Where to find Christian philosophy?: Spatiality in John Chrysostom’s counter to Greek Paideia

This article examines the use of the concept philosophia in the writings and homilies of John Chrysostom. Although Chrysostom in his discussion of intellectual achievements draws on a long-standing tradition of Christian apologetics, he lends a new direction to the debate by highlighting the spatiality of philosophy. He not only counters Hellenic paideia with Christian wisdom, but locates these two types of philosophy in the city and the countryside, respectively. The article argues that the spatial dimension is vital to Chrysostom’s view of philosophy as he aims to extend the rural ideal of asceticism to the polis to create a healthy Christian community within the city

Independence, Relative Randomness, and PA Degrees

We study pairs of reals that are mutually Martin-L\"{o}f random with respect to a common, not necessarily computable probability measure. We show that a generalized version of van Lambalgen's Theorem holds for non-computable probability measures, too. We study, for a given real $A$, the \emph{independence spectrum} of $A$, the set of all $B$ so that there exists a probability measure $\mu$ so that $\mu\{A,B\} = 0$ and $(A,B)$ is $\mu\times\mu$-random. We prove that if $A$ is r.e., then no $\Delta^0_2$ set is in the independence spectrum of $A$. We obtain applications of this fact to PA degrees. In particular, we show that if $A$ is r.e.\ and $P$ is of PA degree so that $P \not\geq_{T} A$, then $A \oplus P \geq_{T} 0'$