Jeffry L. Hirst

Weak comparability of well orderings and reverse mathematics

Abstract Two countable well orderings are weakly comparable if there is an order preserving injection of one into the other. We say the well orderings are strongly comparable if the injection is an isomorphism between one ordering and an initial segment of the other. In [5], Friedman announced that the statement “any two countable well orderings are strongly comparable” is equivalent to ATR 0 . Simpson provides a detailed proof of this result in Chapter 5 of [13]. More recently, Friedman has proved that the statement “any two countable well orderings are weakly comparable” is equivalent to ATR 0 . The main goal of this paper is to give a detailed exposition of this result.

The polarized Ramsey’s theorem

We study the effective and proof-theoretic content of the polarized Ramsey’s theorem, a variant of Ramsey’s theorem obtained by relaxing the definition of homogeneous set. Our investigation yields a new characterization of Ramsey’s theorem in all exponents, and produces several combinatorial principles which, modulo bounding for

Reverse Mathematics and Uniformity in Proofs without Excluded Middle

{\Sigma^0_2}

Reverse Mathematics and Recursive Graph Theory

formulas, lie (possibly not strictly) between Ramsey’s theorem for pairs and the stable Ramsey’s theorem for pairs.

Connected components of graphs and reverse mathematics

Abstract We study the complexity of generic reals for computable Mathias forcing in the context of computability theory. The n-generics and weak n-generics form a strict hierarchy under Turing reducibility, as in the case of Cohen forcing. We analyze the complexity of the Mathias forcing relation, and show that if G is any n-generic with n ≥ 2 then it satisfies the jump property G ( n − 1 ) ≡ T G ′ ⊕ ∅ ( n ) . We prove that every such G has generalized high Turing degree, and so cannot have even Cohen 1-generic degree. On the other hand, we show that every Mathias n-generic real computes a Cohen n-generic real.

Hilbert versus Hindman

We show that when certain statements are provable in subsystems of constructive analysis using intuitionistic predicate calculus, related sequential statements are provable in weak classical subsystems. In particular, if a

Reverse mathematics and rank functions for directed graphs

\Pi^1_2

Reverse mathematics and homeomorphic embeddings

sentence of a certain form is provable using E-HA