2009/03/28 by Itaï Ben Yaacov · 1 citation
Mathematics · #math.LO #msc:03C35 #msc:03C90 #msc:03C95
paper · pdf · doi:10.1142/s0219061308000737
published as Journal of Mathematical Logic 8, 2 (2008) 197-223
arxiv created 2009/03/28 · arxiv updated 2010/04/22
We present an adaptation of continuous first order logic to unbounded metric structures. This has the advantage of being closer in spirit to C. Ward Henson's logic for Banach space structures than the unit ball approach (which has been the common approach so far to Banach space structures in continuous logic), as well as of applying in situations where the unit ball approach does not apply (i.e., when the unit ball is not a definable set). We also introduce the process of single point emboundment (closely related to the topological single point compactification), allowing to bring unbounded structures back into the setting of bounded continuous first order logic. Together with results from \citeBenYaacov:Perturbations regarding perturbations of bounded metric structures, we prove a Ryll-Nardzewski style characterisation of theories of Banach spaces which are separably categorical up to small perturbation of the norm. This last result is motivated by an unpublished result of Henson.