Ordinals I: basic notions

The generalization of the concept of natural integers by means of sum, subtraction, product and quotient successively yielded to the definition of relative integers and subsequently to rational numbers. Then reals and complexes were introduced. All these constructions result from a study of natural integers which is essentially algebraic. At the end of the last century, Cantor generalized the notion of natural integer by considering the order structure of integers instead of algebraic means. According to the idea of "counting more and more", he introduced transfinite numbers over natural numbers. Thus an ordinal may be simply defined as a natural number or a transfinite number. How can we go beyond the infinite of natural numbers: we can represent the natural numbers as points of a semi-line, and juxtapose another semi-line and so on... However this construction can be done for any well-ordered set. Therefore, we will first define well-orderings and show that two well-orderings are always comparable: either these well-orderings are isomorphic or there is an embedding from one to the other. Moreover we will define typical well-ordered sets: ordinals. The isomorphic relation between well-orderings is an equivalence relation and we show that each ordinal is a representant of an equivalence class. Since the collection of all ordinals is itself well-ordered, we can define transfinite induction on this collection. Actually, finite ordinals are natural numbers and ordinal arithmetic extends usual one. The ordinals up to epsilon_0 can be described in an intuitive way (with 0, 1, omega and the sum, product and exponentiation operations). In contrast, it cannot be done for ordinals beyond epsilon_0. In order to describe greater ordinals, Veblen, we introduced a normal functions hierarchy (i.e. strictly increasing and continuous) from ordinals into ordinals. This hierarchy is built by taking at each step fixed points of the function defined at the previous step and, at the limit steps, the function which enumerates the common points of the images of the preceding functions. This construction allows us to describe ordinals until Gamma_0, We show how each ordinal smaller than Gamma_0 can be described using ordinals preceding Gamma_0. Therefore Gamma_0 and epsilon_0 have a similar role if Veblen functions are used for Gamma_0 in addition to the previous operations