Abstract
We show that the tree property, stationary reflection and the failure of approachability at \(\kappa ^{++}\) are consistent with \(\mathfrak {u}(\kappa )= \kappa ^+ < 2^\kappa \), where \(\kappa \) is a singular strong limit cardinal with the countable or uncountable cofinality. As a by-product, we show that if \(\lambda \) is a regular cardinal, then stationary reflection at \(\lambda ^+\) is indestructible under all \(\lambda \)-cc forcings (out of general interest, we also state a related result for the preservation of club stationary reflection).
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Notes
However, to our knowledge there is no publication which surveys which cardinal invariants patterns at \(\omega \) can be realized with compactness at \(\omega _2\).
Suppose \(\kappa \) is singular. For every node \(t\in T\) and every regular \(\theta < \kappa \), there are unboundedly many levels of \(T_t\) with size at least \(\theta \), arguing as we did in the proof of (i) here. If \(\gamma < \kappa ^+\) is an ordinal of cofinality \(\mathrm {cf}(\kappa )\) such that for every regular \(\theta < \kappa \), there is a level of \(T_t\) below \(\gamma \) of size at least \(\theta \), then the level \(\gamma \) of \(T_t\) must have size \(\ge \theta \) for every such \(\theta \) (and so size \(\kappa \)) because T is well-pruned.
This is strictly stronger than requiring \(\le _J\) by (2). Note that if \(\prod A\) has no maximal elements, then the clause \(f = g\) can be omitted.
In our notational convention, \(\mathbb {Q}\) is \(\kappa ^+\)-distributive if \(\mathbb {Q}\) does not add new sequences of length \(<\kappa ^+\).
In fact, only two specific sequences in V are actually required for the main theorem (the sequence of measurables \(\bar{\kappa }\) satisfying \(\mathsf GCH\) and its successors; see below).
The proof of this abuses the notation and treats \(\mathbb {P}_\delta \) as defined with respect to this single \(\bar{\theta }\); to handle all the relevant \(\bar{\theta }\)’s, we need to ensure that each \(\bar{\theta }\) appears cofinally often below \(\delta \) in \(\mathbb {P}_\delta \).
Since we will have \(2^\kappa = \kappa ^{++}\) in the final model, \(\kappa ^+\) is the only interesting value of \(\mathfrak {u}(\kappa )\). But in principle, the forcing can be iterated up to any ordinal with cofinality \(>\kappa \).
The proof of (iii) was suggested to us by Menachem Magidor.
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Both authors were supported by FWF/GAČR grant Compactness principles and combinatorics (19-29633L).The second author acknowledges the institutional support RVO 67985840.
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Honzik, R., Stejskalová, Š. Small \(\mathfrak {u}(\kappa )\) at singular \(\kappa \) with compactness at \(\kappa ^{++}\). Arch. Math. Logic 61, 33–54 (2022). https://doi.org/10.1007/s00153-021-00776-5
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DOI: https://doi.org/10.1007/s00153-021-00776-5