Abstract

It is a well-known story that Russell's discovery of his paradox shook the foundations of Frege's logical system for arithmetic. But there is another route to this paradox. Hilbert pointed out to Frege that he had already found other even more convincing contradictions which he communicated to Zermelo, thereby initiating Zermelo's independent discovery of Russell's paradox. In this paper, we follow this less familiar route and analyze three paradoxes, namely Hilbert's paradox, Zermelo's version of Russell's paradox and Schröder's paradox of 0 and 1. Furthermore, tradition in which these paradoxes were found is reconsidered. We examine Schröder's place in the foundational study and criticize an alleged dichotomy between the algebraic and logistic traditions.