Genericity and arbitrariness

In this article we compare the notions of genericity and arbitrariness on the basis of the realist import of the method of forcing. We argue that Cohen's Theorem, similarly to Cantor's Theorem, can be considered a meta-theoretical argument in favor of the existence of uncountable collections. Then we discuss the effects of this meta-theoretical perspective on Skolem's Paradox. We conclude discussing how the connection between arbitrariness and genericity can be used to argue in favor of Forcing Axioms.