Magari's theorem via the recession frame (Q1090325)

In a paper of \textit{G. Boolos} and \textit{G. Sambin} [ibid. 14, 351-358 (1985; Zbl 0589.03005)] it has already been proved that the normal modal system GH obtained from the basic calculus K by addition of the axiom \[ H\quad L(p\equiv Lp)\supset Lp \] is incomplete in the following sense: there is a formula, viz. the well-known characteristic axiom of S4: \[ (4)\quad Lp\supset LLp, \] which is not a theorem of GH although (4) is valid on all frames \(<W,R>\) which validate H. This paper offers another incompleteness-proof which makes use of the so- called recession-frame \(<W^*,R^*>\) in which \(W^*\) is the set of natural numbers and the accessibility relation \(R^*\) is defined by: \(nR^*m\) iff \(n\leq m+1\). While axiom H is valid in every allowable model based on this frame, formula (4) turns out not to hold in a certain model \(<W^*,R^*,V>\).