\(\square \) on the singular cardinals. (English) Zbl 1166.03022

Say that \(\square\) holds on the singular cardinals if there exists \(\langle C_\nu: \nu\) a singular cardinal\(\rangle\) such that (a) \(C_\nu\) is a club subset of \(\nu \cap \text{Card}\) of order-type less than \(\nu\), (b) the limit points of \(C_\nu\) are singular cardinals, and (c) \(C_\tau = C_\nu \cap\tau\) whenever \(\tau\) is a limit point of \(C_\nu\). The authors prove the following three theorems:
If it is consistent with ZFC that \(\square\) on the singular cardinals up to \(\mu\) fails for some cardinal \(\mu\), then it is consistent with ZFC that there is an inaccessible cardinal which is a stationary limit of cardinals of uncountable Mitchell order.
If \(\kappa\) is \(\mu\)-supercompact for some inaccessible cardinal \(\mu > \kappa\), then \(\square\) fails on the singular cardinals less than \(\mu\).
There is a class forcing extension of \(V\) in which (i) every almost inaccessibly hyperstrong cardinal \(\kappa\) retains that property (where \(\kappa\) is said to be almost inaccessibly hyperstrong if it is the critical point of an elementary embedding \(j : V\rightarrow M\) with \(V_\lambda \subseteq M\) for some \(M\)-inaccessible \(\lambda > j(\kappa)\)), and (ii) \(\square\) holds on the singular cardinals.


03E05 Other combinatorial set theory
03E35 Consistency and independence results
03E55 Large cardinals
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