 Concrete and abstract realizability semantics for a wide variety of formal systems have been developed and studied by logicians and computer scientists; cf. Troelstra [1998] and van Oosten [2002] and [2008]. Variations of the basic notions are especially useful for establishing relative consistency and relative independence of the nonlogical axioms in theories based on intuitionistic logic; some examples are Moschovakis [1971], Lifschitz [1979], and the realizability notions for constructive and intuitionistic set theories developed by Rathjen [2006, 2012] and Chen [2012]. Early abstract realizability notions include the slashes of Kleene [1962, 1963] and Aczel [1968], and Läuchli [1970]. Kohlenbach, Avigad and others have developed realizability interpretations for parts of classical mathematics.