Theodore A. Slaman

On the Strength of Ramsey's Theorem for Pairs

We study the proof–theoretic strength and effective content of the infinite form of Ramseys theorem for pairs. Let RT k n denote Ramseys theorem for k –colorings of n –element sets, and let RT n denote (∀ k )RT k n . Our main result on computability is: For any n ≥ 2 and any computable (recursive) k –coloring of the n –element sets of natural numbers, there is an infinite homogeneous set X with X ″ ≤ T 0 ( n ) . Let I Σ n and B Σ n denote the Σ n induction and bounding schemes, respectively. Adapting the case n = 2 of the above result (where X is low 2 ) to models of arithmetic enables us to show that RCA 0 + I Σ 2 + RT 2 2 is conservative over RCA 0 + I Σ 2 for Π 1 1 statements and that RCA 0 + I Σ 3 + RT 2 is Π 1 1 -conservative over RCA 0 + I Σ 3 . It follows that RCA 0 + RT 2 2 does not imply B Σ 3 . In contrast, J. Hirst showed that RCA 0 + RT 2 does imply B Σ 3 , and we include a proof of a slightly strengthened version of this result. It follows that RT 2 is strictly stronger than RT 2 2 over RCA 0 .

Randomness and Recursive Enumerability

One recursively enumerable real

Relative to any nonrecursive set

\alpha

Interpretability and Definability in the Recursively Enumerable Degrees

dominates another one

Extremes in the degrees of inferability

\beta

The atomic model theorem and type omitting

if there are nondecreasing recursive sequences of rational numbers

The Metamathematics Of Stable Ramsey’s Theorem For Pairs

(a[n]:n\in\omega)

Complementation in the Turing Degrees

approximating

When oracles do not help

\alpha

Measures and Their Random Reals

and