Oswald Veblen

A system of axioms for geometry

CHAPTER I. The axioms and their independence. Introductory statement of the axioms . . .1 Categorical and disjunctive systems . . . ? 2 Independence proofs and historical remarks on axioms IX-XII. ? 3-6 Finite systems proving the independence of I-VIII . . . ? 7-14 CHAPTER II. General properties of space. Definition and properties of line, plane, space . . . . . ? 1-3 Generalizations of oider. Regiolns. ? 4 Continuity .? 5 Parallel lines .? 6 CHAPTER III. Projective and metric geometry. Ideal elements defined independently of XI and XII . . . . ? 1-2 Fundanmental theorems of projective geometry .? 3-6 Definition of congruence relations . . . . ? 7-8 The axiom system as categorical. . . . . ? 9

Continuous increasing functions of finite and transfinite ordinals

A continuous increasing function of a set of ordinal numbers is analogous to a progressively continuous increasing function of the real variable. Some of its properties J are developed below, especially such as bear on the notion of a derived function of the atth degree (cf. § 3) and its extensions (cf. § 4). They are nearly all generalizations of properties discovered by Cantor § for particular functions and so may be used to simplify some of his proofs and generalize some of his results. In particular they extend his theory of e-numbers. One of the most interesting problems in the theory of transfinite numbers arises in connection with Hardys || scheme for obtaining a subset of the continuum of type il. The success of his method depends on determining for each ordinal number a(a +. ß + l)oi the second class a unique fundamental sequence Sa= {<*„} such that Lav = a. For each number a (a 4= ß + 1 ) of the second class there evidently exists an infinitude of such sequences, of which, in any special case, one may be selected. But no one has as yet given a method of determining a set of sequences {Sa} such that for each a. (a 4= ß + 1, eo = a < il) there exists one and only one Sa. If a is restricted to be less than e0, the first e-number, the problem of determining { Sa} is very easily solved. For every number of the second kind^[ in the second number-class can be written uniquely in the form,**