Abstract: Kim B Bruce gave a system of axioms for second order logic L(Q). H J Keisler in 1977 published a book called "Model Theory for Infinitary Logit" to give a system of axioims for infinitary logic. This article combines the two ideas of the two system of axioms, applying the consistency property to give a system of axioms and model theory for infinitary second order logic L_ω1ω(Q). This part of the article is mainly the proofs of Scott's isomorphism theorem and Craig interpolation theorem for L_ω1ω(Q).