Scott induction and closure under \(\omega\)-sups
The paper presents a attempt to giving a semantic analysis of Scott's induction axiom of LCF, the logic of continuous functionals, \[ (\forall f)[R(f,\perp)\wedge \forall x[R(f,x)\to R(f,f(x))]\to R(f,Yf)]. \] This axiom is valid for relations \(R\subseteq [P\to P]\times P\) satisfying the condition \[ (*)\quad (\forall f)\{x: R(f,x)\} \] is closed under \(\omega\)-sups, which has often been used as a method for proving that certain definable R satisfies Scott's induction axiom. The paper proves that (*) is not a necessary condition, and hence the proof method is not complete. From certain definability and effectiveness aspects of the relation between being Scott-induction and closed under \(\omega\)-sups, some interesting classes of \(\omega\)-cpo's are shown to be finitely axiomatized in the language of LCF.
scientific article; zbMATH DE number 3988696