It is generally claimed that infinite idealizations are required for explaining phase transitions within statistical mechanics (e.g., Batterman 2011). Nevertheless, Menon and Callender (2013) have outlined theoretical approaches that describe phase transitions without using the infinite limit. This paper closely investigates one of these approaches, which consists of studying the complex zeros of the partition function (Borrmann et al. 2000). Based on this theory, we argue for the plausibility for eliminating the infinite limit for studying phase transitions. We offer a new account for phase transitions in finite systems, and we argue for the use of the infinite limit as an approximation for studying phase transitions in large systems.