The metamathematics of scattered linear orderings (Q1106846)

Pursuing the proof-theoretic program of H. Friedman and S. 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, \(\Pi^ 1_ 1\)-CA\({}_ 0\) 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, arithmetical transfinite recursion \(ATR_ 0\) 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.