Is Set theory Justified: the Effectiveness and Sufficiency of Set Theory as a Solution to the Mathematical Crisis

This article presents an objection to the claim that the iterative conception of the set (ICS) is a potential solution to the justification of the foundation of mathematics. The beginning of this article provides an overview of ICS, which is thought to be the paradigm underlying contemporary set theory, and the stage theory, an axiomatized account of ICS. Then, using Boolos’ research as support, an argument is made to defend ICS’s failure to uphold the axioms of choice, extensionality, and the axiom schema of replacement. After that, a response based on Alexander Paseau’s work was given to Boolos’ objection, claiming that a second conception of set, the Frege-von Neumann conception (FN), justifies the remaining axioms which ICS failed. This article concludes by concluding that the commonly taken-for-granted set theory is unjustified, calling for people’s attention to this significant topic.