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The Kuznetsov-Index of a modal logic is the least cardinal µ such that any consistent formula has a Kripke-model of size  µ if it has a Kripke-model at all. The Kuznetsov-Spectrum is the set of all Kuznetsov-Indices of modal logics with countably many operators. It has been shown by Thomason that there are tense logics with Kuznetsov-Index i!+!. Futhermore, Chagrov has constructed an extension of K4 with Kuznetsov-Index i!. We will show here that for each countable ordinalthere are logics with Kuznetsov-Index i. Furthermore, we show that the Kuznetsov-Spectrum is identical to the spectrum of indices for 11-theories, which is likewise defined. A particular consequence is the following. If inaccessible (weakly compact, measurable) cardinals exist, then the least inaccessible (weakly compact, measurable) cardinal is also a Kuznetsov-Index. Suppose ϕ is an elementary formula and that ϕ is consistent with an elementary theory T in a countable language. Then there exists a countable T -model for ϕ. Furthermore, in any infinite cardinalitythere exists a T -model for ϕ. For other languages this does not need to hold, for example for second-order logic. Modal logic also has first-order structures, namely Kripke-frames, but the language is a fragment of monadic second order predicate logic. Moreover, modal logics neither necessarily define first-order classes of frames nor is every first-order definable class of frames modally definable (see (1)). The same is true for intermediate logics. Therefore, Hosoi and Ono (9) raised the following question: Do there exist intermediate logics Λ such that Λ is complete but not complete with respect to countable Kripke-frames?

Modal Logics That Need Very Large Frames