User:Googleaarex/My ordinal notation

Up to \(\varepsilon_0\)
1. \(O(0)\) = 1

2. \(O(\alpha)[0]\) = 0

3. \(O(\alpha + 1)[n+1]\) = \(O(\alpha) + O(\alpha + 1)[n+1]\)

Up to \(\varphi(\omega,0)\)
4. \(O(0...0,0,\alpha + 1 \#)[n+1]\) = \(O(0...0,O(0...0,0,\alpha + 1 \#)[n],\alpha \#)\)

5. \(O(\# 0)\) = \(O(\#)\)

Up to BHO
6. \(O(0...0,\#[\alpha + 1] \#)\) = \(O(0...0,0,\#[\alpha] \#)\)

7. \(O(\# \#[0] \#)\) = \(O(\# \# \#)\)