In this paper we calibrate the exact proof-theoretic strength of Kruskal's theorem, thereby giving, in some sense, the most elementary proof of Kruskal's theorem. Furthermore, these investigations give rise to ordinal analyses of restricted bar induction.

Pursuing the proof-theoretic program of Friedman and Simpson, we begin the study of the metamathematics of countable linear orderings by proving two main results. Over the weak base system consisting of arithmetic comprehension, II 1 1 -CA0 is equivalent to Hausdorff's theorem concerning the canonical decomposition of countable linear orderings into a sum over a dense or singleton set of scattered linear orderings. Over the same base system, ATR0 is equivalent to a version of the Continuum Hypothesis for linear orderings, (...) which states that every countable linear ordering either has countably many or continuum many Dedekind cuts. (shrink)

Two linear orderings are equimorphic if each can be embedded into the other. We prove that every hyperarithmetic linear ordering is equimorphic to a recursive one. On the way to our main result we prove that a linear ordering has Hausdorff rank less than $\omega _{1}^{\mathit{CK}}$ if and only if it is equimorphic to a recursive one. As a corollary of our proof we prove that, given a recursive ordinal α, the partial ordering of equimorphism types of linear orderings of (...) Hausdorff rank at most α ordered by embeddablity is recursively presentable. (shrink)

A rank function for a directed graph G assigns elements of a well ordering to the vertices of G in a fashion that preserves the order induced by the edges. While topological sortings require a one-to-one matching of vertices and elements of the ordering, rank functions frequently must assign several vertices the same value. Theorems stating basic properties of rank functions vary significantly in logical strength. Using the techniques of reverse mathematics, we present results that require the subsystems ${\ensuremath{\vec{RCA}_0}}$ , (...) ${{\vec{ACA}_0}}$ , ${{\vec{ATR}_0}}$ , and ${{\vec{\Pi^1_1 - CA}_0}}$. (shrink)