The semantic integration of manufacturing systems has been impeded by the lack of rigorous ontologies for specific domains of manufacturing processes and resources. In this paper we present a cutting process ontology for 2D shapes such as sheet metal parts, axiomatized in first-order logic. This ontology is an extension of the ontology of ISO 18629 (Process Specification Language) and an earlier shape ontology first used to support object recognition. The full ontology consists of an axiomatization of all possible ways to change a surface as the result of a cutting process and a taxonomy of cutting processes. All component ontologies are verified using representation theorems.