The basis of this book is two monographs by R. L. Goodstein: "Recursive number theory" and "Recursive analysis." The monograph "Recursive Number Theory" contains a systematic and detailed description and study of Goodstein's calculus p. p. Equalities and some modifications of this calculus; it also describes and studies some "superstructures" over the calculus of equalities, using certain extensions of the language of the calculus of equalities and allowing "translations" into the calculus of equalities