It is widely known that one of the major tasks of 'Foundations' is to construct a formal system which can he said to contain the whole of mathematics. For various reasons axiomatic set theory is a very suitable choice for such a system and it is one which has proved acceptable to both logicians and mathematicians. The particular demands of mathematicians and logicians, however, are not the same. As a result there exist at the moment two different formulations of set theory which can be roughly said to cater for logicians and mathematicians respectively. It is these systems which are the subject of this dissertation. The system of set theory constructed for logicians is by P. Bernays. This will be discussed in chapter II. For mathematicians No Bourbaki has constructed a system of set theory within which he has already embedded a large part of mathematics. This system will be discussed in chapter III. Chapter I is historical and contains some of Cantor's original ideas. The relationship between Bernays' system and (essentially) Bourbaki's system is commented upon in chapter IV. <p
