For a cardinal of the form κ = בκ, Shelah’s logic L 1 has characterisation as maximal above $${ \cup _{\lambda < \kappa }}{L_{\lambda ,\omega }}$$ satisfying strengthening undefinability well-order. Karp’s chain [20] is known to satisfy well-order and interpolation. We prove that if singular countable cofinality, . Moreover, we show strong limit cardinals $$L_{ , }^c{ }}L_{\lambda ,\lambda }^c$...