Woodin asked in his Fall 2005 seminar whether supercompactness of a cardinal implies HOD-supercompactness, and Sargsyan (2008) proved the answer is no: after Laver preparation and a length-κ\kappa Reverse Easton iteration whose iterands code newly added subsets into the continuum function, followed by the forcing Add(κ,1)\mathrm{Add}(\kappa,1) adding a single Cohen subset of κ\kappa, a supercompact κ\kappa can fail to be HOD-supercompact because the required agreement between HOD and its images under supercompactness embeddings is destroyed at κ\kappa itself. The same paper closes with three questions that remain open and form the subject of this problem. First, does Woodin's Global Universality Theorem — under the HOD Hypothesis, if δ\delta is HOD-supercompact then every elementary embedding j:HOD∩Vγ+1→HOD∩Vj(γ)+1j:\mathrm{HOD}\cap V_{\gamma+1}\to\mathrm{HOD}\cap V_{j(\gamma)+1} with critical point at least δ\delta belongs to HOD — remain valid when HOD-supercompactness is weakened to ordinary supercompactness? Second, is that failure consistent: can there be a supercompact κ\kappa, a proper class of singular strong limit cardinals δ\delta of uncountable cofinality with (δ+)HOD=δ+(\delta^+)^{\mathrm{HOD}}=\delta^+, and an elementary embedding of an initial segment of HOD with critical point at least κ\kappa that does not lie in HOD? Third, is it consistent with a supercompact cardinal κ\kappa that HOD fails to compute the successor of some strong limit cardinal δ>κ\delta>\kappa of uncountable cofinality, that is, (δ+)HOD<δ+(\delta^+)^{\mathrm{HOD}}<\delta^+? The three questions are open; they mark the exact threshold between the two sides of Woodin's HOD dichotomy in the presence of a supercompact cardinal.

Background

Let HOD\mathrm{HOD} denote the class of hereditarily ordinal definable sets — sets definable from ordinal parameters, with every member of their transitive closure so definable as well — a transitive inner model of ZFC that is itself definable without parameters. A cardinal κ\kappa is supercompact if for every λ>κ\lambda>\kappa there is an elementary embedding j:V→Mj:V\to M into a transitive class MM with critical point κ\kappa (the least ordinal moved by jj) such that j(κ)>λj(\kappa)>\lambda and Mλ⊆MM^\lambda\subseteq M. For a proper class inner model NN, Woodin's Suitable Extender Models I defines κ\kappa to be NN-supercompact if for arbitrarily large λ\lambda such an embedding can be chosen to additionally satisfy j(N)∩Vλ=N∩Vλj(N)\cap V_\lambda=N\cap V_\lambda, where VλV_\lambda is the λ\lambda-th rank-initial segment of the cumulative hierarchy; κ\kappa is HOD-supercompact when N=HODN=\mathrm{HOD}: the embeddings must correctly compute NN up to VλV_\lambda. A regular uncountable cardinal λ\lambda is ω\omega-strongly measurable in HOD if for some κ<λ\kappa<\lambda with (2κ)HOD<λ(2^\kappa)^{\mathrm{HOD}}<\lambda there is no partition of the ordinals of cofinality ω\omega below λ\lambda into κ\kappa stationary sets that belongs to HOD; the HOD Hypothesis asserts a proper class of regular cardinals that are not ω\omega-strongly measurable in HOD, and it is implied by the concrete condition that a proper class of singular cardinals of uncountable cofinality are singular in HOD with their successors computed correctly. A cardinal δ\delta is strong limit if 2λ<δ2^\lambda<\delta for every λ<δ\lambda<\delta, and extendible if for every α\alpha there is an elementary embedding j:Vδ+α→Vβj:V_{\delta+\alpha}\to V_\beta into some rank-initial segment VβV_\beta with critical point δ\delta. Woodin's HOD Dichotomy Theorem states that if δ\delta is extendible then either every singular γ>δ\gamma>\delta is singular in HOD with (γ+)HOD=γ+(\gamma^+)^{\mathrm{HOD}}=\gamma^+, or every regular cardinal above δ\delta is measurable in HOD; Goldberg later showed the dichotomy already holds above a strongly compact cardinal κ\kappa: either every singular λ>κ\lambda>\kappa is singular in HOD with (λ+)HOD=λ+(\lambda^+)^{\mathrm{HOD}}=\lambda^+, or all sufficiently large regular cardinals are measurable in HOD. Woodin's Global Universality Theorem (Suitable Extender Models I, Theorem 201) shows that if the HOD Hypothesis holds and δ\delta is HOD-supercompact, then every elementary embedding j:HOD∩Vγ+1→HOD∩Vj(γ)+1j:\mathrm{HOD}\cap V_{\gamma+1}\to\mathrm{HOD}\cap V_{j(\gamma)+1} with crit(j)≥δ\mathrm{crit}(j)\ge\delta lies in HOD, so in particular there is no nontrivial elementary embedding from HOD to HOD with critical point at least δ\delta. In his Fall 2005 seminar Woodin asked whether every supercompact cardinal is HOD-supercompact. Sargsyan answered negatively: starting from a supercompact κ\kappa carrying a Laver function — a function f:κ→Vκf:\kappa\to V_\kappa that can predict any target set along some supercompactness embedding, used to organize a forcing preparation that makes supercompactness indestructible — he forced with a length-κ\kappa Reverse Easton iteration, an iteration indexed by the cardinals below κ\kappa that at selected measurable cardinals α\alpha adds a Cohen subset of α\alpha and codes it into the continuum function λ↦2λ\lambda\mapsto 2^\lambda far above, followed by the forcing Add(κ,1)\mathrm{Add}(\kappa,1) adding a single Cohen subset of κ\kappa. In the extension κ\kappa remains supercompact, by Laver's original indestructibility argument, while by Hamkins's closure-point theorem — if the forcing admits a closure point below the critical point, every ultrapower embedding of the extension restricts to an embedding definable in the ground model — together with the homogeneity of Cohen forcing, the coding breaks the agreement j(HOD)∩Vλ=HOD∩Vλj(\mathrm{HOD})\cap V_\lambda=\mathrm{HOD}\cap V_\lambda at κ\kappa itself, so κ\kappa is not HOD-supercompact. This separation leaves open how much of the structural theory of HOD survives under mere supercompactness, and Sargsyan's paper closes with the three questions that constitute this problem.

Problem Statement

The problem is to determine how much of the structural theory of HOD survives when the hypothesis of HOD-supercompactness is weakened to ordinary supercompactness. Two questions, posed at the conclusion of Sargsyan's separation of the two notions (the second, covering question is attributed by Sargsyan to Woodin), remain open and share this objective. The two items below jointly cover the three open questions posed at the conclusion of Sargsyan's paper: item (1) poses the universality question, whose 'must' form and consistency form (the source's Question 2) are equivalent restatements, and item (2) poses the source's Question 3, the covering question.

(1) Global Universality at a merely supercompact cardinal. Assume the HOD Hypothesis — in the source question taken in the concrete form of a proper class of singular strong limit cardinals δ\delta with cf(δ)>ω\mathrm{cf}(\delta)>\omega and (δ+)HOD=δ+(\delta^+)^{\mathrm{HOD}}=\delta^+, which implies it — and let κ\kappa be supercompact but not necessarily HOD-supercompact, where (δ+)HOD(\delta^+)^{\mathrm{HOD}} denotes the successor of δ\delta as computed inside HOD. Must every elementary embedding j:HOD∩Vγ+1→HOD∩Vj(γ)+1j:\mathrm{HOD}\cap V_{\gamma+1}\to\mathrm{HOD}\cap V_{j(\gamma)+1} with crit(j)≥κ\mathrm{crit}(j)\ge\kappa belong to HOD? In consistency form, the source's Question 2: is it consistent that κ\kappa is supercompact, such a proper class of correctly computed singular strong limit cardinals exists, and yet for some γ>κ\gamma>\kappa there is an elementary embedding j:HOD∩Vγ+1→HOD∩Vj(γ)+1j:\mathrm{HOD}\cap V_{\gamma+1}\to\mathrm{HOD}\cap V_{j(\gamma)+1} with crit(j)≥κ\mathrm{crit}(j)\ge\kappa such that j∉HODj\notin\mathrm{HOD}? A positive answer to the consistency question would show that Woodin's Global Universality Theorem genuinely requires HOD-supercompactness; a negative answer would extend the theorem, and with it the rigidity of HOD, to all supercompact cardinals.

(2) HOD covering above a supercompact cardinal. Is it consistent that κ\kappa is supercompact and yet for some strong limit cardinal δ>κ\delta>\kappa with cf(δ)>ω\mathrm{cf}(\delta)>\omega one has (δ+)HOD<δ+(\delta^+)^{\mathrm{HOD}}<\delta^+? The witnessing δ\delta may be singular or regular (a regular strong limit being inaccessible), and the singular case is the one directly tied to the HOD dichotomy: by Goldberg's strengthening of the dichotomy at strongly compact cardinals, a positive answer witnessed by a singular δ\delta necessarily makes all sufficiently large regular cardinals measurable in HOD — the far side of the HOD dichotomy — in a model that still contains a supercompact. Conversely, a negative answer, establishing (δ+)HOD=δ+(\delta^+)^{\mathrm{HOD}}=\delta^+ for every strong limit δ>κ\delta>\kappa of uncountable cofinality from supercompactness alone, would be the first covering theorem for HOD from supercompactness, a key step toward the near-side picture underlying Woodin's HOD conjecture.

References

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