  We propose a new method for constructing Turing ideals satisfying principles
of reverse mathematics below the Chain-Antichain Principle (CAC). Using this
method, we are able to prove several new separations in the presence of Weak
Konig's Lemma (WKL), including showing that CAC+WKL does not imply the thin set
theorem for pairs, and that the principle "the product of well-quasi-orders is
a well-quasi-order" is strictly between CAC and the Ascending/Descending
Sequences principle, even in the presence of WKL.

Logic