Definiteness in Early Set Theory

Abstract

The notion of definiteness has played a fundamental role in the early developments of set theory. We consider its role in work of Cantor, Zermelo and Weyl. We distinguish two very different forms of definiteness. First, a condition can be definite in the sense that, given any object, either the condition applies to that object or it does not. We call this intensional definiteness. Second, a condition or collection can be definite in the sense that, loosely speaking, a totality of its instances or members has been circumscribed. We call this extensional definiteness. Whereas intensional definiteness concerns whether an intension applies to objects considered one by one, extensional definiteness concerns the totality of objects to which the intension applies. We discuss how these two forms of definiteness admit of precise mathematical analyses. We argue that two main types of explication of extensional definiteness are available. One is in terms of completability and coexistence (Cantor), the other is based on a novel idea due to Hermann Weyl and can be roughly expressed in terms of proper demarcation. We submit that the two notions of extensional definiteness that emerges from our investigation enable us to identify and understand some of the most important fault lines in the philosophy and foundations of mathematics.