In this paper I discuss Husserl's solution of the problem of imaginary elements in mathematics as presented in the drafts for two lectures hegave in Göttingen in 1901 and other related texts of the same period,a problem that had occupied Husserl since the beginning of 1890, whenhe was planning a never published sequel to Philosophie der Arithmetik(1891). In order to solve the problem of imaginary entities Husserl introduced,independently of Hilbert, two notions of completeness (definiteness in Husserl'sterminology) for a formal axiomatic system. I present and discuss these notionshere, establishing also parallels between Husserl's and Hilbert's notions ofcompleteness.