Ordinals are important things in googology. Most of them are infinite, so they’re not just numbers. But we can use them to present some “levels”. Here I bring 4 ordinal notations I would use in this site – Cantor’s normal form, binary φ function, θ function, and Taranovsky’s notation.

# Definition of ordinals

Ordinals are **something** that fits these properties. First we need a binary relation “<” – that’s “less than” – on ordinals.

- Any ordinal a and b must be and only be one of those 3 relations: a < b, a = b, or b < a.
- a = a – that means a < a is always false.
- Any sets or classes of ordinals must have a minimum.

We also write a < b as b > a. And means a < b or a = b.

# Basic notations

Here’s a very basic notation. The minimum of all ordinals is 0, and is the successor of .

We call an ordinal x *successor ordinal* iff it’s the successor of some ordinal. x is a limit ordinal iff it’s neither 0 nor successor ordinal. We call the least limit ordinal ω – it’s the supremum of all natural numbers.

Then there’re some arithmetic on ordinals. First is the addition.

- for limit ordinal β

Then multiplication.

- for limit ordinal β

Then exponentiation.

- for limit ordinal β

Then ordinals have such properties:

Finally, we have a notation called Cantor’s normal form.

- Ordinal is in Cantor’s normal form.
- Ordinal where n and all k
_{i}‘s are positive integers and are all in Cantor’s normal form is in Cantor’s normal form.

If we just use Cantor’s normal form built from 0, 1 and ω, we have such a limit – .

# Binary φ function

Binary φ function is defined as follows:

The second line means, is the (1+β)-th common fixed point that for all δ < α.

Explanation

– that’s just a simple Cantor’s normal form.

is the least ordinal such that . We say it’s the first fixed point of . The second fixed point is , the third is , and so on. The ω-th is .

There’re also some ordinals such that – the fixed points of . The first fixed point is , the second is , and so on.

also has fixed points. They’re .

Generally speaking, is the (1+β)-th fixed point of . This works for successor ordinal α.

For limit case, such as ω, is a fixed point of , also , also , also , and so on. It’s the **common** fixed point of for all δ < ω, and it’s the (1+β)-th ordinal have such property. Notice that this “common fixed point” also works for successor case, so we get the definition.

Using 0, 1, ω, by addition and φ function, we have such a limit – it’s the first fixed point of – that’s . Generally, we use to present the (1+β)-th fixed point of .