Math Conjectures

HOD Conjecture

OPENLandmarkConjectureProposed c. 2010 · Standard version

Canonical statement

Assume there is an extendible cardinal, meaning a cardinal δ\delta such that for every ordinal λ>δ\lambda>\delta there are an ordinal θ\theta and an elementary embedding j:Vλ→Vθj:V_\lambda\to V_\theta with critical point δ\delta and j(δ)>λj(\delta)>\lambda. Then there is a proper class of regular cardinals κ\kappa that are not ω\omega-strongly measurable in HOD\mathrm{HOD}. Here HOD\mathrm{HOD} is the class of hereditarily ordinal-definable sets, and a regular κ\kappa is ω\omega-strongly measurable in HOD\mathrm{HOD} if there is η<κ\eta<\kappa with (2η)HOD<κ(2^\eta)^{\mathrm{HOD}}<\kappa such that HOD\mathrm{HOD} has no partition of Sωκ={α<κ:cf⁡(α)=ω}S^\kappa_\omega=\{\alpha<\kappa:\operatorname{cf}(\alpha)=\omega\} into η\eta stationary sets.
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Assume there is an extendible cardinal, meaning a cardinal \(\delta\) such that for every ordinal \(\lambda>\delta\) there are an ordinal \(\theta\) and an elementary embedding \(j:V_\lambda\to V_\theta\) with critical point \(\delta\) and \(j(\delta)>\lambda\). Then there is a proper class of regular cardinals \(\kappa\) that are not \(\omega\)-strongly measurable in \(\mathrm{HOD}\). Here \(\mathrm{HOD}\) is the class of hereditarily ordinal-definable sets, and a regular \(\kappa\) is \(\omega\)-strongly measurable in \(\mathrm{HOD}\) if there is \(\eta<\kappa\) with \((2^\eta)^{\mathrm{HOD}}<\kappa\) such that \(\mathrm{HOD}\) has no partition of \(S^\kappa_\omega=\{\alpha<\kappa:\operatorname{cf}(\alpha)=\omega\}\) into \(\eta\) stationary sets.
LOG-SET-007:[Woodin2010Suitable][WoodinDavisRodriguez2017][AguileraEtAl2025BeyondHOD]
Logic and foundations

Notes

The HOD conjecture arose around 2010 in Woodin's work on suitable extender models [Woodin2010Suitable]. It concerns HOD\mathrm{HOD}, the inner model of hereditarily ordinal-definable sets, and asks how faithfully it approximates the set-theoretic universe VV. Assuming an extendible cardinal exists, the conjecture asserts that there is a proper class of regular cardinals that are not ω\omega-strongly measurable in HOD\mathrm{HOD} — informally, cardinals at which HOD\mathrm{HOD} correctly witnesses the splitting of the cofinality-ω\omega stationary set into many stationary pieces.

The central structural result is Woodin's HOD dichotomy [WoodinDavisRodriguez2017]: in the presence of an extendible cardinal, HOD\mathrm{HOD} is either close to VV or very far from it, with no intermediate behavior, and this dichotomy already carries powerful consequences. The conjecture asserts that the close alternative always holds. Recent work on large cardinals beyond HOD\mathrm{HOD}, including the exacting cardinals, tests the boundary of the conjecture from above [AguileraEtAl2025BeyondHOD].

At present neither a proof nor a countermodel derived from accepted stronger hypotheses is known, and the conjecture remains a central open question of the large-cardinal program.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.