Mitchell rank for supercompactness

This paper defines a Mitchell rank for supercompact cardinals. If $κ$ is a $θ$-supercompact cardinal then $o_{θ-sc}(κ) = \sup { o_{θ-sc}(μ) + 1 \ | \ μ\in m(κ)}$, where $m(κ)$ is the collection of normal fine measures on $P_κθ$. We show how to force to kill the degree of a measurable cardinal $κ$ to any specified degree which is less than or equal to the degree of $κ$ in the ground model. We will also show how to softly kill the Mitchell rank for supercompactness of any supercompact cardinal so that in the forcing extension it is any desired degree less than or equal to its degree in the ground model, along with some results concerning strongly compact cardinals.

💡 Research Summary

The paper introduces a Mitchell‑type rank for supercompact cardinals and shows how to manipulate this rank by forcing. The author first recalls the classical Mitchell rank for measurable cardinals: given the collection m(κ) of normal measures on κ, the Mitchell order μ ◁ ν holds exactly when μ∈M_ν (the ultrapower by ν), and the rank o(κ) is defined as
 o(κ)=sup{ o(μ)+1 | μ∈m(κ) }.
For a θ‑supercompact cardinal κ the paper defines an analogous collection m(κ) consisting of the normal fine measures on P_κθ. The “supercompact Mitchell rank” is then set to
 o_{θ‑sc}(κ)=sup{ o_{θ‑sc}(μ)+1 | μ∈m(κ) },
where o_{θ‑sc}(μ) is the rank of the measure μ as computed inside the ultrapower M_μ. Lemma 8 proves that o_{θ‑sc}(μ)=o_{θ‑sc}(κ)^{M_μ}, establishing the internal‑external correspondence needed for the later forcing arguments.

The core technical contribution is a family of forcing constructions that can lower the Mitchell rank of a measurable or supercompact cardinal to any prescribed value not exceeding its original rank. The first main result (Theorem 6) shows how to force a measurable κ to have Mitchell rank exactly 1. The forcing is an Easton‑support iteration of length κ. At each inaccessible stage γ<κ a forcing Q_γ adds a club C_γ⊂γ consisting only of non‑measurable cardinals; at non‑inaccessible stages the iteration does nothing. Because each Q_γ is ≤β‑closed for every β<γ and has size ≤γ, the whole iteration preserves cardinals and cofinalities. Moreover, using Hamkins’s approximation and cover lemmas, the iteration does not create new measurable cardinals. The author lifts the original ultrapower embedding j:V→M_μ through the iteration and the final club‑adding forcing, showing that κ remains measurable in the extension but that every normal measure now concentrates on a set of non‑measurables, forcing o(κ)≤1. Since o(κ)≥1 already, the rank is exactly 1.

The second major theorem (Theorem 11) generalises this to any ordinal α. Working in a model of ZFC+GCH, the author defines an ORD‑length Easton‑support iteration. At each inaccessible γ the iteration forces a club C_γ⊂γ such that every δ∈C_γ satisfies o(δ)<α (the Mitchell rank of δ in the ground model). After forcing, every cardinal κ>α satisfies o(κ)≤α; if the original rank was at least α, then equality holds, so the rank is “softly killed’’ to α. The iteration again preserves all measurable cardinals and creates none, because the same approximation/cover arguments apply. Lemma 9 shows that under the δ‑approximation and δ‑cover properties the Mitchell rank cannot increase, while Lemma 10 shows that if every normal ultrapower embedding lifts, the rank cannot decrease. Together these lemmas guarantee that the forcing precisely forces the desired rank.

The paper also discusses the relationship between supercompactness and strong compactness. It cites results of Hamkins–Shelah (showing that one can force a κ that is θ‑supercompact but not (θ+)-supercompact) and of Magidor (producing a strongly compact cardinal that is not supercompact). These illustrate that the hierarchy of large‑cardinal notions is robust under forcing, and that the newly defined supercompact Mitchell rank behaves analogously to the classical Mitchell rank for measurables.

In summary, the author provides:

1. A natural definition of Mitchell rank for θ‑supercompact cardinals, extending the classical notion for measurables.
2. A forcing construction that reduces the rank of a measurable cardinal to 1 while preserving all measurables and creating none.
3. A general forcing scheme that, for any ordinal α, reduces the supercompact Mitchell rank of every cardinal above α to exactly min{α, original rank}.
4. Technical lemmas ensuring that the rank does not inadvertently increase during forcing, using the approximation and cover properties.
5. Connections to strong compactness, showing that the methods fit into the broader landscape of large‑cardinal manipulation.

These results give set theorists a new tool for calibrating the strength of supercompactness in forcing extensions, mirroring the well‑studied manipulation of Mitchell ranks for measurable cardinals.