On the naturalness of new axioms in set theory

In this article we analyze the notion of natural axiom in set theory. To this aim we review the intrinsic-extrinsic dichotomy, finding both the- oretical and practical difficulties in its use. We will describe and discuss a theoretical framework, that we will call conceptual realism, where the standard justification strategy is usually placed. In outlining our view, we suggest that the extensive use of naturalness calls for a revision of the standard strategy, in favor of a justification that takes into account also the historical process that lead to the formalization of set theory. Specifically we will argue that an axiom can be considered natural when it helps the clarification of the notion of arbitrary set.