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, $\omega$ . Ordinals can be added and multiplied. The sum of two ordinals $a$ and $b$ 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 $1+\omega=\omega$ , while $\omega+1>\omega$ .

Every ordinal characterizes the order type of the ordered ordinals less than it. For example, $0,1,2,\dotsb,\omega$ has order type $\omega+1$ .

The smallest ordinal that can't be constructed from $\omega$ by addition, multiplication, and exponentiation is $\varepsilon_0$ , the first fixed point of the map $\alpha\mapsto\omega^\alpha$ .

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User:Evin/Draft:Ordinal