User:Evin-/Draft:Ordinal
Ordinals are an extension of the natural numbers. Ordinals can be used to describe the order type of a set. The order type of the natural numbers is the first infinite ordinal, . Ordinals can be added and multiplied. The sum of two ordinals and is the ordinal that describes the order type of a set with order type a concatenated with one of order type b. Warning! Ordinal addition is not commutative. For example , while .
Every ordinal characterizes the order type of the ordered ordinals less than it. For example, has order type .
The smallest ordinal that can't be constructed from by addition, multiplication, and exponentiation is , the first fixed point of the map .
The Veblen functions
Based on the definition of , Oswald Veblen in 1908 introduced an ordinal-indexed hierarchy of functions. and enumerates the common fixed points of for all . , is the smallest ordinal inaccessible from the ordinals. It is called . The set of all ordinals accessible from the functions, addition, multiplication, and exponentiation is well-ordered. It has order type , the Feferman–Schütte ordinal. It is the first fixed point of the map .