The Axiomatic Method: An Introduction to Mathematical Logic

This book is written primarily for the student of mathematics who possesses some measure of mathematical ability, has a working knowledge of the axiomatic approach to mathematics, and in particular has been exposed to the axiomatic method as applied to the study of modern abstract algebra. It is the axiomatic method itself that is under scrutiny here. Consider the statement: “‘Euclid’s fifth postulate is a logical consequence of his other four postulates.” There are two possible interpretations of this statement, depending upon the viewpoint. To a working mathematician, a proposition is a logical consequence of a postulate-set if the given proposition is true about each mathematical system for which the postulates are true. To a logician, a proposition is a logical consequence of a postulate-set provided that the given proposition can be deduced from the postulate-set by applying the laws and procedures of a particular system of logic. Notice that in the latter approach mathematical systems—the realizations of the given postulateset—are not involved. On the one hand, then, the axiomatic approach is concerned with mathematical systems and with demonstrating that given propositions (i.e., theorems) are true in these mathematical systems. The other side of the axiomatic method is the purely logical side in which the theorems of the system are established by applying a completely formalized Theory of Deduction to the given postulate-set. There is a striking difference in viewpoint: the first approach emphasizes the mathematical systems characterized by the given postulate-set, whereas the second approach considers only the logical apparatus, which is applied directly to the postulate-set. A theory of deduction may be regarded as a black box: feed in a set of propositions (the given postulate-set), turn a crank, and out come the theorems of the system. Our plan is to study both approaches to the axiomatic method and to demonstrate that they are indeed two aspects of the same thing. To be precise, we shall prove that, under a suitable theory of deduction, each logical consequence of a given postulate-set under the first interpretation is also a logical consequence of the given postulate-set under the second interpretation.