Ordinals I: Basic Notions.

At the end of the last century, Cantor generalized the notion of natural integer by considering the order structure of integers. 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, and ordinal arithmetic extends usual one. The ordinals up to ε0 can be described in an intuitive way (with 0, 1, ω and the sum, product and exponentiation operations). In contrast, it cannot be done for ordinals beyond ε0. In order to describe greater ordinals, Veblen introduced a normal functions hierarchy (i.e. strictly increasing and continuous) from ordinals into ordinals. This construction allows us to describe ordinals until Γ0, using ordinals preceding Γ0. Therefore, Γ0 and ε0 have a similar role if Veblen functions are used for Γ0 in addition to the previous operations.