In Reverse Mathematics, the axiom system DNR, asserting the existence of diagonally non-recursive functions, is strictly weaker than WWKL0.

A recursively enumerable splitting of an r.e. setAis a pair of r.e. setsBandCsuch thatA=B∪CandB∩C= ⊘. Since for such a splitting degA= degB∪ degC, r.e. splittings proved to be a quite useful notion for investigations into the structure of the r.e. degrees. Important splitting theorems, like Sacks splitting [S1], Robinson splitting [R1] and Lachlan splitting [L3], use r.e. splittings.Since each r.e. splitting of a set induces a splitting of its degree, it is natural to study the relation between the degrees of (...) r.e. splittings and the degree splittings of a set. We say a setAhas thestrong universal splitting property if each splitting of its degree is represented by an r.e. splitting of itself, i.e., if for degA=b∪cthere is an r.e. splittingB, CofAsuch that degB=band degC=c. The goal of this paper is the study of this splitting property.In the literature some weaker splitting properties have been studied as well as splitting properties which imply failure of the SUSP. (shrink)

We show that, in the partial ordering of the computably enumerable computable Lipschitz degrees, there is a degree a>0a>0 such that the class of the degrees which do not cup to a is not bounded by any degree less than a. Since Ambos-Spies [1] has shown that, in the partial ordering of the c.e. identity-bounded Turing degrees, for any degree a>0a>0 the degrees which do not cup to a are bounded by the 1-shift a+1a+1 of a where a+1 (...) follows that the elementary theories of the partial orderings and differ. (shrink)

Ambos-Spies, K. and R.A. Shore, Undecidability and 1-types in the recursively enumerable degrees, Annals of Pure and Applied Logic 63 3–37. We show that the theory of the partial ordering of recursively enumerable Turing degrees is undecidable and has uncountably many 1-types. In contrast to the original proof of the former which used a very complicated O''' argument our proof proceeds by a much simpler infinite injury argument. Moreover, it combines with the permitting technique to get similar results for (...) any ideal of the r.e. degrees. (shrink)

We show that the theory of the partial ordering of the computably enumerable degrees in any given nontrivial interval is undecidable and has uncountably many 1-types.

The changes that have taken place in Ukrainian Protestantism since the independence of the country have repeatedly attracted the attention of domestic religious scholars. V. Yelensky, V. Lyubashchenko, O. Nazarkina, Y. Reshetnikov, V. Titarenko, V. Franchuk, P. Yarotsky, TV, addressed the problem in their works. Further analysis, however, is greatly complicated by the lack of empirical material that would be accompanied by primary interpretations.

We explore the complexity of Sacks’ Splitting Theorem in terms of the mind change functions associated with the members of the splits. We prove that, for any c.e. set A, there are low computably enumerable sets $A_0\sqcup A_1=A$ splitting A with $A_0$ and $A_1$ both totally $\omega ^2$ -c.a. in terms of the Downey–Greenberg hierarchy, and this result cannot be improved to totally $\omega $ -c.a. as shown in [9]. We also show that if cone avoidance is added then there (...) is no level below $\varepsilon _0$ which can be used to characterize the complexity of $A_1$ and $A_2$. (shrink)

Name der Zeitschrift: Zeitschrift für Religionswissenschaft Jahrgang: 25 Heft: 2 Seiten: 263-286.

We show that the partial order of Σ0 3-sets under inclusion is elementarily definable with parameters in the semilattice of r.e. wtt-degrees. Using a result of E. Herrmann, we can deduce that this semilattice has an undecidable theory, thereby solving an open problem of P. Odifreddi.

We prove that every Turing degree a bounding some non-GL₂ degree is recursively enumerable in and above (r.e.a.) some 1-generic degree.

It is shown that, if a, b are recursively enumerable degrees such that 0

Logic and Philosophy of Logic

Logic and Philosophy of Logic, Miscellaneous in Logic and Philosophy of Logic

Model Theory in Logic and Philosophy of Logic

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