Stephen G. Simpson

A Dual Form of Ramsey's Theorem

Abstract Let k ϵ ω , where ϵ is the set of all natural numbers. Ramseys 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 (respectively infinitely many) blocks. Given X ϵ ( ω ) ω let ( X ) k be the set of all Y ϵ ( ω ) k such that Y is coarser than X. Dual Ramsey Theorem . If ( ω ) k = C 0 ∪ … ∪ C t −1 where each C i is Borel then there exists X ϵ ( ω ) ω such that ( X ) k ⊆ C i for some i l . Dual Galvin-Prikry Theorem . Same as before with k replaced by ω. We also obtain dual forms of theorems of Ellentuck and Mathias. Our results also provide an infinitary generalization of the Graham-Rothschild “parameter set” theorem [ Trans. Amer. Math. Soc. 159 (1971), 257–292] and a new proof of the Halpern-Lauchli Theorem [ Trans. Amer. Math. Soc. 124 (1966), 360–367].

Partial Realizations of Hilbert's Program

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 on December 29, 1985 in Washington, D. C. The panelists were Solomon Feferman, Dag Prawitz and myself. The moderator was Wilfried Sieg. The research which I discuss here was partially supported by NSF Grant DMS-8317874. I am grateful to the organizers of this timely symposium on an important topic. As a mathematician I particularly value the opportunity to address an audience consisting largely of philosophers. It is true that I was asked to concentrate on the mathematical aspects of Hilbert’s Program. But since Hilbert’s Program is concerned solely with the foundations of mathematics, the restriction to mathematical aspects is really no restriction at all. Hilbert assigned a special role to a certain restricted kind of mathematical reasoning known as finitistic. The essence of Hilbert’s Program was to justify all of set-theoretical mathematics by means of a reduction to finitism. It is

Which Set Existence Axioms are Needed to Prove the Cauchy/Peano Theorem for Ordinary Differential Equations?

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 context of the formal system RCA 0 whose principal axioms are comprehension and induction. Our main result is that, over RCA 0 , the Cauchy/Peano Theorem is provably equivalent to weak Konigs lemma, i.e. the statement that every infinite {0, 1}-tree has a path. We also show that, over RCA 0 , the Ascoli lemma is provably equivalent to arithmetical comprehension, as is Osgoods theorem on the existence of maximum solutions. At the end of the paper we digress to relate our results to degrees of unsolvability and to computable analysis.

Nonprovability of Certain Combinatorial Properties of Finite Trees

Abstract 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 systems, ATR 0 and Π 1 1 -CA 0 , are well known as relatively strong fragments of second order arithmetic. The unprovable combinatorial theorems are concerned with embeddability properties of finite trees. Friedmans methods are based in part on the existence of a close relationship between finite trees on the one hand, and systems of ordinal notations which occur in Gentzen-style proof theory on the other.

Measure theory and weak König's lemma

We develop measure theory in the context of subsystems of second order arithmetic with restricted induction. We introduce a combinatorial principleWWKL (weak-weak Königs lemma) and prove that it is strictly weaker thanWKL (weak Königs lemma). We show thatWWKL is equivalent to a formal version of the statement that Lebesgue measure is countably additive on open sets. We also show thatWWKL is equivalent to a formal version of the statement that any Borel measure on a compact metric space is countably additive on open sets.

Degrees of Unsolvability: A Survey of Results

Publisher Summary The structure of the degrees is essentially a classification of all sets of integers according to complexity. The recursive sets of integers comprise the lowest level of complexity. A “degree” is just an equivalence class of sets of integers, under the following equivalence relation: S 1 is recursive relative to S 2 , and S 2 is recursive relative to S 1 . This chapter presents the theorems whose statements highlight the structure and uses of degrees. The chapter also discusses the structure of the degrees without jump, the jump operator, and degrees of complete theories.

Almost everywhere domination and superhighness

Let ! denote the set of natural numbers. For functions f,g : ! ! !, we say that f is dominated by g if f(n) < g(n) for all but finitely many n 2 !. We consider the standard “fair coin” probability measure on the space 2 ! of infinite sequences of 0’s and 1’s. A Turing oracle B is said to be almost everywhere dominating if, for measure one many X 2 2 ! , each function which is Turing computable from X is dominated by some function which is Turing computable from B. Dobrinen and Simpson have shown that the almost everywhere domination property and some of its variant properties are closely related to the reverse mathematics of measure theory. In this paper we exposit some recent results of Kjos-Hanssen, Kjos-Hanssen/Miller/Solomon, and others concerning LR-reducibility and almost everywhere domination. We also prove the following new result: If B is almost everywhere dominating, then B is superhigh, i.e., 0 00 is truth-table computable from B 0 , the Turing jump of B.

Which set existence axioms are needed to prove the separable Hahn-Banach theorem?

Abstract We work in the context of weak subsystems of second order arithmetic. RCA0 is the system with Δ10 comprehension and Σ10 induction on the natural numbers. WKL0 is RCA0 plus weak Konigs lemma for trees of finite sequences of 0s and 1s. Within RCA0 we encode a separable Banach space  as a countable normed space A over Q . Points of  are Cauchy sequences from A which converge at the rate of at least 2−n. We show that the Hahn-Banach theorem for separable Banach spaces is provably equivalent to WKL0 over RCA0. Thus, once again, WKL0 is revealed as mathematically powerful, despite being proof theoretically equivalent to primitive recursive arithmetic.

A degree-theoretic definition of the ramified analytical hierarchy☆

Let D be the set of all (Turing) degrees, < the usual partial ordering of D and j the (Turing) jump operator on D. The following relations are shown to be first-order definable in the structure D = 〈D, ⩽, j〉 : d1 is hyperarithmetical in d2, d1 is the hyperjump of d2, d1 is ramified analytical in d2 (Corollaries 4.6,4.13,4.16). A first-order, degree theoretic definition of the ramified analytical hierarchy is obtained (Theorem 5.6). A first-order sentence is found which is true in D if the universe is (a generic extension of) L, and false in D if 0# exists (Corollary 4.7). The question of whether the notion of uniform upper bound is degree theoretically definable is investigated (Section 6). Exact pairs of upper bounds are used to replace analytical definability by arithmetical definability (Theorem 3.1).

Located sets and reverse mathematics

LetX be a compact metric space. A closed setK X is located if the distance functiond(x;K )e xists as ac ontinuous real- valued function on X; weakly located if the predicate d(x;K) >r is 0 allowing parameters. The purpose of this paper is to explore the concepts of located and weakly located subsets of a compact separable metric space in the context of subsystems of second order arithmetic such as RCA0, WKL0 and ACA0. We also give some applications of these concepts by discussing some versions of the Tietze extension theorem. In particular we prove an RCA0 version of this result for weakly located closed sets. 1. Introduction and Summary of Results This paper is part of the program known as Reverse Mathematics. This program investigates what set existence axioms are needed in or- der to prove specic mathematical theorems. It consists of establishing the weakest subsystem of second order arithmetic in which a theorem of ordinary mathematics can be proved. The basic reference for this program is Simpsons monograph (17) while an overview can be found in (15). In this paper we carry out a Reverse Mathematics study of the con- cept of located subsets of a compact complete separable metric space. This concept arises naturally in the context of metric spaces. Even if with a dierent aim, it plays a fundamental role in the work of Bishop and Bridges (1). Bishop and Bridges proved a constructive version of the well known Tietze extension theorem for located closed sets in a compact space and uniformly continuous functions with modulus of uniform continuity. In this paper we prove an RCA0 version of this result for weakly located closed sets. The version of Tietzes theorem for continuous functions and non-compact spaces has been studied by Brown in (2).