Set Theories and OWL Full

Abstract

Set theories are the theoretical foundation of Semantic Web languages, RDF and OWL, as they are the foundation of mathematical theories. The W3C document of RDF Semantics mentions Zermelo-Fraenkel (ZF) Set Theory, and the W3C document of OWL Semantics invokes the comprehension principle in order to materialize RDF entities in OWL. However, this invocation arose from misunderstandings of set theories, and what is worse, it caused misdirected criticisms against RDF and OWL Full under the pretense that 'comprehension principle allows the paradox to invade upon systems'. Aiming to rescue RDF and OWL Full theory from such theoretical disorder, this paper firstly reviews the history of comprehension principle in set theories, and gives an overview of set theories of Cantor, ZF, and additionally KIF 3.0. Then, the theoretical foundation of OWL Full is introduced with the discussion of Russell's Ramified Type Theory.