Is the Ultrapower Axiom consistent with an extendible cardinal? (Goldberg's Conjecture 2.1.8)

The Ultrapower Axiom (UA) of Goldberg is the abstract combinatorial trace of the Comparison Lemma of inner model theory: any two ultrapower embeddings of the universe of sets admit a comparison by internal ultrapowers. UA holds in every known canonical inner model, and under a supercompact cardinal it yields a coherent structural theory (linear Mitchell order, GCH above the least supercompact, the least strongly compact cardinal being supercompact, V as an HOD-generic extension), expected to hold in the still-unbuilt canonical inner models for supercompact cardinals. Goldberg's Conjecture 2.1.8 — posed as a precise test question for the inner model problem for supercompact cardinals — asks whether UA is consistent with an extendible cardinal. No model of UA is known to contain even a single supercompact cardinal, and no conflict between UA and extendibility is known; the theorems proved about UA under a supercompact cardinal are conditional on the hypothesis that UA is consistent with supercompact cardinals. The problem: prove or refute that ZFC + UA + 'there exists an extendible cardinal' is consistent.

Background

An ultrafilter $U$ on a set $X$ is countably complete if it is closed under countable intersections; these are exactly the ultrafilters whose ultrapowers of the universe are wellfounded. To each countably complete ultrafilter $U$ is associated the ultrapower embedding $j_U : V \to M_U$, the canonical elementary embedding of the set-theoretic universe $V$ into the transitive model $M_U = \mathrm{Ult}(V,U)$. An elementary embedding $k : M \to N$ between transitive models of ZFC is an internal ultrapower embedding if $N$ is the ultrapower of $M$ by an ultrafilter that is an element of $M$ — informally, an ultrapower that $M$ can carry out on its own.

Large cardinal axioms are calibrated by such embeddings; the critical point of an elementary embedding is the least ordinal it moves. A cardinal $\kappa$ is $\lambda$-supercompact if some embedding $j : V \to M$ with critical point $\kappa$ satisfies $j(\kappa) > \lambda$ and $M^\lambda \subseteq M$ (that is, $M$ is closed under $\lambda$-sequences); $\kappa$ is supercompact if it is $\lambda$-supercompact for every $\lambda$. A cardinal $\kappa$ is extendible if for every $\alpha > \kappa$ there is an elementary embedding $j : V_\alpha \to V_\beta$ with critical point $\kappa$, where $V_\alpha$ is the $\alpha$-th stage of the cumulative hierarchy of sets; extendibility is far stronger than supercompactness.

The Ultrapower Axiom (UA) states: every pair of ultrapower embeddings $j_0 : V \to M_0$ and $j_1 : V \to M_1$ of the universe admits a comparison — internal ultrapower embeddings $k_0 : M_0 \to N$ and $k_1 : M_1 \to N$ into a common transitive model $N$ with $k_0 \circ j_0 = k_1 \circ j_1$. Here an ultrapower embedding of the universe is an elementary embedding $j : V \to M$ into a transitive class $M$ that is generated over $j[V]$ by a single element, equivalently an embedding isomorphic to $j_U : V \to \mathrm{Ult}(V,U)$ for some countably complete ultrafilter $U$ on a set. Goldberg introduced UA in his dissertation The Ultrapower Axiom (2020; book version 2022) by abstracting the Comparison Lemma, the fundamental tool of inner model theory — the study of canonical, L-like inner models of ZFC (mice): the Comparison Lemma allows any two such models to be compared by iterating them against each other, and its combinatorial essence is precisely UA. Consequently UA holds in all known canonical inner models, such as the extender models $L[E]$, the canonical inner models built from sequences of extenders (fine-structural codes for elementary embeddings), which accommodate large cardinals up to the current frontier of the theory, and it is expected to hold in any canonical inner model that will ever be built.

Two classical open problems frame the conjecture. First, the inner model problem for supercompact cardinals: construct a canonical inner model of ZFC containing a supercompact cardinal. No such model is known; current constructions reach, even conditionally on iterability hypotheses, only the level of superstrong cardinals (a superstrong cardinal is a cardinal $\kappa$ carrying an elementary embedding $j : V \to M$ with critical point $\kappa$ and $V_{j(\kappa)} \subseteq M$). Second, the questions of Solovay–Reinhardt–Kanamori on the multiplicity of normal ultrafilters, for instance whether a $2^\kappa$-supercompact cardinal must carry more than one normal ultrafilter concentrating on nonmeasurable cardinals; a measurable cardinal is one carrying a nonprincipal $\kappa$-complete ultrafilter, and a normal ultrafilter on $\kappa$ is a $\kappa$-complete ultrafilter closed under diagonal intersections, equivalently one derived from an elementary embedding with critical point $\kappa$.

UA organizes this landscape. UA implies the Mitchell order on normal ultrafilters is linear (the Mitchell order is defined by $U \triangleleft W$ iff $U \in M_W$, and its linearity is characteristic of canonical inner models); UA is equivalent to the linearity of the Ketonen order, a wellfounded order on all countably complete ultrafilters on ordinals; and, assuming a supercompact cardinal, UA implies the Generalized Continuum Hypothesis above the least supercompact cardinal, that the least strongly compact cardinal is supercompact, and that $V$ is a generic extension of HOD, the inner model of hereditarily ordinal-definable sets. In place of building models, the dissertation develops this axiom-driven theory as evidence that the unbuilt canonical inner models for supercompact cardinals are coherent — evidence, in Goldberg's words, for Conjecture 2.1.8.

Conjecture 2.1.8 isolates the precise consistency question: the Ultrapower Axiom is consistent with an extendible cardinal. Goldberg expects that neither it nor the weaker Conjecture 2.1.4 (consistency of a linear Mitchell order with an extendible cardinal) will be proved without first solving the inner model problem. A negative answer — a ZFC proof that an extendible cardinal refutes UA — would show that no canonical inner model with an extendible cardinal can exist, since the Comparison Lemma implies UA. A stronger related form, the UA Conjecture (Conjecture 8.4.32 of the dissertation), asserts that ZFC proves that if $\delta$ is extendible then there is an inner model of UA with the $\delta$-cover and $\delta$-approximation properties (Woodin's conditions on how closely an inner model must cover and approximate $V$); Woodin's Ultimate-L Conjecture (2017) would imply it. Goldberg's follow-up paper The uniqueness of elementary embeddings (2024) proves that the Weak Ultrapower Axiom — the same comparison statement, except that the two embeddings need only have a common internal ultrapower target, without the commutativity requirement $k_0 \circ j_0 = k_1 \circ j_1$ — under an extendible cardinal already yields most of the theory of UA, which makes the gap between Weak UA and full UA in an extendible universe the exact locus of the open question.

Problem Statement

Prove or refute Goldberg's Conjecture 2.1.8: the Ultrapower Axiom is consistent with an extendible cardinal.

Concretely, determine whether the theory ZFC + UA + 'there exists an extendible cardinal' is consistent, where ZFC is Zermelo–Fraenkel set theory with the Axiom of Choice and $V$ is the universe of all sets. UA is the statement that every pair of ultrapower embeddings of the universe admits a comparison by internal ultrapower embeddings: for any two ultrapower embeddings $j_0 : V \to M_0$ and $j_1 : V \to M_1$, there are internal ultrapower embeddings $k_0 : M_0 \to N$ and $k_1 : M_1 \to N$ into a common transitive model $N$ with $k_0 \circ j_0 = k_1 \circ j_1$. Here an ultrapower embedding of the universe is an elementary embedding $j : V \to M$ into a transitive class $M$ that is generated over $j[V]$ by a single element, equivalently an embedding isomorphic to $j_U : V \to \mathrm{Ult}(V, U)$ for some countably complete ultrafilter $U$ on a set; and an internal ultrapower embedding of a transitive model $M$ of ZFC is an elementary embedding whose target is the ultrapower of $M$ by a countably complete ultrafilter that is an element of $M$. A cardinal $\kappa$ is extendible if for every $\alpha > \kappa$ there is an elementary embedding $j : V_\alpha \to V_\beta$ with critical point $\kappa$ (the least ordinal moved by $j$), where $V_\alpha$ denotes the $\alpha$-th stage of the cumulative hierarchy of sets.

References

1. Gabriel Goldberg, The Ultrapower Axiom, Ph.D. dissertation, Harvard University (2020), arXiv:2006.03293, https://arxiv.org/abs/2006.03293; book version: The Ultrapower Axiom, De Gruyter Series in Logic and its Applications vol. 10, De Gruyter (2022), DOI 10.1515/9783110719734, https://doi.org/10.1515/9783110719734 (source of Conjecture 2.1.8)
2. Gabriel Goldberg, The uniqueness of elementary embeddings, Journal of Symbolic Logic (2024), pages 1-25, DOI 10.1017/jsl.2024.60, https://doi.org/10.1017/jsl.2024.60; preprint arXiv:2103.13961, https://arxiv.org/abs/2103.13961
3. Tom Benhamou, Gabriel Goldberg, Applications of the Magidor iteration to ultrafilter theory, Forum of Mathematics, Sigma 14 (2026), DOI 10.1017/fms.2026.10211, https://doi.org/10.1017/fms.2026.10211; preprint arXiv:2412.09683, https://arxiv.org/abs/2412.09683
4. Eyal Kaplan, A note on the Ketonen order and Lipschitz reducibility between ultrafilters, Annals of Pure and Applied Logic 177, article 103805 (2026), DOI 10.1016/j.apal.2026.103805, https://doi.org/10.1016/j.apal.2026.103805; preprint arXiv:2512.12835, https://arxiv.org/abs/2512.12835
5. Arthur W. Apter, Forcing level by level equivalence and a consequence of UA, Bollettino dell'Unione Matematica Italiana 18 (2024), pages 431-444, DOI 10.1007/s40574-024-00419-6, https://doi.org/10.1007/s40574-024-00419-6
6. Tom Benhamou, Gabriel Goldberg, The Galvin property under the ultrapower axiom, Canadian Journal of Mathematics 77 (2024), pages 1686-1717, DOI 10.4153/S0008414X2400052X, https://doi.org/10.4153/S0008414X2400052X; preprint arXiv:2306.15078, https://arxiv.org/abs/2306.15078
7. Arthur W. Apter, Eyal Kaplan, Alejandro Poveda, The number of measures on very large measurable cardinals, arXiv:2603.11710 (2026), https://arxiv.org/abs/2603.11710
8. Tom Benhamou, Saturation properties of ultrafilters in canonical inner models, Journal of Mathematical Logic 26(2), article 2550005 (published online 2025; issue dated 2026), DOI 10.1142/S0219061325500059, https://doi.org/10.1142/S0219061325500059; preprint arXiv:2212.14096 (2022), https://arxiv.org/abs/2212.14096
9. Gabriel Goldberg, Some combinatorial properties of Ultimate L and V, arXiv:2007.04812 (2020), https://arxiv.org/abs/2007.04812
10. W. Hugh Woodin, In search of Ultimate-L: the 19th Midrasha Mathematicae Lectures, Bulletin of Symbolic Logic 23(1) (2017), 1-109, DOI 10.1017/bsl.2016.34, https://doi.org/10.1017/bsl.2016.34
11. Gabriel Goldberg, Grigor Sargsyan, Benjamin Siskind, Ultrapowers of determinacy models as iteration trees on HOD, arXiv:2603.20951 (2026), https://arxiv.org/abs/2603.20951