Gitik and Rinot :1771–1795, 2012) proved assuming the existence of a supercompact that it is consistent to have a strong limit cardinal \ of countable cofinality such that \, there is a very good scale at \, and \ fails along some reflecting stationary subset of \\). In this paper, we force over Gitik and Rinot’s model but with a modification of Gitik–Sharon :311, 2008) diagonal Prikry forcing to get this result for \.

Suppose that λ is the successor of a singular cardinal μ whose cofinality is an uncountable cardinal κ. We give a sufficient condition that the club filter of λ concentrating on the points of cofinality κ is not λ+-saturated.1 The condition is phrased in terms of a notion that we call weak reflection. We discuss various properties of weak reflection. We introduce a weak version of the ♣-principle, which we call ♣*−, and show that if it holds on a stationary (...) subset S of λ, then no normal filter on S is λ+-saturated. Under the above assumptions, ♣*− is true for any stationary subset S of λ which does not contain points of cofinality κ. For stationary sets S which concentrate on points of cofinality κ, we show that ♣*− holds modulo an ideal obtained through the weak reflection. (shrink)

We strengthen the revised GCH theorem by showing, e.g., that for , for all but finitely many regular κ ω implies that the diamond holds on λ when restricted to cofinality κ for all but finitely many .We strengthen previous results on the black box and the middle diamond: previously it was established that these principles hold on for sufficiently large n; here we succeed in replacing a sufficiently large n with a sufficiently large n.The main theorem, concerning the accessibility (...) of λ on cofinality κ, Theorem 3.1, implies as a special case that for every regular λ>ω, for some κ<ω, we can find a sequence such that , , and we can fix a finite set of “exceptional” regular cardinals θ<ω so that if Aλ satisfies A<ω, there is a pair-coloring so that for every -monochromatic BA with no last element, letting δ:=supB it holds that —provided that is not one of the finitely many “exceptional” members of. (shrink)

Assume ZFC is consistent then for everyB⫅ω there is a generic extension of the ground world whereB is recursive in the monadic theory ofω 2.