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1. Relational systems A formal system in which there is a single primitive predicate and that a binary one we shall call a relational system. Such a system in which the primitive predicate has the properties of equality-reflexiveness, symmetry, transitiveness, and certain properties of replacement which we shall discuss in Chapter 2-will be called an equational system; one in which the primitive relation is a quasi-ordering (see 9 2D)a quasi-ordered systetn. In most systems of ordinary mathematics equality is taken for granted.

L]. The general analysis of E. H. Moore [IFG], which was expressed in a symbolism showing strongly the influence of Peano, used postulational and epitheoretical methods extensively. Naturally there were similar developments in other countries; but they did not have the same direct influence on the present work. , one taking logic for granted. But one can hardly formulate thenotionof an axiomatic system without asking almost immediately how logic is to be formulated. ) Consequently the two types of formalization developed almost simultaneously.

The dispute with the intuitionists is no doubt responsible for sharpening some of the ideas; for claims of the latter in this connection see Brouwer [IBF], Heyting [MGL]. For an intelligible account of the ideas of intuitionism see Wilder [IFM] Chapter X ; cf. also Heyting [MGL]. Hilbertism developed into the syntactical conception of formal methods, which is currently the most fashionable (see 3 2 ) . The notion of formal system expounded in this work is that developed in a series of papers by Curry.

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