THE KUZNETSOV-GERČIU AND RIEGER-NISHIMURA LOGICS The Boundaries of the Finite Model Property

We give a systematic method of constructing extensions of the Kuznetsov-Gerčiu logic KG without the finite model property (fmp for short), and show that there are continuum many such. We also introduce a new technique of gluing of cyclic intuitionistic descriptive frames and give a new simple proof of Gerčiu’s result [9, 8] that all extensions of the RiegerNishimura logic RN have the fmp. Moreover, we show that each extension of RN has the poly-size model property, thus improving on [9]. Furthermore, for each function f : ω → ω, we construct an extension Lf of KG such that Lf has the fmp, but does not have the f -size model property. We also give a new simple proof of another result of Gerčiu [9] characterizing the only extension of KG that bounds the fmp for extensions of KG. We conclude the paper by proving that RN.KC = RN + (¬p ∨ ¬¬p) is the only pre-locally tabular extension of KG, introduce the internal depth of an extension L of RN, and show that L is locally tabular if and only if the internal depth of L is finite.