As remarked above, if KF proves \(T\ulcorner \phi \urcorner\) for some sentence \(\phi\) then \(\phi\) holds in all Kripke fixed point models. In particular, there are \(2^{\aleph_0}\) fixed points that form a model of the internal theory of KF. Thus from the perspective of KF, the least fixed point (from which Kripke’s theory is defined) is not singled out. Burgess (2014) provides an expansion of KF, named \(\mu\)KF, that attempts to capture the minimal Kripkean fixed point. KF is expanded by additional axioms that express that the internal theory of KF is the smallest class closed under the defining axioms for Kripkean truth. This can be formulated as a single axiom schema that states, for each open formula \(\phi\), 