Axiomatic Recursive Function Theory

Abstract

Publisher Summary This chapter discusses the axiomatic recursive function theory. The results of Wagner-Strong theory are general and they hold for all basic recursive function theories (BRFT's). The chapter focuses on the BRFT and shows that any collection of partial functions satisfying the Kleene enumeration theorem must contain all partial recursive functions. A minimality theorem is obtained for the hyperarithmetic functions. A generalization of the relative categoricity is presented. Relative categoricity is sensitive to that consideration of nonprojectibility. Ordinary recursive function theory and the theory of forcing with finite conditions are discussed. A transparent necessary and sufficient condition on monadic and binary partial functions in order for them to be conservatively extended to a BRFT is given. A related problem considered is to give a transparent necessary and sufficient condition on monadic and binary partial functions together with a distinguished binary partial function for this structure to be conservatively extended to a BRFT.