The present treatise belongs to the metamathematics of Algebra. The fact that this subject has in recent years attracted the active attention of various eminent logicians has encouraged the present author to contribute to it yet another monograph. A general survey of the contents will be found in the first section of the first chapter. Except where references to earlier papers are given, the methods and results described here were obtained within the last twelve months. Some of the work was reported upon in lectures in Zurich (June, 1954} and Jerusalem (July, 1954) and at the Symposium on the mathematical interpretation, of formal systems, which was held in conjunction with the International Congress of Mathematicians (Amsterdam, September, 1954).