ABSTRACT
A prime reason for the increase in importance of mathematical logic in the twentieth century was the discovery of the paradoxes of set theory and the need for a revision of intuitive (and contradictory) set theory. Many different axiomatic theories have been proposed to serve as a foundation for set theory but, no matter how they may differ at the fringes, they all have as a common core the fundamental theorems that mathematicians require for their daily work. We make no claim about the superiority of the system we shall use except that, from a notational and conceptual standpoint, it is a convenient basis for present-day mathematics. We shall describe a first-order theory NBG, which is basically a system
of the same type as one originally proposed by J. von Neumann (1925, 1928) and later thoroughly revised and simplified by R. Robinson (1937), Bernays (1937-1954), and Gödel (1940) (We shall follow Godel’s monograph to a great extent, although there will be some significant differences.)* NBG has a single predicate letter A22 but no function letter or individual
constants.y In order to conform to the notation in Bernays (1937-1954) and Gödel (1940), we shall use capital italic letters X1,X2,X3, . . . as variables instead of x1, x2, x3, . . .. (As usual, we shall useX,Y,Z, . . . to represent arbitrary variables.) We shall abbreviate A22(X,Y) by X 2 Y, and :A22(X,Y) by X=2Y. Intuitively, 2 is to be thought of as the membership relation and the values
of the variables are to be thought of as classes. Classes are certain collections of objects. Some properties determine classes, in the sense that a property P may determine a class of all those objects that possess that property. This ‘‘interpretation’’ is as imprecise as the notions of ‘‘collection’’ and ‘‘property.’’ The axioms will reveal more about what we have in mind. They will provide us with the classes we need in mathematics and appear modest enough so that contradictions are not derivable from them. Let us define equality in the following way.
