Hilbert and Set Theory

was the preeminent mathematician of the early decades of the 20th Century, 1 a mathematician whose pivotal and penetrating results, emphasis on central problems and conjectures, and advocacy of programmatic approaches greatly expanded mathematics with new procedures, initiatives, and contexts. With the emerging exten-sional construal of mathematical objects and the development of abstract structures, set-theoretic formulations and operations became more and more embedded into the basic framework of mathematics. And Hilbert speciically championed Cantorian set theory, declaring 1926: 170]: \From the paradise that Cantor has created for us no one will cast us out." On the other hand, Hilbert did not make direct mathematical contributions toward the development of set theory. Although he liberally used non-constructive arguments, his were still the concerns of mainstream mathematics, and he stressed concrete approaches and the eventual solvability of every mathematical problem. After its beginnings as the study of the transsnite numbers and deenable collections of reals, set theory was becoming an open-ended, axiomatic investigation of arbitrary collections and functions. For Hilbert this was never to be a major concern, but he nonetheless exerted a strong innuence on this development both through his broader mathematical approaches and through his speciic attempt to establish the Continuum Hypothesis. What follows is a historical and episodic account of Hilbert's results and initiatives and their ramiications and extensions, in so far as they bear on set theory and its development. 2 The emphasis on set theory presents a tangential view of Hilbert's main mathematical endeavors , but one that illuminates their larger themes and motivations. Because of its basic interplay with set theory, we deal at length with Hilbert's program for establishing the consistency of mathematics by \\nitary reasoning". Section 1 discusses Hilbert's use of non-constructive existence proofs, with the focus on his rst major result; section 2 discusses his axiomatization of Euclidean geometry, with the focus on his Completeness Axiom; and then section 3 discusses questions about the real numbers and their arithmetic that Hilbert would later approach through his proof theory. With this as a backdrop, section 4 considers Hilbert's involvement in the early development of set theory, and section 5 considers both his mathematical logic as a reaction to Russell's and the two crucial new questions that Hilbert raised. Section 6 describes Hilbert's approach to establishing the consistency of mathematics , and section 7 its application to the Continuum Hypothesis. Then section 8 discusses GG …