Woodin's Ultimate-$L$ Conjecture: existence of a weakly $\Sigma_2$-definable weak extender model for supercompactness satisfying $V=\text{Ultimate-}L$

The Ultimate-$L$ Conjecture of W. Hugh Woodin predicts that if $\delta$ is an extendible cardinal, then there is a weak extender model $N$ for the supercompactness of $\delta$ — a transitive inner model of ZFC that inherits supercompactness measures on $\mathcal{P}_\delta(\lambda)$ in the sense that for every $\lambda>\delta$ some normal fine $\delta$-complete measure on $\mathcal{P}_\delta(\lambda)$ contains $N\cap\mathcal{P}_\delta(\lambda)$ and satisfies $U\cap N\in N$ — which is weakly $\Sigma_2$-definable, contained in HOD, and satisfies the axiom $V=\text{Ultimate-}L$ (a proper class of Woodin cardinals plus $\Sigma_2$-reflection to the HODs of determinacy models). An affirmative answer would produce the first canonical fine-structural inner model at the level of a supercompact cardinal — the central goal of the inner model program since the 1960s — and establish $V=\text{Ultimate-}L$, which implies the Continuum Hypothesis, as a candidate axiom compatible with all large cardinals. The conjecture is open: the surrounding results (the consistency strength of Sealing, the HOD Dichotomy and Ultrapower Axiom work, and the exacting-cardinal results of references [15] and [16]) are conditional or adjacent and do not settle it. The refutation of Woodin's HOD and Ultimate-L conjectures in reference [15] is conditional on the consistency of ZFC with an exacting cardinal above an extendible cardinal — an assumption that is unproven and known only to follow from the consistency of certain large cardinals beyond choice — so the conjecture is not known to fail.

Background

The inner model program is the project, begun with Gödel's constructible universe $L$ (the inner model consisting of all sets definable from ordinal parameters), of building canonical transitive inner models — proper classes $M\subseteq V$ containing all ordinals and satisfying ZFC — that can host ever stronger large cardinals. A cardinal $\delta$ is supercompact if for every $\lambda>\delta$ there is an elementary embedding $j:V\to M$ with critical point $\delta$, $j(\delta)>\lambda$, and $M$ closed under $\lambda$-sequences; $\delta$ is extendible (Definition 3.11 of the source lectures) if for every $\lambda>\delta$ there is an elementary embedding $j:V_{\lambda+1}\to V_{j(\lambda)+1}$ with critical point $\delta$ and $j(\delta)>\lambda$. Successive generations of canonical inner models — the mice and extender models of modern fine structure theory, where an extender is a combinatorial fragment of an elementary embedding coding large-cardinal strength and a mouse is a countable transitive set built level-by-level from a sequence of extenders — have reached Woodin cardinals, but no canonical inner model with a supercompact cardinal has been constructed, and this is the outstanding gap of the program.

The lectures — W. Hugh Woodin's "In Search of Ultimate-L: The 19th Midrasha Mathematicae Lectures" (Bulletin of Symbolic Logic, 2017), the primary source of this problem — introduce the key auxiliary notion. A transitive class $N\models\mathrm{ZFC}$ is a weak extender model for "$\delta$ is supercompact" (Definition 3.5) if for every $\lambda>\delta$ there is a $\delta$-complete normal fine measure $U$ on $\mathcal{P}_\delta(\lambda)$ (the set of $\delta$-sized subsets of $\lambda$) such that $N\cap\mathcal{P}_\delta(\lambda)\in U$ and $U\cap N\in N$: $N$ correctly computes enough of the supercompactness structure of $\delta$ to inherit the $\delta$-covering property. Weak extender models need not be fine-structural, so the notion isolates what any canonical model at the supercompact level must achieve without presupposing the construction.

The definability clause of the conjecture is likewise defined in the lectures (Definitions 4.34/4.35). A sequence $\langle N_\alpha : \alpha\in\mathrm{Ord}\rangle$ is weakly $\Sigma_2$-definable if there is a formula $\varphi(x)$ such that (1) for all $\gamma<\delta_1<\delta_2<\delta_3$, if the initial segments up to $\gamma$ of $(N_\varphi)^{V_{\delta_1}}$ and $(N_\varphi)^{V_{\delta_3}}$ agree, then the initial segment up to $\gamma$ of $(N_\varphi)^{V_{\delta_2}}$ agrees with them as well, and (2) for every $\alpha\in\mathrm{Ord}$, $N_\alpha$ equals the initial segment up to $\alpha$ of $(N_\varphi)^{V_\delta}$ for all sufficiently large $\delta$, where for any transitive set $W$, $(N_\varphi)^W=\{a\in W : W\models\varphi[a]\}$. A transitive inner model $N\models\mathrm{ZFC}$ is weakly $\Sigma_2$-definable if the sequence $\langle N\cap V_\alpha : \alpha\in\mathrm{Ord}\rangle$ is weakly $\Sigma_2$-definable in this sense. (Every $\Sigma_2$-definable inner model, and in particular HOD, is weakly $\Sigma_2$-definable.)

Two further ingredients frame the conjecture. HOD is the class of hereditarily ordinal-definable sets, the largest canonical inner model definable without parameters; the HOD Dichotomy Theorem (from an extendible cardinal $\delta$) says that either every regular cardinal $\kappa\geq\delta$ is $\omega$-strongly measurable in HOD (a HOD-internal strengthening of measurability, Definition 3.35), or no regular cardinal $\kappa\geq\delta$ is $\omega$-strongly measurable in HOD — and in the second case HOD is itself a weak extender model for the supercompactness of $\delta$, so that HOD is "close" to $V$. Second, the proposed axiom $V=\text{Ultimate-}L$ (Definition 7.14) asserts that there is a proper class of Woodin cardinals (Woodin cardinals being the large cardinals governing the determinacy of infinitely long games on the reals) and that every true $\Sigma_2$-sentence reflects to the HOD of some determinacy model $L(A,\mathbb{R})$ with $A$ universally Baire (that is, every continuous preimage of $A$ has the Baire property in any topological space; under a proper class of Woodin cardinals these sets have absoluteness properties across set-generic extensions). $\mathrm{AD}^+$ is the strengthening of the Axiom of Determinacy (every set of reals determines an infinite two-player game on the integers) under which the HODs of the models $L(A,\mathbb{R})$ carry a rich inner-model-theoretic structure.

The lectures also identify the technical obstructions. Fine-structural extender models are traditionally built in the nonstrategic-extender hierarchy, where each level is constructed from an extender sequence alone and iterability is proved externally; the comparison obstruction shows this hierarchy cannot reach the supercompact level. Woodin argues the construction must instead take place in the strategic-extender hierarchy, where each model carries a second predicate for its own iteration strategy (the strategy guiding iterated ultrapowers through well-founded branches of iteration trees), with iterability established inductively through connections to the HODs of $\mathrm{AD}^+$-models. The supporting global iteration hypotheses — the Weak $(\omega_1+1)$-Iteration Hypothesis and the Weak Unique Branch Hypothesis (Definitions 4.13/4.14), which assert that countable coarse premice (the simplified premice of "Suitable Extender Models I", given by an extender sequence without the full fine structure) have iteration strategies of order $\omega_1+1$ for $0$-strongly closed maximal iteration trees on them, and that such trees have at most one cofinal well-founded branch — cannot simply be assumed: Woodin predicts, on the basis of an unpublished 419-page manuscript, that natural backgrounded constructions resting on them are vacuous, so any proof must find the technical reason for the forced transition to the strategic hierarchy.

Problem Statement

Ultimate-$L$ Conjecture (Woodin; Conjecture 7.41 of the 2017 Bulletin of Symbolic Logic lectures "In Search of Ultimate-L", stated there as a slight weakening of the original version, which is implicit in "Suitable Extender Models I" and formally defined in the unpublished manuscript "Fine structure at the finite levels of supercompactness", reference [24] of the lectures): Suppose that $\delta$ is an extendible cardinal. Then there exists a weak extender model $N$ for the supercompactness of $\delta$ such that (1) $N$ is weakly $\Sigma_2$-definable and $N\subseteq\mathrm{HOD}$, and (2) $N\models V=\text{Ultimate-}L$.

The task is to prove or refute this conjecture. The lectures relate two further versions of the conjecture to it, of different logical strength:

(a) the "core elements" form predicted in the lectures (p. 5-6): if $\kappa$ is a huge cardinal, then there is a transitive set $M$ with $\mathrm{Ord}^M=\kappa$, $M\subseteq\mathrm{HOD}^{V_\kappa}$, $M\models V=\text{Ultimate-}L$, and $(V_\kappa,M)\models$ "$M$ is a weak extender model for $\delta$ is supercompact" for some $\delta<\kappa$ — a statement about $V_{\kappa+1}$ in which the weak $\Sigma_2$-definability clause is dropped (footnote 2). The lectures describe this as the goal of the in-preparation manuscript "The Ultimate-L Conjecture" (reference [25] of the lectures): "if $\kappa$ is a huge cardinal then the Ultimate-L Conjecture holds in $V_{\kappa+1}$ in a very slightly weakened form". This form is weaker than Conjecture 7.41 — it assumes a stronger large-cardinal hypothesis, yields only a local model in $V_{\kappa+1}$, and drops the definability clause — so a proof of (a) is substantial partial progress but does not by itself settle Conjecture 7.41.

(b) the original version of the conjecture, formally defined in the unpublished manuscript "Fine structure at the finite levels of supercompactness" (reference [24] of the lectures, from which Conjecture 7.41 is obtained by dropping one clause), which adds the requirement that there is an extender sequence $\tilde E$ of length $\delta$ in $V$ whose restriction to $N$ both belongs to $N$ and witnesses in $N$ that $\delta$ is a Woodin cardinal (footnote 4). This version implies Conjecture 7.41, and the lectures state that "Proving either conjecture would show in a decisive fashion the transcendence of the strategic-extender hierarchy": a proof of (b) therefore establishes Conjecture 7.41. Since (b) is strictly stronger, a refutation of (b) would not by itself refute Conjecture 7.41.

The conjecture is the central open goal of the Ultimate-$L$ program, and the same lectures pose the directly supporting open questions that constitute natural intermediate milestones: the HOD-Ultrafilter Conjecture (every countably complete ultrafilter in $\mathrm{HOD}^{L(A,\mathbb{R})}$, for $A\subseteq\mathbb{R}$ with $L(A,\mathbb{R})\models\mathrm{AD}^+$, generates a countably complete filter; Definition 7.19); whether $V=\text{Ultimate-}L$ implies weak comparison (Question 7.39; weak comparison, Definition 6.36, requires that the transitive collapses of any two finitely generated $\Sigma_2$-elementary submodels of a weak extender model $N\models V=\mathrm{HOD}$ that agree on theory and reals can be elementarily embedded into a common transitive set by elementary embeddings that are close to their domains, in the sense of Definition 5.29 of the lectures); the rigorous truth, falsity, or falsifying consistency of the Weak $(\omega_1+1)$-Iteration Hypothesis and the Weak Unique Branch Hypothesis (Definitions 4.13/4.14); the technical reason why the construction must transition from the nonstrategic-extender hierarchy to the strategic-extender hierarchy; and the representability and comparison hypotheses for HODs of determinacy models (including Hod Pair Capturing) needed to carry out the strategic-extender induction. Resolving these satellites is valuable partial progress but does not by itself settle the conjecture.

References

1. 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
2. W. Hugh Woodin, "Suitable Extender Models I", Journal of Mathematical Logic 10:101-339 (2010), DOI 10.1142/S021906131000095X, https://doi.org/10.1142/S021906131000095X
3. W. Hugh Woodin, "Suitable Extender Models II: Beyond $\omega$-Huge", Journal of Mathematical Logic 11:115-436 (2011), DOI 10.1142/S021906131100102X, https://doi.org/10.1142/S021906131100102X
4. Joan Bagaria, Peter Koellner, W. Hugh Woodin, "Large Cardinals Beyond Choice", The Bulletin of Symbolic Logic 25(3):283-318 (2019), DOI 10.1017/bsl.2019.28, https://doi.org/10.1017/bsl.2019.28
5. Grigor Sargsyan, Nam Trang, "Sealing of the Universally Baire Sets", The Bulletin of Symbolic Logic 27(3):254-266 (2021), DOI 10.1017/bsl.2021.29, https://doi.org/10.1017/bsl.2021.29
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9. Omer Ben-Neria, Yair Hayut, "On $\omega$-Strongly Measurable Cardinals", Forum of Mathematics, Sigma 11:e19 (2023), DOI 10.1017/fms.2023.15, https://doi.org/10.1017/fms.2023.15
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12. W. Hugh Woodin, "The Equivalence of Axiom $(\ast)^+$ and Axiom $(\ast)^{++}$", Journal of Mathematical Logic 25(3):2450020 (2025), DOI 10.1142/S021906132450020X, https://doi.org/10.1142/S021906132450020X
13. Joan Bagaria, Claudio Ternullo, "Steel's Programme: Evidential Framework, the Core and Ultimate-L", The Review of Symbolic Logic 16(3):788-812 (2023), DOI 10.1017/S1755020321000447, https://doi.org/10.1017/S1755020321000447
14. Sandra Müller, Grigor Sargsyan, "Gödel's Program in Set Theory", arXiv:2412.07325 (2024), https://arxiv.org/abs/2412.07325
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16. Juan Pablo Aguilera, Joan Bagaria, Gabriel Goldberg, Philipp Lücke, "Large Cardinals Beyond HOD", arXiv:2509.10254 (2025), https://arxiv.org/abs/2509.10254
17. Gabriel Goldberg, "Some Combinatorial Properties of Ultimate L and V", arXiv:2007.04812 (2020), https://arxiv.org/abs/2007.04812
18. Gabriel Goldberg, Dan Hathaway, "On $(\Sigma^2_1)^{uB}$ Absoluteness Between V and HOD", arXiv:2505.01393 (2025), https://arxiv.org/abs/2505.01393