Axiomatic set theory was created by German mathematician Zermelo as a strategy for addressing and resolving paradoxes in the field of mathematical study. By adopting this strategy, the axiomatic technique will be applied to set theory. The person argues that Cantor's failure to impose restrictions on the idea of a set is the cause of the dilemma. They also claim that Cantor's definition of a set is unclear. Both of these arguments are predicated on the idea that Cantor neglected to place limitations on the idea of a set. Zermelo hypothesized that the condensed version of the axioms would make it easier to define a set and elaborate on its properties, and he was correct in his prediction. The creation of an axiomatization for set theory is the first of this research's main goals. Second, the investigation of various set theory development methods in comparison to one another. Even though there are intriguing puzzles in the field of set theory that have not yet been solved, the axiomatization of set theory is widely regarded as a significant accomplishment in the field.