Small measurable cardinals
Pith reviewed 2026-05-24 10:28 UTC · model grok-4.3
The pith
The first measurable cardinal can be the first weakly critical cardinal or the first Mahlo cardinal from the assumption of a single measurable cardinal.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
By refining the definition of a j-decomposable system, the authors obtain a lifting criterion that is sufficient to extend weakly compact embeddings to symmetric extensions. This allows constructions in which the first measurable cardinal is also the first weakly critical cardinal, and in which it is also the first Mahlo cardinal, both from the assumption of a single measurable cardinal. However, the paper proves that if the first inaccessible cardinal coincides with the first measurable cardinal, then in a suitable inner model this cardinal has Mitchell order at least two.
What carries the argument
The refined j-decomposable system, which supplies an improved lifting criterion for elementary embeddings through symmetric extensions that preserves weakly compact cardinals.
If this is right
- The first measurable cardinal can be made to equal the first weakly critical cardinal using only one measurable cardinal in the ground model.
- The first measurable cardinal can be made to equal the first Mahlo cardinal using only one measurable cardinal in the ground model.
- If the first measurable cardinal is also the first inaccessible cardinal, then some inner model contains a measurable cardinal of Mitchell order at least two.
- The refined lifting criterion allows preservation of weakly compact embeddings where earlier definitions did not suffice.
Where Pith is reading between the lines
- The same refinement technique may permit collapsing other pairs of large cardinal properties under minimal assumptions.
- These results tighten the consistency-strength relations between measurable, weakly critical, and Mahlo cardinals.
- The limitation on Mitchell order when measurability and inaccessibility coincide may extend to other combinations of large cardinal properties.
Load-bearing premise
The refined definition of j-decomposable systems supplies a lifting criterion sufficient to preserve weakly compact embeddings in the symmetric extensions constructed in the paper.
What would settle it
A symmetric extension in which the first measurable cardinal is the first Mahlo cardinal yet the weakly compact embedding fails to lift under the refined criterion would show that the improvement is not doing the claimed work.
read the original abstract
We continue the work from [8] and make a small -- but significant -- improvement to the definition of $j$-decomposable system. This provides us with a better lifting of elementary embeddings to symmetric extensions. In particular, this allows us to more easily lift weakly compact embeddings and thus preserve the notion of weakly critical cardinals. We use this improved lifting criterion to show that the first measurable cardinal can be the first weakly critical cardinal or the first Mahlo cardinal, both relative to the existence of a single measurable cardinal. However, if the first inaccessible cardinal is the first measurable cardinal, then in a suitable inner model it has Mitchell order of at least $2$.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper improves the definition of j-decomposable systems from prior work to obtain a stronger lifting criterion for elementary embeddings into symmetric extensions, with particular attention to weakly compact embeddings. This is used to prove, relative to a single measurable cardinal, that the first measurable cardinal can be the first weakly critical cardinal or the first Mahlo cardinal. The paper also shows that if the first inaccessible cardinal coincides with the first measurable cardinal, then in a suitable inner model the measurable has Mitchell order at least 2.
Significance. If the lifting criterion is valid, the results establish new consistency facts about the possible positions of the least measurable cardinal among other small large-cardinal notions, using only a single measurable as hypothesis. This clarifies the hierarchy of small large cardinals and supplies information on Mitchell order in inner models under an additional inaccessibility assumption. The technical improvement to j-decomposable systems is a modest but potentially useful advance for symmetric-extension arguments.
major comments (2)
- [§3 and §4] §3 (refined definition of j-decomposable system) and the lifting argument in §4: the claim that the new definition supplies a lifting criterion sufficient to preserve weakly compact embeddings in the symmetric extensions is load-bearing for both consistency theorems. The text does not explicitly verify that the criterion applies to the specific embeddings generated by the forcing constructions without extra Mitchell-order hypotheses; if the lifting fails for these embeddings, the consistency results do not follow from a single measurable.
- [final theorem] Theorem on the Mitchell-order lower bound (the final result): the argument that the first inaccessible = first measurable implies Mitchell order ≥2 in an inner model relies on the same lifting criterion. A failure of the lifting in this case would leave the negative result unproved at the stated strength.
minor comments (2)
- [§2] Notation for j-decomposable systems is introduced without a clear comparison table to the definition in [8]; adding such a table would help readers track the improvement.
- [abstract and §1] The abstract states the results are 'relative to the existence of a single measurable cardinal,' but the text should explicitly restate the exact large-cardinal assumption used in each theorem.
Simulated Author's Rebuttal
We thank the referee for the detailed report and for identifying the need for explicit verification of the lifting criterion's applicability. We address the two major comments below. The improvements to the j-decomposable system definition are intended to work for the embeddings arising from a single measurable cardinal, but we agree that an explicit check strengthens the presentation.
read point-by-point responses
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Referee: [§3 and §4] §3 (refined definition of j-decomposable system) and the lifting argument in §4: the claim that the new definition supplies a lifting criterion sufficient to preserve weakly compact embeddings in the symmetric extensions is load-bearing for both consistency theorems. The text does not explicitly verify that the criterion applies to the specific embeddings generated by the forcing constructions without extra Mitchell-order hypotheses; if the lifting fails for these embeddings, the consistency results do not follow from a single measurable.
Authors: The embeddings in the constructions of §4 are the canonical ones induced by the single measurable cardinal κ (via its normal measure), and the refined j-decomposable system in Definition 3.4 is formulated exactly to accommodate the weak compactness of these embeddings under the symmetric extensions used. No additional Mitchell order is required because the decomposition conditions (i)–(iii) in the new definition are satisfied directly by the ultrapower embeddings at a measurable cardinal. We will add an explicit verification lemma (new Lemma 4.3) confirming that the criterion applies verbatim to the embeddings arising in both consistency proofs, without extra hypotheses. revision: yes
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Referee: [final theorem] Theorem on the Mitchell-order lower bound (the final result): the argument that the first inaccessible = first measurable implies Mitchell order ≥2 in an inner model relies on the same lifting criterion. A failure of the lifting in this case would leave the negative result unproved at the stated strength.
Authors: The final theorem likewise uses only the embeddings from the single measurable cardinal (now assumed to be the least inaccessible). The same lifting criterion applies, and the inner-model argument proceeds by deriving a contradiction from Mitchell order 1 via the lifted embedding. As with the consistency results, we will insert an explicit applicability check (new Remark 5.2) to confirm that no extra Mitchell-order assumption is used. revision: yes
Circularity Check
No significant circularity; derivation is self-contained via explicit forcing constructions
full rationale
The paper begins from a model with a single measurable cardinal, improves the definition of j-decomposable systems (building on but not reducing to [8]), proves a new lifting criterion for weakly compact embeddings in symmetric extensions, and applies this criterion in explicit forcing constructions to obtain the stated consistency results. No step equates a derived claim to its inputs by construction, renames a fitted quantity as a prediction, or relies on a load-bearing self-citation whose content is unverified outside the paper. The self-citation to prior work supplies only the base technique; the central claims rest on the novel improvement and standard set-theoretic arguments that are externally falsifiable.
Axiom & Free-Parameter Ledger
axioms (2)
- standard math ZFC set theory
- domain assumption Existence of a measurable cardinal
discussion (0)
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