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Mathematical analysis of the analytic hierarchy process (AHP) led to the development of a mathematical function, usually called the inconsistency index, which has the center role in measuring the inconsistency of the judgements in AHP. Inconsistency index is a mathematical function which maps every pairwise comparison matrix (PCM) into a real number. An inconsistency index can be considered more trustworthy when it satisfies a set of suitable properties. Therefore, the research community has been trying to postulate a set of desirable rules (axioms, properties) for inconsistency indices. Subsequently, various axiomatic frameworks for these functions have been proposed independently. However, the existing literature remains fragmented and lacks a unifying framework. Therefore, the objective of this article is twofold. Over the past decade (2014-2024), significant progress has been made in the axiomatization of inconsistency indices' properties. In this article, we first provide a comprehensive review of these advancements. We then critically evaluate and compare the aforementioned axiomatic structures, discussing future research directions.