Marcus Rossberg

Cantor on Frege's Foundations of Arithmetic: Cantor's 1885 Review of Frege's Die Grundlagen der Arithmetik

In 1885, Georg Cantor published his review of Gottlob Freges Grundlagen der Arithmetik. In this essay, we provide its first English translation together with an introductory note. We also provide a translation of a note by Ernst Zermelo on Cantors review, and a new translation of Freges brief response to Cantor. In recent years, it has become philosophical folklore that Cantors 1885 review of Freges Grundlagen already contained a warning to Frege. This warning is said to concern the defectiveness of Freges notion of extension. The exact scope of such speculations varies and sometimes extends as far as crediting Cantor with an early hunch of the paradoxical nature of Freges notion of extension. William Tait goes even further and deems Frege ‘reckless’ for having missed Cantors explicit warning regarding the notion of extension. As such, Cantors purported inkling would have predated the discovery of the Russell–Zermelo paradox by almost two decades. In our introductory essay, we discuss this alleged implicit (or even explicit) warning, separating two issues: first, whether the most natural reading of Cantors criticism provides an indication that the notion of extension is defective; second, whether there are other ways of understanding Cantor that support such an interpretation and can serve as a precisification of Cantors presumed warning.

Somehow Things Do Not Relate: On the Interpretation of Polyadic Second-Order Logic

Boolos has suggested a plural interpretation of second-order logic for two purposes: (i) to escape Quine’s allegation that second-order logic is set theory in disguise, and (ii) to avoid the paradoxes arising if the second-order variables are given a set-theoretic interpretation in second-order set theory. Since the plural interpretation accounts only for monadic second-order logic, Rayo and Yablo suggest an new interpretation for polyadic second-order logic in a Boolosian spirit. The present paper argues that Rayo and Yablo’s interpretation does not achieve the goal.