Chain logic and Shelah’s Infinitary logic

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 $$L_{ , }^c{ }}L_{\lambda ,\lambda }^c$$ with models version then gives partial solution Problem 1.4 from [28], which asked whether for cofinality there was strictly between $${L_{{\kappa ^ + },\omega },{\kappa }}}$$ having modulo accepting upper bound model class Lκ, κ, satisfies required properties. In addition, this not κ-compact, question have on various occasions. contribute further development by proving Union Lemma identifying chain-independent fragment logic, showing it still considerable expressive power. conclusion, shown simply defined emulates in interpolation, maximality respect it, Lemma. addition completeness theorem.