K. Fine introduced this frame to study incompleteness of S4 logics. V. Shehtman later adapted it to obtain an incomplete superintuitionistic logic. It has served as a useful source of counterexamples: for instance, G. Bezhanishvili, D. Gabelaia, and M. Jibladze employed it in their negative solution to Kuznetsov’s problem for bi-Heyting algebras.
Details of the construction
The Fine poset consists of four countable subsets: A,B,C,D, labelled accordingly as ai,bi,ci,di, etc. They have the following roles: the B and C parts are descending ladder-like constructions (reminiscent of the Mardaev frame and Rieger-Nishimura); the A part forms an antichain, which covers progressively deeper parts of B∪C; and D forms an infinite ascending chain, with designated points below A, and all of them below B∪C. In detail, the cover relations are:
bi+2 < ci, ci+2 < bi
ai < bi+1, ci+1
di < ai, di+1
The Fine general frame adds the point ∞, in the center, which compactifies all three of the above components, and sits between D and B∪C.
p = {c₀}, q = {b₀}
↑di = {dj, aj : j ≥ i} ∪ B ∪ C ∪ {∞}
The algebra is 2-generated, being generated by the two maximal singletons.
The algebra enjoys many interesting features: it is a so-called KM-algebra, a bi-Heyting algebra, and validates the De Jongh-Gabbay axiom bb2. This latter axiom is important in Shehtman's incompleteness proof and the Bezhanishvili-Gabelaia-Jibladze proof, since it is what makes candidate Kripke frames or complete bi-Heyting algebras validate axioms which the Fine general frame does not.
References
- An incomplete logic containing S4.Theoria 40(1), 23–29.DOI
- On Neighbourhood Semantics 30 years later.In S. N. Artëmov, H. Barringer, A. S. d’Avila Garcez, L. C. Lamb, and J. Woods (eds.), We Will Show Them! Essays in Honour of Dov Gabbay, Volume Two. College Publications, pp. 663–692.Author’s bibliography
- A negative solution of Kuznetsov’s problem for varieties of bi-Heyting algebras.Journal of Mathematical Logic 22(3), article 2250013, 21 pp.DOI