This article is dedicated to Professor Burton Dreben on his coming of age. I owe him particular thanks for his careful reading and numerous suggestions for improvement. My thanks go also to Jose Ruiz and the referee for their helpful comments. Parts of this account were given at the 1995 summer meeting of the Association for Symbolic Logic at Haifa, in the Massachusetts Institute of Technology logic seminar, and to the Paris Logic Group. The author would like to express his thanks to the various organizers, as well as his gratitude to the Hebrew University of Jerusalem for its hospitality during the preparation of this article in the autumn of 1995.

this paper, the seminal results of set theory are woven together in terms of a unifying mathematical motif, one whose transmutations serve to illuminate the historical development of the subject. The motif is foreshadowed in Cantor's diagonal proof, and emerges in the interstices of the inclusion vs. membership distinction, a distinction only clarified at the turn of this century, remarkable though this may seem. Russell runs with this distinction, but is quickly caught on the horns of his well-known paradox, an early expression of our motif. The motif becomes fully manifest through the study of functions f :

In 1873 C. Hermite [6] proved that the number e is transcendental. In his paper he explains in a very clear way how he found his proof. He starts with an analogy between simultaneous diophantine approximation of real numbers on the one hand and analytic complex functions of one variable on the other. He first solves the analytic problem by constructing explicitly what is now called Padé approximants for the exponential function. In fact there are two types of such approximants, they are now called type I and type II, and what Hermite did in 1873 was to compute Padé approximants of type II. He also found those of type I in 1873 and studied them later in 1893. K. Mahler was the first in the mid’s 1930 to relate the properties of the two types of Padé’s approximants and to use those of type I in order to get a new proof of Hermite’s transcendence Theorem (and also of the generalisation by Lindemann and Weierstraß as well as quantitative refinements). See [2] Chap. 2 § 3. In the analogy with number theory, Padé approximants of type II are related with the simultaneous approximation of real numbers ϑ1,..., ϑm by rational numbers pi/q with the same denominator q (one does not require that the fractions are irreducible), which means that we wish to bound from below max 1≤i≤m ∣ϑi − pi q ∣ in terms of q, while type I is related with the study of lower bounds for linear combinations |a0 + a1ϑ1 + · · · + amϑm| when a0,..., am are rational integers, not all of which are 0, in terms of the number max0≤i≤m |ai|. After Hermite’s seminal work, F. Lindemann was able to extend the argument and to prove the transcendence of π (hence he solved the old greek problem 8

Abstract. I address the historical emergence of the mathematical infinite, and how we are to take the infinite in and out of mathematics. The thesis is that the mathematical infinite in mathematics is a matter of method. The infinite, of course, is a large topic. At the outset, one can historically discern two overlapping clusters of concepts: (1) wholeness, completeness, universality, absoluteness. (2) endlessness, boundlessness, indivisibility, continuousness. The first, the metaphysical infinite, I shall set aside. It is the second, the mathematical infinite, that I will address. Furthermore, I will address mathematical infinite by considering its historical emergence in set theory and how we are to take it in and out of mathematics. Insofar as physics and, more broadly, science deals with the mathematical infinite through mathematical language and techniques, my remarks should be subsuming and consequent. The main underlying point is that how the mathematical infinite is approached, assimilated, and applied in mathematics is not a matter of “ontological commitment”, of coming to terms with whatever that might mean, but rather of epistemological articulation, of coming to terms through knowledge. The mathematical infinite in mathematics is a matter of method. How we deal with the specific individual issues involving the infinite turns on the narrative we present about how it fits into methodological mathematical frameworks established and being established. The first section discusses the mathematical infinite in historical context, and the second, set theory and the emergence of the mathematical infinite. The third section discusses the infinite in and out of mathematics, and how it is to be taken. §1. The Infinite in Mathematics What role does the infinite play in modern mathematics? In modern mathematics, infinite sets abound both in the workings of proofs and as subject matter in statements, and so do universal statements, often of ∀ ∃ “for all there exists” form, which are indicative of direct engagement with the infinite. In many ways the role of the infinite is importantly “second-order ” in the sense that Frege regarded number generally, in that the concepts of modern mathematics are understood as having infinite instances over a broad range. 1 But

We describe a HOL Light formalization of Hermite’s proof that the base of the natural logarithm e is transcendental. This is the first time a proof of this fact has been formalized in a theorem prover. 1