Axiomatization and Completeness for Some Restricted Second-Order Languages
John L. Bell
This paper is drawn from two unpublished chapters of my 1969 Oxford D.Phil dissertation Problems in Mathematical Logic: model theoretic and axiomatization results for certain restricted second-order languages. In what follows I formulate complete Hilbert -style infinitary axiomatizations of two systems of (classical) second-order logic: weak second-order logic WSOL1 and definable subset logic DSL. In WSOL second – order variables are construed as ranging over finite sets of individuals, in DSL over first-order definable sets of individuals. The completeness of the two axiomatizations is proved by Boolean algebraic methods.
Keywords: Restricted second-order languages, axiomatization, boolean algebras