Is the Ultrapower Axiom consistent with a supercompact cardinal?

The Ultrapower Axiom (UA) asserts that any two countably complete ultrafilters can be compared, i.e., that their ultrapowers can be iterated to a common target; it is a direct generalization of the linearity of the Mitchell order on normal ultrafilters and holds in every canonical inner model produced by current inner-model-theoretic methodology, which cannot produce models in which it fails. Goldberg's 2017 announcement 'Strong Compactness and the Ultrapower Axiom' poses, as a precise test question for the inner model program, whether UA is consistent with the existence of a supercompact cardinal: a positive answer is expected to require constructing the canonical inner model for a supercompact cardinal, while a negative answer would show that no canonical inner model with a supercompact cardinal can exist. Subsequent work resolved the announcement's neighboring sub-question by proving under UA that every strongly compact cardinal is either supercompact or a measurable limit of supercompact cardinals, and derived many structural consequences of UA, but the consistency question itself remains open as of 2026.

Background

A cardinal $\kappa$ is measurable if it carries a $\kappa$-complete nonprincipal ultrafilter (an ultrafilter closed under intersections of fewer than $\kappa$ members). Stronger large cardinal notions are defined through the spaces $P_\kappa(\lambda)=\{\sigma\subseteq\lambda:|\sigma|<\kappa\}$. A cardinal $\kappa$ is strongly compact if for every $\lambda$, $P_\kappa(\lambda)$ carries a fine $\kappa$-complete ultrafilter, where fine means that each $a\in\lambda$ belongs to $\kappa$-almost all $\sigma$; this is equivalent to the compactness theorem for the infinitary logic $L_{\kappa,\omega}$ (first-order logic extended to permit conjunctions and disjunctions over fewer than $\kappa$ formulas). The cardinal $\kappa$ is supercompact if for every $\lambda$ the ultrafilter can moreover be taken normal, meaning any function $f$ with $f(\sigma)\in\sigma$ almost everywhere is constant almost everywhere. Every supercompact cardinal is strongly compact, and a longstanding conjecture of Solovay (recounted in Goldberg's papers) asserted the converse that every strongly compact cardinal is supercompact; Menas refuted this in 1975 by showing that every measurable limit of strongly compact cardinals is strongly compact but never supercompact. Whether strongly compact and supercompact cardinals are equiconsistent, in the sense that the consistency of either hypothesis implies the consistency of the other, is a prominent open question equivalent in difficulty to the inner model problem for supercompact cardinals, one of the central open problems of modern set theory: constructing a canonical, iterable inner model with a supercompact cardinal.

The Ultrapower Axiom is an abstract formulation of the comparison process that underlies inner model theory. If $U$ is a countably complete ultrafilter (closed under countable intersections, the completeness needed for the ultrapower to be well-founded), the ultrapower construction yields an elementary embedding $j_U:V\to M_U$ of the universe into a transitive inner model. The Mitchell order on normal ultrafilters is defined by $U\triangleleft W$ if and only if $U\in M_W$; its linearity is a basic feature of canonical inner models. The Ultrapower Axiom (UA) generalizes this: for any countably complete ultrafilters $U_0$ and $U_1$, there exist countably complete ultrafilters $W_0\in M_{U_0}$ and $W_1\in M_{U_1}$ such that the iterated ultrapowers agree, $M^{M_{U_0}}_{W_0}=M^{M_{U_1}}_{W_1}$, and the corresponding embeddings commute, $j^{M_{U_0}}_{W_0}\circ j_{U_0}=j^{M_{U_1}}_{W_1}\circ j_{U_1}$; briefly, any two ultrapowers can be compared to a common target. UA follows from Woodin's Weak Comparison principle, itself an immediate consequence of the basic comparison lemma used to construct and analyze canonical inner models, and the methodology of inner model theory cannot produce models in which UA fails, so every extender model (an inner model built from extenders, systems of ultrafilters that code elementary embeddings) constructed by current methods satisfies it. Inner model constructions conditional on iterability (the requirement that the comparison process does not break down) have reached only the finite levels of supercompactness, and these models satisfy UA.

Goldberg's announcement 'Strong Compactness and the Ultrapower Axiom' (arXiv:1710.03586, 2017) derived striking consequences of UA in the presence of strong compactness: the generalized continuum hypothesis (GCH, the assertion $2^\lambda=\lambda^+$) holds above the least strongly compact cardinal; if there is a strongly compact cardinal then the universe is a generic extension of HOD, the inner model of hereditarily ordinal-definable sets; and, most notably, the least strongly compact cardinal is supercompact, a canonical form of the equiconsistency of strong compactness and supercompactness. The announcement then poses Question 1, 'Is UA consistent with a supercompact cardinal?', calling it a precise test question for the inner model program: since the inner-model methodology cannot produce models failing UA, a refutation of UA by a supercompact cardinal would suggest that no canonical model with a supercompact cardinal exists, while the coherence of the UA theory is evidence that such a model does exist. The announcement also poses the neighboring question whether the second strongly compact cardinal must be supercompact, and Conjecture 9: assuming UA, the Ground Axiom (the assertion that the universe is not a nontrivial set-forcing extension of any inner model), and a supercompact cardinal, the universe equals Ultimate $L$, Woodin's hypothesized canonical inner model accommodating all large cardinals (Woodin, Bulletin of Symbolic Logic, 2017).

Subsequent work resolved the neighboring sub-question and sharpened the surviving core. Goldberg's companion paper (arXiv:1810.05058) and memoir (Contemporary Mathematics 752, 2020; De Gruyter book edition, 2022) prove that under UA every strongly compact cardinal is either supercompact or a measurable limit of supercompact cardinals, so that the only counterexamples to Solovay's conjecture are Menas's; by Menas's theorem this dichotomy is best possible. The memoir restates the consistency core as Conjecture 2.1.8, that UA is consistent even with an extendible cardinal (a cardinal $\delta$ such that for every $\lambda$ there is an elementary embedding $j:V_{\lambda+1}\to V_{j(\lambda)+1}$ with critical point $\delta$, i.e., least ordinal moved by $j$, satisfying $j(\delta)>\lambda$), and as the UA Conjecture, that ZFC proves the existence of an inner model of UA with the $\delta$-covering and $\delta$-approximation properties (two closeness conditions on an inner model from Woodin's Ultimate-$L$ program, requiring that the model cover every subset of $\delta$ by one of its own small sets and that any subset of $\delta$ whose small fragments all lie in the model already lies in the model) whenever $\delta$ is extendible, expressing the expectation that neither will be proved without first solving the inner model problem. The memoir also introduces the Complete Ultrapower Axiom (CUA), requiring that for every uncountable $\kappa$ any two ultrapower embeddings with critical point at least $\kappa$ admit a comparison whose component embeddings also have critical point at least $\kappa$, and asks whether even CUA is consistent with two cardinals $\kappa<\lambda$ that are both $\lambda^+$-supercompact, since the direct generalization of the least-strongly-compact analysis to higher strongly compact cardinals requires principles beyond UA. Recent work (2023-2026) treats UA as a working hypothesis and its compatibility with supercompactness as unsettled, applying consequences of UA to the Galvin property (a covering-type property of normal fine ultrafilters), to Magidor iterations (a forcing construction iterating Prikry-type forcings, which change cofinalities to $\omega$), and to the number of normal measures on large measurable cardinals.

Problem Statement

Is the Ultrapower Axiom consistent with the existence of a supercompact cardinal? Precisely: is there a model of ZFC satisfying both (i) the Ultrapower Axiom, i.e., for any two countably complete ultrafilters $U_0$ and $U_1$ there exist countably complete ultrafilters $W_0\in M_{U_0}$ and $W_1\in M_{U_1}$ with $M^{M_{U_0}}_{W_0}=M^{M_{U_1}}_{W_1}$ and $j^{M_{U_0}}_{W_0}\circ j_{U_0}=j^{M_{U_1}}_{W_1}\circ j_{U_1}$, and (ii) the assertion that some cardinal $\kappa$ is supercompact, i.e., $P_\kappa(\lambda)$ carries a normal fine $\kappa$-complete ultrafilter for every cardinal $\lambda\geq\kappa$? In the standard relative-consistency reading: assuming ZFC plus 'there is a supercompact cardinal' is consistent, is ZFC plus UA plus 'there is a supercompact cardinal' consistent? The question is Goldberg's Question 1 from 'Strong Compactness and the Ultrapower Axiom' (arXiv:1710.03586).

References

1. Gabriel Goldberg, "Strong Compactness and the Ultrapower Axiom," arXiv:1710.03586 (2017). https://arxiv.org/abs/1710.03586
2. Gabriel Goldberg, "The Ultrapower Axiom and the equivalence between strong compactness and supercompactness," arXiv:1810.05058 (2018). https://arxiv.org/abs/1810.05058
3. Gabriel Goldberg, "Strong compactness and the ultrapower axiom I: the least strongly compact cardinal," Journal of Mathematical Logic 22 (2022), DOI 10.1142/s0219061322500052. https://doi.org/10.1142/s0219061322500052
4. Gabriel Goldberg, "The Ultrapower Axiom," Contemporary Mathematics 752, American Mathematical Society (2020), DOI 10.1090/conm/752/15131; book edition, De Gruyter (2022), DOI 10.1515/9783110719734; arXiv:2006.03293. https://arxiv.org/abs/2006.03293
5. Gabriel Goldberg, "Compactness and Comparison," arXiv:1801.09153 (2018). https://arxiv.org/abs/1801.09153
6. Gabriel Goldberg, "Strongly compact cardinals and ordinal definability," Journal of Mathematical Logic 23 (2023), DOI 10.1142/s0219061322500106; arXiv:2107.00513. https://arxiv.org/abs/2107.00513
7. Gabriel Goldberg, "Some combinatorial properties of Ultimate L and V," arXiv:2007.04812 (2020). https://arxiv.org/abs/2007.04812
8. Arthur W. Apter, "Forcing level by level equivalence and a consequence of UA," Bollettino dell'Unione Matematica Italiana 18(2):431-444 (2024), DOI 10.1007/s40574-024-00419-6. https://doi.org/10.1007/s40574-024-00419-6
9. Tom Benhamou and Gabriel Goldberg, "Applications of the Magidor iteration to ultrafilter theory," Forum of Mathematics, Sigma 14 (2026) e61, DOI 10.1017/fms.2026.10211; arXiv:2412.09683. https://arxiv.org/abs/2412.09683
10. Tom Benhamou and Ben-Zion Weltsch, "Supercompact Measures and the Galvin Property," Journal of Symbolic Logic (2026), DOI 10.1017/jsl.2026.10216; arXiv:2510.07669. https://arxiv.org/abs/2510.07669
11. Arthur W. Apter, Eyal Kaplan, and Alejandro Poveda, "The number of measures on very large measurable cardinals," arXiv:2603.11710 (2026). https://arxiv.org/abs/2603.11710
12. Gabriel Goldberg, Grigor Sargsyan, and Benjamin Siskind, "Ultrapowers of determinacy models as iteration trees on HOD," arXiv:2603.20951 (2026). https://arxiv.org/abs/2603.20951
13. W. Hugh Woodin, "In search of Ultimate-L: the 19th Midrasha Mathematicae Lectures," The Bulletin of Symbolic Logic 23(1):1-109 (2017), DOI 10.1017/bsl.2016.34. https://doi.org/10.1017/bsl.2016.34
14. Telis K. Menas, "On strong compactness and supercompactness," Annals of Mathematical Logic 7(4):327-359 (1975), DOI 10.1016/0003-4843(75)90009-1. https://doi.org/10.1016/0003-4843(75)90009-1