A computability structure on a metric space is a set of sequences which satisfy certain conditions. Of a particular interest are those computability structures which contain a dense sequence, so called separable computability structures. In this paper we observe maximal computability structures which are more general than separable computability structures and we examine their properties. In particular, we examine maximal computability structures on subspaces of Euclidean space, we give their characterization and we investigate conditions under which a maximal computability structure on such a space is unique. We also give a characterization of separable computability structures on a segment.

This is a companion to a paper by the authors entitled “Gödel’s natural deduction,” which presented and made comments about the natural deduction system in Gödel’s unpublished notes for the elementary logic course he gave at the University of Notre Dame in 1939. In that earlier paper, which was itself a companion to a paper that examined the links between some philosophical views ascribed to Gödel and general proof theory, one can find a brief summary of Gödel’s notes for the Notre Dame course. In order to put the earlier paper in proper perspective, a more complete summary of these interesting notes, with comments concerning them, is given here.

The isomorphism invariance criterion of logical nature has much to commend it. It can be philosophically motivated by the thought that logic is distinctively general or topic neutral. It is capable of precise set-theoretic formulation. And it delivers an extension of ‘logical constant’ which respects the intuitively clear cases. Despite its attractions, the criterion has recently come under attack. Critics such as Feferman, MacFarlane and Bonnay argue that the criterion overgenerates by incorrectly judging mathematical notions as logical. We consider five possible precisifications of the overgeneration argument and find them all unconvincing.

This paper surveys the interactions between Russell and Gödel, both personal and intellectual. After a description of Russell’s influence on Gödel, it concludes with a discussion of Russell’s reaction to the incompleteness theorems.