Abstract. We discuss the work of Paul Bernays in set theory, mainly his axiomatization and his use of classes but also his higher-order reflection principles. Paul Isaak Bernays (1888–1977) is an important figure in the development of mathematical logic, being the main bridge between Hilbert and Gödel in the intermediate generation and making contributions in proof theory, set theory, and the philosophy of mathematics. Bernays is best known for the two-volume 1934,1939 Grundlagen der Mathematik [39, 40], written solely by him though Hilbert was retained as first author. Going into many reprintings and an eventual second edition thirty years later, this monumental work provided a magisterial exposition of the work of the Hilbert school in the formalization of first-order logic and in proof theory and the work of Gödel on incompleteness and its surround, including the first complete proof of the Second Incompleteness Theorem. 1 Recent re-evaluation of Bernays ’ role actually places him at the center of the development of mathematical logic and Hilbert’s program. 2 But starting in his forties, Bernays did his most individuated, distinctive mathematical work in set theory, providing a timely axiomatization and later applying higher-order reflection principles, and produced a stream of

Husserl repeatedly has claimed that (1) mathematics without a philosophical foundation is not a science but a mere technique; (2) philosophical considerations may lead to the rejection of parts of mathematical practice; but (3) they cannot lead to mathematical innovations. My thesis is that Husserl’s third claim is wrong, by his own standards. To explain this thesis, let me first introduce the term ‘revisionism’. It is understood here, following Crispin Wright, as the term that applies to ‘any philosophical standpoint which reserves the potential right to sanction or modify pure mathematical practice ’ [Wright 1980, p.117]. I want to make a distinction between weak and strong revisionism. The point of reference is the actual practice of mathematics. Weak revisionism then potentially sanctions a subset of this practice, while strong revisionism potentially not only limits but extends it, in different directions. In strong revisionism, certain combinations of limitation and extension may lead to a mathematics that is no longer compatible with the unrevised one. ‘May lead’, not ‘necessarily leads’: it is all a matter of reserving rights; whether there is occasion to exercise them is a further question. To illustrate these categories, let me give examples of non-revisionism, weak revisionism, and strong revisionism. Non-revisionism can be found in Wittgenstein’s Philosophische Untersuchungen, where philosophy can neither change nor ground mathematics: Die Philosophie darf den tatsächlichen Gebrauch der Sprache in keiner Weise antasten, sie kann ihn am Ende also nur beschreiben. Denn sie kann ihn auch nicht begründen. Sie läßt alles wie es ist. Sie läßt auch die Mathematik wie sie ist, und keine mathematische

According to Curtis Franks ’ preface, this book bundles his historical, philosophical and logical research to center around what he thinks are “the most important and most overlooked aspects of Hilbert’s program … a glaring oversight of one truly unique aspect of Hilbert’s thought, ” namely that “questions about mathematics that arise in philosophical reflection⎯questions about how and why its methods work⎯might best be addressed mathematically … Hilbert’s program was primarily an effort to demonstrate that. ” The standard “well-rehearsed ” story for these oversights is said to be that Hilbert’s philosophical vision was dashed by Gödel’s incompleteness theorems. But Franks argues to the contrary that Gödel’s remarkable early contributions to metamathematics instead drew “significant attention to the then fledgling discipline, ” a field that has since proved to be exceptionally productive scientifically (even through the work of such logicians as Tarski, who mocked Hilbert’s program). One may well ask how the author’s effort to put this positive face on the patent failure of Hilbert’s program can possibly succeed in showing that mathematical knowledge is autonomous, that mathematics has only to look to itself for its proper foundations. Let us see.