Set Theory and Third Mathematical Crisis: A Series of Events Reflected Upon the Mathematical Foundation

The basis of mathematics remains a highly debated and prominent subject among the mathematical community. The absence of a solid mathematical basis gives rise to a range of occurrences, such as mathematical crises. There is a multitude of arguments pertaining to the Mathematical Foundation. This research aims to elucidate the need for a foundation in mathematics and examine the origins of mathematical crises via an analysis of Classical Mathematics. Subsequently, an examination of three significant mathematical crises and the subsequent evolution of Mathematics over the 19th and 20th centuries will be undertaken. The Third Mathematical Crisis is a significant component of this work. Subsequently, a comprehensive exposition will be presented on the three prominent educational institutions that emerged during the Third Mathematical Crisis. This paper aims to provide a comprehensive analysis of the Third Mathematical Crisis, with a particular focus on the axiomatization of Set Theory. The process and outcome of this crisis will be thoroughly examined and discussed. There are counterexamples that challenge the notion of Set Theory as a basic theory. This paper will mostly concentrate on the subject of Category Theory. This paper provides an introduction to the genesis and identity of Category Theory, along with a concise explanation why the widespread interest among mathematicians in this field and its potential as a foundational framework for Mathematics. As a consequence, Category may serve as a fundamental basis to a significant degree.