Modal Logics That Need Very Large Frames

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 iω+ω. Futhermore, Chagrov has constructed an extension of K4 with Kuznetsov–Index iω . We will show here that for each countable ordinal λ there are logics with Kuznetsov–Index iλ. Furthermore, we show that the Kuznetsov–Spectrum is identical to the spectrum of indices for Π 1 –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.