A computability theoretic equivalent to Vaught's conjecture

We prove that, for every theory $T$ which is given by an ${\mathcal L}_{\omega_1,\omega}$ sentence, $T$ has less than $2^{\aleph_0}$ many countable models if and only if we have that, for every $X\in 2^\omega$ on a cone of Turing degrees, every $X$-hyperarithmetic model of $T$ has an $X$-computable copy. We also find a concrete description, relative to some oracle, of the Turing-degree spectra of all the models of a counterexample to Vaught's conjecture

Pure Σ2\Sigma_2-Elementarity beyond the Core

We display the entire structure ${\cal R}_2$ coding $\Sigma_1$- and $\Sigma_2$-elementarity on the ordinals. This will enable the analysis of pure $\Sigma_3$-elementary substructures.Comment: Extensive rewrite of the introduction. Mathematical content of sections 2 and 3 unchanged, extended introduction to section 2. Removed section 4. Theorem 4.3 to appear elsewhere with corrected proo

Levels of discontinuity, limit-computability, and jump operators

We develop a general theory of jump operators, which is intended to provide an abstraction of the notion of "limit-computability" on represented spaces. Jump operators also provide a framework with a strong categorical flavor for investigating degrees of discontinuity of functions and hierarchies of sets on represented spaces. We will provide a thorough investigation within this framework of a hierarchy of $\Delta^0_2$-measurable functions between arbitrary countably based $T_0$-spaces, which captures the notion of computing with ordinal mind-change bounds. Our abstract approach not only raises new questions but also sheds new light on previous results. For example, we introduce a notion of "higher order" descriptive set theoretical objects, we generalize a recent characterization of the computability theoretic notion of "lowness" in terms of adjoint functors, and we show that our framework encompasses ordinal quantifications of the non-constructiveness of Hilbert's finite basis theorem

Investigations of subsystems of second order arithmetic and set theory in strength between Pi-1-1-CA and delta-1-2-CA+BI: part I

This paper is the rst of a series of two. It contains proof{theoretic investigations on subtheories of second order arithmetic and set theory. Among the principles on which these theories are based one nds autonomously iterated positive and monotone inductive de ni- tions, 1 1 trans nite recursion, 1 2 trans nite recursion, trans nitely iterated 1 1 dependent choices, extended Bar rules for provably de nable well-orderings as well as their set-theoretic counterparts which are based on extensions of Kripke-Platek set theory. This rst part intro- duces all the principles and theories. It provides lower bounds for their strength measured in terms of the amount of trans nite induction they achieve to prove. In other words, it determines lower bounds for their proof-theoretic ordinals which are expressed by means of ordinal representation systems. The second part of the paper will be concerned with ordinal analysis. It will show that the lower bounds established in the present paper are indeed sharp, thereby providing the proof-theoretic ordinals. All the results were obtained more then 20 years ago (in German) in the author's PhD thesis [43] but have never been published before, though the thesis received a review (MR 91m#03062). I think it is high time it got published

Semi-continuous Sized Types and Termination

Some type-based approaches to termination use sized types: an ordinal bound for the size of a data structure is stored in its type. A recursive function over a sized type is accepted if it is visible in the type system that recursive calls occur just at a smaller size. This approach is only sound if the type of the recursive function is admissible, i.e., depends on the size index in a certain way. To explore the space of admissible functions in the presence of higher-kinded data types and impredicative polymorphism, a semantics is developed where sized types are interpreted as functions from ordinals into sets of strongly normalizing terms. It is shown that upper semi-continuity of such functions is a sufficient semantic criterion for admissibility. To provide a syntactical criterion, a calculus for semi-continuous functions is developed.Comment: 33 pages, extended version of CSL'0