This paper is divided into two parts. Part I provides a resumé of the evolution of the notion of predicativity. Part II describes our own work on the subject.Part I§1. Conceptions of sets.Statements about sets lie at the heart of most modern attempts to systematize all (or, at least, all known) mathematics. Technical and philosophical discussions concerning such systematizations and the underlying conceptions have thus occupied a considerable portion of the literature on the foundations of mathematics.

Previously published: Princeton University Press, 1940.

We began by distinguishing Tarskian and Fitchean notions of universality in such a way that the claim that no language is universal in the sense of Tarski is compatible with accepting Fitchean universality. Then we examined a proposal involving two truth concepts — one that fit the Fitchean notion and another that followed Tarski's views on truth — finding little advantage in such generosity. We attempted a reformulation of Herzberger's argument for the negative view — the view that no language (...) is universal in Tarski's sense — but found it unsuccessful when the language of the argument's formulation was brought under consideration. A more persuasive argument for EI was found, free of the defect of the previous one. EI was then shown to have unsettling consequences, prompting us to inquire about avoiding it. We found this possible, noting that EI is itself a solution to the semantic paradoxes, to which there are alternatives that avoid the unwelcome aspects of EI. However, whether any such alternative is ultimately preferable to EI remains to be seen. (shrink)