Genericity and arbitrariness
In this talk 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, may be viewed as a meta-theoretic argument for the existence of uncountable collections. We then discuss the effect of this meta-theoretical perspective on Skolem’s Paradox. In analyzing the connection between genericity and arbitrariness we will also study a class of posets whose elements consist of generic extensions and whose order is induced by the relation of generic extension. We then show that there are different degrees of genericity among sets. We conclude discussing the effect of our arguments on different multiverse positions.