The HOL-Omega logic A new logic is posited for the widely used HOL theorem prover, as an extension of the existing higher order logic of the HOL4 system. The logic is extended to three levels, adding kinds to the existing levels of types and terms. New types include type operator variables and universal types as in System $F$. Impredicativity is avoided through the stratification of types by ranks according to the depth of universal types. The new system, called HOL-Omega or $mathrm{HOL} _{omega }$, is a merging of HOL4, HOL2P, and major aspects of System $F _{omega }$ from Chapter 30 of [{it B. C. Pierce}, Types and programming languages. Cambridge: MIT Press (2002)]. This document presents the abstract syntax and semantics for the kinds, types, and terms of the logic, as well as the new fundamental axioms and rules of inference. As the new logic is constructed according to the design principles of the LCF approach, the soundness of the entire system depends critically and solely on the soundness of this core.