Abstract. We study the reverse mathematics and computability-theoretic strength of (stable) Ramsey’s Theorem for pairs and the related principles COH and DNR. We show that SRT 2 2 implies DNR over RCA0 but COH does not, and answer a question of Mileti by showing that every computable stable 2-coloring of pairs has an incomplete ∆ 0 2 infinite homogeneous set. We also give some extensions of the latter result, and relate it to potential approaches to showing that SRT 2 2 does not imply RT 2 2. 1.

We investigate the complexity of several classical model theoretic theorems about prime and atomic models and omitting types. Some are provable in RCA0, others are equivalent to ACA0. One, that every atomic theory has an atomic model, is not provable in RCA0 but is incomparable with WKL0, more than Π1 1 conservative over RCA0 and strictly weaker than all the combinatorial principles of Hirschfeldt and Shore [2007] that are not Π1 1 conservative over RCA0. A prior-ity argument with Shore blocking shows that it is also Π 1 1-conservative over BΣ2. We also provide a theorem provable by a finite injury priority argument that is conservative over IΣ1 but implies IΣ2 over BΣ2, and a type omitting theorem that is equivalent to the principle that for every X there is a set that is hyperimmune relative to X. Finally, we give a version of the atomic model theorem that is equivalent to the principle that for every X there is a set that is not recursive in X, and is thus in a sense the weakest possible natural principle not true in the ω-model consisting of the recursive sets.

Abstract. We study combinatorial principles weaker than Ramsey’s theorem for pairs over the system RCA0 (Recursive Comprehension Axiom) with Σ0 2-bounding. It is shown that the principles of Cohesiveness (COH), Ascending and Descending Sequence (ADS), and Chain/Antichain (CAC) are all Π1 1-conservative over Σ0 2-bounding. In particular, none of these principles proves Σ0 2-induction. Key words. Reverse mathematics, Π1 1-conservation, RCA0, Σ0 2-

We present some abstract theorems showing how domination properties equivalent to not being GL2 or array recursive can be used to construct sets generic for different notions of forcing. These theorems are then applied to give simple proofs of several old results. We also give a direct uniform proof of a recent result of Ambos-Spies, Ding, Wang and Yu [2009] that every degree above any not in GL2 is recursively enumerable in a 1-generic degree strictly below it. Our major new result is that every array nonrecursive degree is r.e. in some degree strictly below it. Our analysis of array nonrecursiveness and construction of generic sequences below ANR degrees also reveal a new level of uniformity in these types of results.

In 1900 the great mathematician David Hilbert laid down a list of 23 mathematical problems [32] which exercised a great influence on subsequent mathematical research. From the perspective of foundational studies, it is noteworthy that Hilbert’s Problems 1 and 2 are squarely in the area of foundations of mathematics, while Problems 10 and 17 turned out to be closely related to mathematical logic.

Abstract. The Rainbow Ramsey Theorem is essentially an “anti-Ramsey ” theorem which states that certain types of colorings must be injective on a large subset (rather than constant on a large subset). Surprisingly, this version follows easily from Ramsey’s Theorem, even in the weak system RCA0 of reverse mathematics. We answer the question of the converse implication for pairs, showing that the Rainbow Ramsey Theorem for pairs is in fact strictly weaker than Ramsey’s Theorem for pairs over RCA0. The separation involves techniques from the theory of randomness by showing that every 2-random bounds an ω-model of the Rainbow Ramsey Theorem for pairs. These results also provide as a corollary a new proof of Martin’s theorem that the hyperimmune degrees have measure one. 1.

Abstract. Hirschfeldt and Shore have introduced a notion of stability for infinite posets. We define an arguably more natural notion called weak stability, and we study the existence of infinite computable or low chains or antichains, and of infinite Π 0 1 chains and antichains, in infinite computable stable and weakly stable posets. For example, we extend a result of Hirschfeldt and Shore to show that every infinite computable weakly stable poset contains either an infinite low chain or an infinite computable antichain. Our hardest result is that there is an infinite computable weakly stable poset with no infinite Π 0 1 chains or antichains. On the other hand, it is easily seen that every infinite computable stable poset contains an infinite computable chain or an infinite Π 0 1 antichain. In Reverse Mathematics, we show that SCAC, the principle that every infinite stable poset contains an infinite chain or antichain, is equivalent over RCA0 to WSCAC, the corresponding principle for weakly stable posets. 1.

The general enterprise of calibrating the strength of classical mathematical theorems in terms of the axioms (typically of set existence) needed to prove them was begun by Harvey Friedman in [1971] (see also [1967]). His goals were both philosophical and foundational. What existence assumptions are really needed to develop classical mathematics

The objective of this paper is to provide a source of open questions in reverse mathematics and to point to areas where there could be interesting developments. The questions I discuss are mostly known and come from somewhere in the literature. My objective was to compile them in one place and discuss them in the context of related work. The list is definitely not comprehensive, and my

We investigate the reverse-mathematical status of several theorems to the effect that the natural number system is second-order categorical. One of our results is as follows. Define a system to be a triple A,i,f such that A is a set and i ∈ A and f: A → A. A subset X ⊆ A is said to be inductive if i ∈ X and ∀a(a ∈ X ⇒ f(a) ∈ X). The system A,i,f is said to be inductive if the only inductive subset of A is A itself. Define a Peano system to be an inductive system such that f is one-to-one and i /∈ the range of f. The standard example of a Peano system is N,0,S where N = {0,1,2,...,n,...} = the set of natural numbers and S: N → N is given by S(n) = n+1 for all n ∈ N. Consider the statement that all Peano systems are isomorphic toN,0,S. We prove that this statement is logically equivalent to WKL0 over RCA ∗ 0. From this and similar equivalences we