Foundations of mathematics is the study of the most basic concepts and logical structure of
mathematics, with an eye to the unity of human knowledge. Almost all of the problems
studied in this book are motivated by an overriding foundational question: What are the …

Abstract § 0. Introduction. What follows is a write-up of my contribution to the symposium
“Hilbert's Program Sixty Years Later” which was sponsored jointly by the American
Philosophical Association and the Association for Symbolic Logic. The symposium was held …

The purpose of this paper is to answer some special cases of the following question: Which
set existence axioms are needed in order to prove the theorems of countable algebra? The
set existence axioms which we consider are formulated in the context of weak subsystems of …

Let kϵω, where ϵ is the set of all natural numbers. Ramsey's Theorem deals with colorings
of the k-element subsets of ω. Our dual form deals with colorings of the k-element partitions
of ω. Let (ω) k (respectively (ω) ω) be the set of all partitions of ω having exactly k …

In this paper we exposit some as yet unpublished results of Harvey Friedman. These results
provide the most dramatic examples so far known of mathematically meaningful theorems of
finite combinatorics which are unprovable in certain logical systems. The relevant logical …

We investigate the provability or nonprovability of certain ordinary mathematical theorems
within certain weak subsystems of second order arithmetic. Specifically, we consider the
Cauchy/Peano existence theorem for solutions of ordinary differential equations, in the …

Recursive functions of ordinals were invented by Takeuti [1] and subsequently specialized
by Kripke [21 and Platek [3] to~ l admissible initial segments of the ordinals, and by Kreisel
and Sacks [41 to the recursive ordinals. Takeuti's principal result was: iff is recursive and/3 is …

Let< D, U> be the semilattice of degrees of recursive unsolvability. The main result of this
paper is that the first-order theory of< D, U> is recursively isomorphic to the truth set of
second-order arithmetic (Corollary 5.6). We also obtain a strong result concerning first-order …

Godel's incompleteness theorem, published in 1931, says that in any given formal system for
mathematics there must exist undecidable number-theoretic statements, ie sentences in the
language of arithmetic which are neither prsvable nor refutable in the given system. This is a …

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