Online research in philosophy

We study the complexity of (finitely-valued and transfinitely-valued) Euclidean functions for computable Euclidean domains. We examine both the complexity of the minimal Euclidean function and any Euclidean function. Additionally, we draw some conclusions about the proof-theoretical strength of minimal Euclidean functions in terms of reverse mathematics.

The goal of this paper is to announce there is a single orbit of the c.e. sets with inclusion, ε, such that the question of membership in this orbit is ${\Sigma _1^1 }$ -complete. This result and proof have a number of nice corollaries: the Scott rank of ε is $\omega _1^{{\rm{CK}}}$ + 1; not all orbits are elementarily definable; there is no arithmetic description of all orbits of ε; for all finite α ≥ 9, there is a properly $\Delta (...) _\alpha ^0 $ orbit (from the proof). (shrink)

Schnorr randomness is a notion of algorithmic randomness for real numbers closely related to Martin-Löf randomness. After its initial development in the 1970s the notion received considerably less attention than Martin-Löf randomness, but recently interest has increased in a range of randomness concepts. In this article, we explore the properties of Schnorr random reals, and in particular the c.e. Schnorr random reals. We show that there are c.e. reals that are Schnorr random but not Martin-Löf random, and provide a new (...) characterization of Schnorr random real numbers in terms of prefix-free machines. We prove that unlike Martin-Löf random c.e. reals, not all Schnorr random c.e. reals are Turing complete, though all are in high Turing degrees. We use the machine characterization to define a notion of "Schnorr reducibility" which allows us to calibrate the Schnorr complexity of reals. We define the class of "Schnorr trivial" reals, which are ones whose initial segment complexity is identical with the computable reals, and demonstrate that this class has non-computable members. (shrink)

Given two incomparable c.e. Turing degrees a and b, we show that there exists a c.e. degree c such that c = (a ⋃ c) ⋂ (b ⋃ c), a ⋃ c | b ⋃ c, and c < a ⋃ b.

We prove that a (recursively) enumerable degree is contiguous iff it is locally distributive. This settles a twenty-year old question going back to Ladner and Sasso. We also prove that strong contiguity and contiguity coincide, settling a question of the first author, and prove that no m-topped degree is contiguous, settling a question of the first author and Carl Jockusch [11]. Finally, we prove some results concerning local distributivity and relativized weak truth table reducibility.