Generic Sets and Arbitrary Objects

In this paper, I investigate whether generic sets, in the context of the theory of forcing, can be understood as arbitrary objects in the sense of Fine or Horsten. I will provide a partially positive and partially negative answer. Specifically, the answer will depend on whether I will consider generic sets as objects or as names. I also compare our perspective with Horsten's view on Boolean‐valued sets as arbitrary objects. I conclude by suggesting that there is a cluster of notions semantically similar to arbitrariness that gives rise to useful objects in mathematics.