The prime number theorem, established by Hadamard and de la Vallée Poussin independently in 1896, asserts that the density of primes in the positive integers is asymptotic to 1 / ln x. Whereas their proofs made serious use of the methods of complex analysis, elementary proofs were provided by Selberg and Erdös in 1948. We describe a formally verified version of Selberg’s proof, obtained using the Isabelle proof assistant. 1

Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of Göttingen and elsewhere in the 1920s

Abstract. This paper explores the set theoretic assumptions used in the current published proof of Fermat’s Last Theorem, how these assumptions figure in the methods Wiles uses, and the currently known prospects for a proof using weaker assumptions. Does the proof of Fermat’s Last Theorem (FLT) go beyond Zermelo Fraenkel set theory (ZFC)? Or does it merely use Peano Arithmetic (PA) or some weaker fragment of that? The answers depend on what is meant by “proof ” and “use, ” and are not entirely known. This paper surveys the current state of these questions and briefly sketches the methods of cohomological number theory used in the existing proof. The existing proof of FLT is Wiles [1995] plus improvements that do not yet change its character. Far from self-contained it has vast prerequisites merely introduced in the 500 pages of [Cornell et al., 1997]. We will say that the assumptions explicitly used in proofs that Wiles cites as steps in his own are “used in fact in the published proof. ” It is currently unknown what assumptions are “used in principle ” in the sense of being proof-theoretically indispensable to FLT. Certainly much less than ZFC is used in principle, probably nothing beyond PA, and perhaps much less than that. The oddly contentious issue is universes, often called Grothendieck universes. 1 On ZFC foundations a universe is an uncountable transitive set U such that 〈U, ∈ 〉 satisfies the ZFC axioms in the nicest way: it contains the powerset of each of its elements, and for any function from an element of U to U the range is also an element of U. This is much stronger than merely saying 〈U, ∈ 〉 satisfies the ZFC axioms. We do not merely say the powerset axiom “every set has a powerset ” is true with all quantifiers relativized to U. Rather, we require “for every set x ∈ U, the powerset of x is also in U ”

Many mathematicians have sought ‘pure ’ proofs of theorems. There are different takes on what a ‘pure ’ proof is, though, and it’s important to be clear on their differences, because they can easily be conflated. In this paper I want to distinguish between two of them. I want to begin with a classical formulation of purity, due to Hilbert: In modern mathematics one strives to preserve the purity of the method, i.e. to use in the proof of a theorem as far as possible only those auxiliary means that are required by the content of the theorem. 1 A pure proof of a theorem, then, is one that draws only on what is “required by the content of the theorem”. I want to continue by distinguishing two ways of understanding “required by the content of [a] theorem”, and hence of understanding what counts as a pure proof of a theorem. I’ll then provide three examples that I think show how these two understandings of contentrequirement, and thus of purity, diverge. 1. Logical purity The first way of understanding purity that I want to consider takes what is “required by the content of [a] theorem ” to be just what suffices for proving that theorem. The ideal is what Hilbert pursued in his Grundlagen der Geometrie: to determine which of the axioms he gave for geometry are sufficient for proving interesting geometric theorems, such that if any of those axioms were left out, the theorem would no longer follow. 2 As an first approximation, then, this ideal can be

statements reformulated to talk about primes

Dedicated to Grigori Mints in honor of his seventieth birthday. Almost from the inception of Hilbert’s program, foundational and structural efforts in proof theory have been directed towards the goal of clarifying the computational content of modern mathematical methods. This essay surveys various methods of extracting computational information from proofs in classical first-order arithmetic, and reflects on some of the relationships between them. Variants of the Gödel-Gentzen doublenegation translation, some not so well known, serve to provide canonical and efficient computational interpretations of that theory. 1