Bahareh Afshari

Post's Programme for the Ershov Hierarchy

This article extends Posts; programme to finite levels of the Ershov hierarchy of Δ2 sets. Our initial characterization, in the spirit of Post (1994, Bulletin of the American Mathematical Society, 50, 284–316), of the degrees of the immune and hyperimmune n-enumerable sets leads to a number of results setting other immunity properties in the context of the Turing and wtt-degrees derived from the Ershov hierarchy. For instance, we show that any n-enumerable hyperhyperimmune set must be co-enumerable, for each n ≥ 2. The situation with regard to the wtt-degrees is particularly interesting, as demonstrated by a range of results concerning the wtt-predecessors of hypersimple sets. Finally, we give a number of results directed at characterizing basic classes of n-enumerable degrees in terms of natural information content. For example, a 2-enumerable degree contains a 2-enumerable dense immune set iff it contains a 2-enumerable r-cohesive set iff it bounds a high enumerable set. This result is extended to a characterization of n-enumerable degrees which bound high enumerable degrees. Furthermore, a characterization for n-enumerable degrees bounding only low2 enumerable degrees is given.

Reverse mathematics and well-ordering principles: A pilot study

Abstract The larger project broached here is to look at the generally Π 2 1 sentence “if X is well-ordered then f ( X ) is well-ordered”, where f is a standard proof-theoretic function from ordinals to ordinals. It has turned out that a statement of this form is often equivalent to the existence of countable coded ω -models for a particular theory T f whose consistency can be proved by means of a cut elimination theorem in infinitary logic which crucially involves the function f . To illustrate this theme, we prove in this paper that the statement “if X is well-ordered then e X is well-ordered” is equivalent to ACA 0 + . This was first proved by Marcone and Montalban [Alberto Marcone, Antonio Montalban, The epsilon function for computability theorists, draft, 2007] using recursion-theoretic and combinatorial methods. The proof given here is principally proof-theoretic, the main techniques being Schutte’s method of proof search (deduction chains) [Kurt Schutte, Proof Theory, Springer-Verlag, Berlin, Heidelberg, 1977] and cut elimination for a (small) fragment of L ω 1 , ω .

A Note on the Theory of Positive Induction, ID 1

The article shows a simple way of calibrating the strength of the theory of positive induction,

Immunity properties and the n -C.E. hierarchy

{{\rm ID}^{*}_{1}}

On the Herbrand content of LK.

. Crucially the proof exploits the equivalence of

Relative computability and the proof−theoretic strength of some theories

{\Sigma^{1}_{1}}