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Magari’s theorem via the recession frame. (English) Zbl 0621.03009
In a paper of G. Boolos and 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 $$<W^*,R^*,V>$$.
Reviewer: W.Lenzen

##### MSC:
 03B45 Modal logic (including the logic of norms)
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