Roads to Infinity: The Mathematics of Truth and Proof

Abstract

ROADS TO INFINITY: THE MATHEMATICS OF TRUTH AND PROOF by John Stillwell A. K. Peters, 2010, 203 pp. ISBN: 978-1-56881-466-7 Roads to Infinity: The Mathematics of Truth and Proof 'is an account of the discovery of the uncountably infinite; the interaction between set theory and logic in the realm of the irifinite; and the mathematical consequences of accepting the infinite levels of infinity. The book follows essentially two roads to infinity: Cantor's diagonal argument and Cantor's construction of the ordinals. Stillwell shows how these two themes intertwine and influence a wide range of mathematical questions including consistency, provability, computability, and existence. Roads to Infinity comprises seven chapters, each based upon a mathematical question. The historical responses to the question are explored and the concepts and theorems resulting from these responses are explained in essentially non-technical language. However, the abiiity to read mathematical symbolism and understand mathematical argumentation is required. Still well begins with Cantor's diagonal argument. His focus is the uncountability of the real continuum and he includes in the discussion the ever-amazing uncountability of transcendental numbers, an application of the diagonal argument to the rate of growth of functions, the paradoxes of set theory, and the axioms of Zermelo-Fraenkel set theory. Next, the book examines the transfinite ordinals, the continuum hypothesis, the axiom of choice and well-ordering, measurability of sets, and Cohen's technique of forcing. Also included here is a discussion of how Cantor's set theory arose from his investigation Fourier series. In the third chapter, Stillwell turns his attention to questions of computability and provability. Here we encounter Godei' s first and second incompleteness theorems, Turing machines, the Halting Problem (which Stillwell finds lurking in Cervantes' Don Quixote^!), and Hubert's Entscheidungsproblem for predicate logic. The chapter leads nicely into Chapter 4 on the consistency and completeness of propositional and predicate logic. A major theme in this chapter on logic is "cut-elimination", a way of inference in logic that replaces modus ponens by falsification trees. Chapter 5 focuses on arithmetic. Here we find a detailed discussion of Peano Aritiimetic, and an infinite extension of Peano Arithmetic and how the extension may be used to prove the consistency of Peano Arithmetic (which cannot prove its own consistency). The diagonal argument theme is reinforced in this chapter as Stillwell shows how the unprovability of consistency of Peano Arithmetic within Peano Arithmetic is related to the argument that 2N" is uncountable. …