K urt Gödel (1906–1978), with his work on the constructible universe L, established the relative consistency of the Axiom of Choice and the Continuum Hypothesis. More broadly, he secured the cumulative hierarchy view of the universe of sets and ensured the ascendancy of first-order logic as the framework for set theory. Gödel thereby transformed set theory and launched it with structured subject matter and specific methods of proof as a distinctive field of mathematics. What follows is a survey… Expand

Analytic philosophy and modern logic are intimately connected, both historically and systematically. Thinkers such as Frege, Russell, and Wittgenstein were major contributors to the early development… Expand

The theory of large cardinals is currently a broad mainstream of modern set theory, the main area of investigation for the analysis of the relative consistency of mathematical propositions and… Expand

Prologue.- 1 The Prehistory of the Axiom of Choice.- 1.1 Introduction.- 1.2 The Origins of the Assumption.- 1.3 The Boundary between the Finite and the Infinite.- 1.4 Cantor's Legacy of Implicit… Expand