Definiteness, and Semantic Completeness. An Analysis of E. Husserl's Gottinger Doppelvortrag

The significance of E. Husserl's Gottingen Lectures of 1901 lies in his attempt to address the problem of the imaginary in mathematics, which he approached through his findings on the concepts of variety and completeness (Vollstandigkeit) of an axiomatic system. According to Husserl, solving the problem of the imaginary involved finding answers to three fundamental questions: 1) when does an element become imaginary from the perspective of a formal axiomatic system? 2) how can the use of imaginary elements in mathematics be justified? and 3) is a syntactically complete axiomatic system compatible with the extension of the concept of number? In this article, I will present Husserl's answers to these questions and provide a partial justification for their correctness. I will divide the content of both lectures into two parts. The first part will present the problem of the imaginary, and the second will examine Husserl's proposal for addressing the aforementioned problems.