INTER-UNIVERSAL TEICHMÜLLER THEORY IV: LOG-VOLUME COMPUTATIONS AND SET-THEORETIC FOUNDATIONS Shinichi Mochizuki April 2020 Abstract. The present paper forms the fourth and final paper in a series of papers concerning “inter-universal Teichmüller theory”. In the first three papers of the series, we introduced and studied the theory surrounding the log- theta-lattice, a highly non-commutative two-di