We give a survey of the study of nonstandard models in recursion theory and reverse mathematics. We discuss the key notions and techniques in effective computability in nonstandard models, and their applications to problems concerning combinatorial principles in subsystems of second order arithmetic. Particular attention is given to principles related to Ramsey’s Theorem for Pairs.

Related Works: Part II: C. T. Chong, Yue Yang. $\Sigma_2$ Induction and Infinite Injury Priority Argument, Part II: Tame $\Sigma_2$ Coding and the Jump Operator. Ann. Pure Appl. Logic, vol. 87, no. 2, 103--116. Mathematical Reviews : MR1490049 Part III: C. T. Chong, Lei Qian, Theodore A. Slaman, Yue Yang. $\Sigma_2$ Induction and Infinite Injury Priority Argument, Part III: Prompt Sets, Minimal Paries and Shoenfield's Conjecture. Mathematical Reviews : MR1818378.

We study the minimal enumeration degree problem in models of fragments of Peano arithmetic () and prove the following results: in any model M of Σ2 induction, there is a minimal enumeration degree if and only if M is a nonstandard model. Furthermore, any cut in such a model has minimal e-degree. By contrast, this phenomenon fails in the absence of Σ2 induction. In fact, whether every Σ2 cut has minimal e-degree is independent of the Σ2 bounding principle.

We give a survey of the study of nonstandard models in recursion theory and reverse mathematics. We discuss the key notions and techniques in effective computability in nonstandard models. and their applications to problems concerning combinatorial principles in subsystems of second order arithmetic. Particular attention is given to principles related to Ramsey's Theorem for Pairs.

We study the problem of existence of maximal chains in the Turing degrees. We show that: 1. ZF+DC+"There exists no maximal chain in the Turing degrees" is equiconsistent with ZFC+"There exists an inaccessible cardinal"; 2. For all a ∈ 2ω.(ω₁)L[a] = ω₁ if and only if there exists a $\Pi _{1}^{1}[a]$ maximal chain in the Turing degrees. As a corollary, ZFC + "There exists an inaccessible cardinal" is equiconsistent with ZFC + "There is no (bold face) $\utilde{\Pi}{}_{1}^{1}$ maximal chain of (...) Turing degrees". (shrink)