Separating principles below
In this paper, we study the Ramsey-type weak Kőnig's Lemma, written , using a technique introduced by Lerman, Solomon, and the second author. This technique uses iterated forcing to construct an ω-model satisfying one principle T1 but not another T2. The technique often allows one to translate a “one step” construction (building an instance of T2 along with a collection of solutions to each computable instance of T1) into an ω-model separation (building a computable instance of T2 together with a Turing ideal where T1 holds but this instance has no solution). We illustrate this translation by separating from (reproving a result of Ambos-Spies, Kjos-Hanssen, Lempp, and Slaman), and then apply this technique to separate from (which has been shown separately by Bienvenu, Patey, and Shafer).