It is proven that the following statement: "there exists a club $C \subseteq \kappa$ such that every α ∈ C is an inaccessible cardinal in L and, for every δ a limit point of C, C ∩ δ is almost contained in every club of δ of L" is equiconsistent with a weakly compact cardinal if κ = ℵ 1 , and with a weakly compact cardinal of order 1 if κ = ℵ 2.