Henkin quantifiers and the definability of truth

Henkin quantifiers have been introduced in Henkin (1961). Walkoe (1970) stu died basic model-theoretical properties of an extension L-1 (H) of ordinary first-order languages in which every sentence is a first-order sentence pr efixed with a Henkin quantifier. In this paper we consider a generalization of Walkoes languages: we close L-1(H)with respect to Boolean operations, a nd obtain the language L-1(H). At the next level, we consider an extension L-2(H) of L-1(H) in which every sentence is an L-1(H)-sentence prefixed wit h a Henkin quantifier. We repeat this construction to infinity. Using the ( un)-definability of truth in N for these languages, we show that this hiera rchy does not collapse. In addition, we compare some of the present results to the ones obtained by Kripke (1975), McGee (1991), and Hintikka (1996).