On the Singular Cardinals

We give upper and lower bounds for the consistency strength of the failure of a combinatorial principle introduced by Jensen, Square on singular cardinals. A combinatorial principle of great importance in set theory is the Global principle of Jensen [6]: Global : There exists 〈Cα | α a singular ordinal 〉 such that for each α, Cα is a closed unbounded subset of α of ordertype less than α, the limit points of Cα are singular ordinals, and Cᾱ = Cα ∩ ᾱ whenever ᾱ is a limit point of Cα. A weakening of Global is the following: We say that holds on the singular cardinals if and only if there exists 〈Cα | α a singular cardinal 〉 such that for each α, Cα is a closed unbounded subset of Card ∩ α of ordertype less than α, the limit points of Cα are singular cardinals, and Cᾱ = Cα ∩ ᾱ whenever ᾱ is a limit point of Cα. Jensen observed that Global is equivalent to the conjunction of on the singular cardinals together with his well-known principles κ for all uncountable cardinals κ. The point is that if α is a singular ordinal then either α is a singular cardinal or α is a limit ordinal with κ < α < κ for a unique cardinal κ: the fragment of Global which assigns a singularising club set Cα to each limit α with κ < α < κ + corresponds to κ. principles are typically used as witnesses to various forms of incompactness or non-reflection: for example if κ holds then there exist κ-Aronszajn trees and non-reflecting stationary subsets of κ. Since large cardinal axioms embody principles of compactness and reflection, it is not surprising that there is some tension between and large cardinals, and this has been extensively investigated. We should distinguish here between the questions “To what extent are large cardinals The first author partially supported by NSF Grants DMS-0400982 and DMS0654046. The second author was supported by FWF Grants P16334-NO5 and P16790NO4.