Abstract: Kim B. Bruce gave a system of axioms lor second order logic L(Q). H. J. Keisler presented a system of axioms for infinilary logic. This article gives a system of axioms and model theory for infinitary second order logic L_W1W(Q), based on the combination of both ideas of the systems and application of the consistency property; proves the model existence theorem and completeness theorem in L_w1w(Q).