We settle a number of questions concerning definability in first order logic with an extra predicate symbol ranging over semi-linear sets. We give new results both on the positive and negative side: we show that in first-order logic one cannot query a semi-linear set as to whether or not it contains a line, or whether or not it contains the line segment between two given points. However, we show that some of these queries become definable if one makes small restrictions on the semi-linear sets considered.
"Definability with a predicate for a semi-linear set." J. Symbolic Logic 68 (1) 319 - 351, March 2003. https://doi.org/10.2178/jsl/1045861516