Some theorems on the expressive limitations of modal languages

Fix a set of predicates Pred, each of a particular number of places, and a countably infinite set of variables Var; for a set of names C form the language L(c) by following the usual formation rules, with the primitives ‘l’, ‘7, ‘Y’ and ‘0’. A term of L(c) is a member of Var U C. Use and mention shall be freely confused. Let 16, (3~)$, Og, and Eu abbreviate 0 E 1, l(Vv)l#, lol@ and (3~) (v = u) where u is a term of L(C) and v is a variable distinct from a; ‘82, ‘v’ and ‘z are defined as usual. We work entirely within the modal logic S5. So we may take a frame to be a pair F = (IU, A), W a non-empty set, A a function on W so that .4(w) is a set for all w E IV, and U{A(w) 1 w E IV) = 1 is non-empty. An F-valuation for L(C) is a function V with domain C U (IV x Pred), V(c) E 1 for c E C; forwEWandPEPred,Pn-place:ifn>l, V(w,P)gZn;forn=O, V(w, P) E {t, f). A structure for L(C) is a triple J/= (W, A, v), Ya (W, A)valuation for L(C). For w E W, we say that w is from &, rf is an assignment fordiff d: Var +A. Let: