On some generalizations of the primitive recursive arithmetic

Formal arithmetical system PRAU is defined as an extension of R.L. Goodstein's system PRA of the primitive recursive arithmetic; it is based on the consideration of functions similar to primitive recursive functions but in general not everywhere defined. It is proved that PRAU is a conservative extension of PRA. Some classes of the program schemes (PRA-schemes and PRAU-schemes) are introduced; it is proved that the classes of functions computable by such schemes coincide with the classes of functions taking part, correspondingly, in PRA and PRAU.