been constructed; the best general lower bound for the combinatorial principles is unchanged since 2009, and PFA's strength remains bracketed between the consistency of AD_R with regular Θ and a supercompact cardinal.

Background

A cardinal κ\kappa is supercompact if for every λ≥κ\lambda \geq \kappa there is an elementary embedding j:V→Mj: V \to M into a transitive class MM with critical point κ\kappa such that j(κ)>λj(\kappa) > \lambda and MM is closed under λ\lambda-sequences; κ\kappa is strongly compact if every κ\kappa-complete filter on any set extends to a κ\kappa-complete ultrafilter. Both notions far exceed measurability, and both admit purely combinatorial reformulations. Write Pκλ={a⊆λ:∣a∣<κ}P_\kappa\lambda = \{a \subseteq \lambda : |a| < \kappa\}. A PκλP_\kappa\lambda-list is a sequence D=⟨da∣a∈Pκλ⟩D = \langle d_a \mid a \in P_\kappa\lambda\rangle with da⊆ad_a \subseteq a. A set d⊆λd \subseteq \lambda is an ineffable branch of DD if d∩a=dad \cap a = d_a for all aa in some stationary subset of PκλP_\kappa\lambda, and a cofinal branch if every a∈Pκλa \in P_\kappa\lambda is contained in some zaz_a with d∩a=dza∩ad \cap a = d_{z_a} \cap a. The list is thin if there is a club C⊆PκλC \subseteq P_\kappa\lambda such that for every c∈Cc \in C the set of traces at cc, {da∩c∣c⊆a∈Pκλ}\{d_a \cap c \mid c \subseteq a \in P_\kappa\lambda\}, has size <κ< \kappa (automatic when κ\kappa is inaccessible, since then 2∣c∣<κ2^{|c|} < \kappa for every c∈Pκλc \in P_\kappa\lambda), and slender if for all sufficiently large θ\theta (where HθH_\theta is the collection of sets whose transitive closure has size <θ< \theta) there is a club C⊆PκHθC \subseteq P_\kappa H_\theta with dM∩λ∩b∈Md_{M \cap \lambda} \cap b \in M for all M∈CM \in C and b∈M∩Pω1λb \in M \cap P_{\omega_1}\lambda. Weiß defined: TP(κ,λ), every thin list has a cofinal branch; SP(κ,λ), every slender list has a cofinal branch; ITP(κ,λ) (the ineffable tree property), every thin list has an ineffable branch; ISP(κ,λ), every slender list has an ineffable branch. At inaccessible κ\kappa these recover the large cardinal hierarchy: κ\kappa is strongly compact iff TP(κ,λ) holds for all λ≥κ\lambda \geq \kappa (a reformulation of a classical characterization via cofinal branches), and κ\kappa is supercompact iff ITP(κ,λ) holds for all λ≥κ\lambda \geq \kappa (Magidor's combinatorial characterization of supercompactness). The point of the definitions is that, unlike the original characterizations, TP/SP/ITP/ISP remain non-vacuous at successor cardinals, so one can ask how much of the combinatorial content of strong compactness and supercompactness can be forced down to small cardinals such as ω2\omega_2. Weiß proved two-sided estimates. Upper bounds by Mitchell-style forcing iterations with the ω1\omega_1-approximation and covering properties (a forcing has the μ\mu-covering property if every set of ordinals of size <μ< \mu in the extension is contained in a ground-model set of size <μ< \mu, and the μ\mu-approximation property if a set of ordinals of the extension is in the ground model whenever its intersections with all μ\mu-small ground-model sets are): if an ineffable cardinal exists then ISP(ω₂,ω₂) is consistent, and if a supercompact cardinal exists then ISP(ω₂,λ) is consistent for every λ≥ω2\lambda \geq \omega_2. Lower bounds by anti-square arguments: when cf(λ)≥κ\mathrm{cf}(\lambda) \geq \kappa, ITP(κ,λ) implies the failure of the weak two-cardinal square principle □cof(<κ)(κ,λ)\Box_{\mathrm{cof}(<\kappa)}(\kappa,\lambda), which asserts a coherent, unthreadable system ⟨Cγ∣γ<λ, cf(γ)<κ⟩\langle \mathcal{C}_\gamma \mid \gamma < \lambda, \ \mathrm{cf}(\gamma) < \kappa \rangle: each Cγ\mathcal{C}_\gamma is a family of fewer than κ\kappa many club subsets of γ\gamma, the system coheres in the sense that C∩β∈CβC \cap \beta \in \mathcal{C}_\beta whenever β\beta is a limit point of C∈CγC \in \mathcal{C}_\gamma, and there is no thread, i.e. no club D⊆λD \subseteq \lambda with D∩δ∈CδD \cap \delta \in \mathcal{C}_\delta for all limit points δ\delta of DD of cofinality <κ< \kappa. The failure of square at a regular λ≥ω3\lambda \geq \omega_3 with ηω<λ\eta^\omega < \lambda for all η<λ\eta < \lambda yields, by the Jensen–Schimmerling–Schindler–Steel 'stacking mice' theorem, an inner model — indeed a sharp — with a proper class of strong cardinals and a proper class of Woodin cardinals. Weiß also showed ITP(κ,κ) implies κ\kappa is ineffable in LL (Gödel's constructible universe), so even the smallest instance carries real strength. The Proper Forcing Axiom (PFA) asserts that for every proper forcing notion P\mathbb{P} (one preserving stationary subsets of [λ]ω[\lambda]^{\omega} for every uncountable cardinal λ\lambda; in particular, proper forcings do not collapse ω1\omega_1) and every family of ℵ1\aleph_1 dense subsets of P\mathbb{P} there is a filter meeting them all; it is forced by collapsing a supercompact cardinal to ω2\omega_2 and is the central forcing axiom of the hierarchy. Viale and Weiß proved that PFA implies ISP(ω₂) — i.e. ISP(ω₂,λ) for all λ≥ω2\lambda \geq \omega_2 — and that any standard proper iteration forcing PFA must start from a universe where the collapsed cardinal is supercompact, heuristic evidence that a supercompact is the true consistency strength of PFA. An inner model of ZFC is a transitive proper class containing all ordinals and satisfying ZFC; the canonical ('fine-structural') ones are the mice — models built level by level from extenders (fragments of elementary embeddings coding large cardinal structure) with a definability structure that supports the comparison method, in which two such models are iterated to a common model without conflict, yielding canonicity and covering theorems. The inner model program has produced canonical inner models with Woodin cardinals and, at its current frontier, a fine-structural theory of mice with long extenders reaching only the level of κ+\kappa^+-supercompactness; the construction of a canonical inner model with a fully supercompact cardinal — Weiß's stated main motivation — remains unsolved, and Woodin has argued (Suitable Extender Models, Ultimate-L) that its solution would yield an 'Ultimate-L' but that current approaches must be fundamentally altered to achieve it.

Problem Statement

The problem is the one stated in the Conclusion (Section 6) of Weiß's 'The combinatorial essence of supercompactness' (2012), which the paper names as 'the main motivation behind the principles we considered'.

Construct an inner model for a supercompact cardinal: build a canonical, fine-structural inner model of ZFC (a mouse / suitable extender model in the sense of the inner model program, i.e. a model built from extenders whose iterates compare without conflict) that contains a supercompact cardinal; equivalently, show that the existence of a supercompact cardinal is witnessed by a canonical inner model, in the sense in which canonical inner models are known for Woodin cardinals.

Related calibration questions — suggested by the source's hedges and remarks, but not stated by the source as open problems and not required parts of a solution of the problem above: (a) determine the exact consistency strength of the principles ITP(κ,λ) and ISP(κ,λ) when κ is a successor cardinal — closing the gap between the forcing upper bounds (an ineffable cardinal for ISP(ω₂,ω₂), a supercompact cardinal for ISP(ω₂,λ) for all λ ≥ ω₂; the source notes that ω₂ serves only as the minimal instance and that one can take successors of larger regular cardinals instead) and the inner-model lower bounds obtained via the failure of weak square and the Jensen–Schimmerling–Schindler–Steel stacking-mice theorem (an inner model with a proper class of strong cardinals and a proper class of Woodin cardinals) — including whether the upper-bound hypotheses are optimal (the source proves they are optimal only relative to its Mitchell-style forcing method, via pullback under the approximation and covering properties); and (b) determine the exact consistency strength of the Proper Forcing Axiom in terms of these two-cardinal ineffability principles — PFA implies ISP(ω₂), every standard proper iteration forcing PFA starts from a supercompact cardinal, and the best current lower bound is the consistency of ZF + AD_R + 'Θ is regular'. Is a supercompact cardinal (or any specific large-cardinal hypothesis) equiconsistent with PFA? Progress on these calibration questions advances the surrounding program; the source's stated problem is the inner model construction above.

References

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