In this paper first order theories for nonmonotone inductive definitions are introduced, and a proof-theoretic analysis for such theories based on combined operator forms a la Richter with recursively inaccessible and Mahlo closure ordinals is given. 1 Introduction Let # be an operator on the power set P (N) of the natural numbers, i.e. a mapping from P (N) to P (N). Then # can be used to generate subsets I # # of the natural numbers if we define I # # := I <# # # #(I <# # ) and I <# # := # {I # # : # < #} by transfinite recursion on the ordinals. Furthermore we let I # # := # {I # # : # an ordinal } be the set of natural numbers inductively defined by #. Obviously there exists a least ordinal # so that I # # = I <# # . We call this ordinal the closure ordinal of the inductive definition generated by # and know that I # # is identical to I # # . The sets I # # are the stages of the inductive definition generated by #. If K is a class of operators, th...

Mahlo universe