S1.0- Systematic Way for Nesting ordinal with Hyper operator (NOHO)

I have been thinking about this for months. Our conventional mathematics uses only first three of hyper operators for ordinal expansion a.k.a arithmetic, i.e.

《1》= addition, e.g. 𝛚+a

《2》 = multiplication, e.g. 𝛚2, 𝛚𝛚

《3》 = power or ↑, e.g. 𝛚³, 𝛚^𝛚

My original idea was to make use of higher hyper operators (《4》 and above) to make the ordinal expansion go much further than just these three, but I was stuck at 𝛚↑↑...↑↑𝛚+a. Then yesterday the "aha!" moment came along and I had this systematic way to solve it.

1- Rules and Description

1. For positive integer b:

• 𝛚《b+1》(𝛚+a+1) = (𝛚《b+1》(𝛚+a))《b》𝛚

1. or in general form:

• Ord《b+1》(𝛚+a+1) = (Ord《b+1》(𝛚+a))《b》Ord (where Ord is any ordinal).

1. Obviously, other rules are the same for higher hyper operators as in current ones.
2. The ordinal arithmetic operation is always from right to left.

2- Fundamental Sequence

i- Cantor Normal Form

Every ordinal number 𝝰 can be uniquely written as

• 𝝰 = $\omega^{\beta_1}c_1$+ $\omega^{\beta_2}c_2$+ ... + $\omega^{\beta_k}c_k$

where k is a natural number,  $c_1$, $c_2$, ... , $c_k$are positive integers, and  $\beta_1$ > $\beta_2$ > ... > $\beta_k$ $\geq$ 0 are ordinal numbers.

ii- NOHO Form

CNF's 𝝰 can be named as 1st level NOHO ordinal and be rewritten as

• $\alpha_1$= ((𝛚《3》$\beta_1$)《2》$c_1$)《1》((𝛚《3》$\beta_2$)《2》$c_2$)《1》 ... 《1》((𝛚《3》$\beta_k$)《2》$c_k$)

and in general form, b-th level ordinal is expressed as

• $\alpha_b$= ((𝛚《b+2》$\beta_1$)《b+1》$\gamma_1$)《b》((𝛚《b+2》$\beta_2$)《b+1》$\gamma_2$)《b》 ... 《b》((𝛚《b+2》$\beta_k$)《b+1》$\gamma_k$)

where $\gamma_1$, $\gamma_2$, ... , $\gamma_k$ can either be ordinals or positive integers.

iii- Fundamental Sequence

Ordinal 𝛚《𝛚》𝛚 can be expressed in this sequence:

• {𝛚《1》1, 𝛚《2》2, 𝛚《3》3, 𝛚《4》4, 𝛚《5》5, 𝛚《6》6, ... }

or in conventional form:

• {𝛚+1, 𝛚2, 𝛚↑3, 𝛚↑↑4, 𝛚↑↑↑5, 𝛚↑↑↑↑6, ... }
• = {𝛚+1, 𝛚2, 𝛚³, 𝛚^𝛚𝛚𝛚, $\epsilon_1$, $\zeta_0$, ... } (some of them were shown in examples below)

A- Examples

1. 𝛚↑↑(𝛚+1) = 𝛚《4》(𝛚+1) = (𝛚《4》𝛚)《3》𝛚 = $\epsilon_0^{\omega}$
2. 𝛚↑↑(𝛚+2) = 𝛚《4》(𝛚+1+1) = (𝛚《4》𝛚+1)《3》𝛚 = $\epsilon_0^{\omega^2}$
3. 𝛚↑↑𝛚2 = 𝛚《4》(𝛚+𝛚) = $\epsilon_0^{\omega^\omega}$
4. 𝛚↑↑𝛚² = 𝛚《4》(𝛚𝛚) = $\epsilon_0^{\omega^{\omega^2}}$
5. 𝛚↑↑𝛚^𝛚 = 𝛚《4》(𝛚^𝛚) = $\epsilon_0^{\omega^{\omega^\omega}}$
6. 𝛚↑↑(𝛚↑↑𝛚) = 𝛚《4》(𝛚《4》𝛚) = $\epsilon_0^{\epsilon_0}$
7. 𝛚↑↑(𝛚↑↑(𝛚↑↑𝛚)) = 𝛚《4》(𝛚《4》(𝛚《4》𝛚)) = $\epsilon_0^{\epsilon_0^{\epsilon_0}}$
   • It can be said that 𝛚↑↑(𝛚+A) = $\epsilon_0^{\omega^A}$(where A denotes an ordinal or a positive integer)
8. 𝛚↑↑↑𝛚 = 𝛚《5》𝛚 = $\epsilon_0^{\epsilon_0^{\epsilon_0^{.^{.^{.}}}}}$= $\epsilon_1$ (Note: I am not sure why Sbiis Sabian reserved 𝛚↑↑↑𝛚 for $\zeta_0$)
9. 𝛚↑↑↑(𝛚+1) = 𝛚《5》(𝛚+1) = (𝛚《5》𝛚)《4》𝛚 = $\epsilon_1^{\epsilon_1^{\epsilon_1^{.^{.^{.}}}}}$= $\epsilon_2$
10. 𝛚↑↑↑𝛚2 = 𝛚《5》(𝛚+𝛚) = $\epsilon_\omega$
11. 𝛚↑↑↑(𝛚2+1) = (𝛚《5》𝛚2)《4》𝛚 = $\epsilon_\omega^{\epsilon_\omega^{\epsilon_\omega^{.^{.^{.}}}}}$= $\epsilon_{\omega+1}$
12. 𝛚↑↑↑𝛚3 = (𝛚《5》(𝛚2+𝛚)) = $\epsilon_{\omega2}$
13. 𝛚↑↑↑(𝛚↑𝛚) = 𝛚《5》(𝛚《3》𝛚) = $\epsilon_{\omega^\omega}$
14. 𝛚↑↑↑(𝛚↑↑𝛚) = 𝛚《5》(𝛚《4》𝛚) = $\epsilon_{\epsilon_0}$
15. 𝛚↑↑↑(𝛚↑↑↑(𝛚↑↑𝛚)) = 𝛚《5》(𝛚《5》(𝛚《4》𝛚)) = $\epsilon_{\epsilon_{\epsilon_0}}$
16. 𝛚↑↑↑↑𝛚 = 𝛚《6》𝛚 $\approx$ $\epsilon_{\epsilon_{._{._{\epsilon_{\epsilon_0}}}}}$= $\zeta_0$
17. 𝛚↑↑↑↑(𝛚+1) = 𝛚《6》(𝛚+1) = (𝛚《6》𝛚)《5》𝛚
    • = (𝛚↑↑↑↑𝛚)↑↑(𝛚↑↑↑↑𝛚)↑↑ ... (𝛚↑↑↑↑𝛚)↑↑(𝛚↑↑↑↑𝛚)
    • = $\zeta_0$↑↑$\zeta_0$↑↑ ... $\zeta_0$↑↑$\zeta_0$ or $\zeta_0$《4》$\zeta_0$《4》 ... $\zeta_0$《4》$\zeta_0$

Side Note

The way I see it, this rule can be applied to any larger ordinal, not just 𝛚, i.e.

• O《b+1》(Ord+a+1) = (O《b+1》(Ord+a))《b》O (where O denotes any ordinal)

Hence the last two terms of Example 17 above can be expressed as

• (𝛚↑↑↑↑𝛚)↑↑(𝛚↑↑↑↑𝛚) = (𝛚↑↑↑↑𝛚)《4》(O+b+2) (where O denotes combinations of epsilon and omega)
  1. = ((𝛚↑↑↑↑𝛚)《4》(O+b+1))《3》(𝛚↑↑↑↑𝛚)
  2. = ((𝛚↑↑↑↑𝛚)《4》(O+b+1))《3》(O+b+2)
  3. = ((𝛚↑↑↑↑𝛚)《4》(O+b+1))^(O+b+1)((𝛚↑↑↑↑𝛚)《4》(O+b+1))
  4. = ((𝛚↑↑↑↑𝛚)《4》(O+b+1))^(O+b+1)(((𝛚↑↑↑↑𝛚)《4》(O+b))《3》(O+b+2))
  5. = ((𝛚↑↑↑↑𝛚)《4》(O+b+1))^(O+b+1)((𝛚↑↑↑↑𝛚)《4》(O+b))^(O+b+1) ... ((𝛚↑↑↑↑𝛚)《4》(𝛚+1))^(O+b+1)((𝛚↑↑↑↑𝛚)《4》𝛚)^(O+b+1)((𝛚↑↑↑↑𝛚)《4》𝛚)
  6. = ($\zeta_0$《4》(O+b+1))^(O+b+1)($\zeta_0$《4》(O+b))^(O+b+1) ... ($\zeta_0$《4》(𝛚+1))^(O+b+1)($\zeta_0$《4》𝛚)^(O+b+1)($\zeta_0$《4》𝛚)
  • (Note: this is the form explained in Section 2.ii, with $\zeta_0$ in place of 𝛚)

3- Ordinal inside Hyper Operator

1. Only two additional rules are needed:
   1. For non-negative integers a and b,
      • O《D+b+1》(Q+a+1) = (O《D+b+1》(Q+a))《D+b》O (where D, O and Q are ordinals)
   2. For positive integer c,
      • O《D+b+1》(c+1) = (O《D+b+1》c)《D+b》O
2. Other rules remain the same.

i- Fundamental Sequence

The 𝛚 based sequence can be expressed as

• 𝛚《《𝛚》》𝛚 = {𝛚《𝛚》𝛚, 𝛚《𝛚《𝛚》𝛚》𝛚, 𝛚《𝛚《𝛚《𝛚》𝛚》𝛚》𝛚, ... } (with central 𝛚 pairs of 《 》's)

In general form: 𝛚《$\alpha_b$》𝛚, where $\alpha_b$is from Section 2.iii above. All rules of 𝛚《《𝛚》》𝛚 shall follow those of 𝛚《𝛚》𝛚.

B- Examples

Define /k\[n] as $f_k(n)$of FGH, then

1. /𝛚《𝛚^𝛚》(𝛚+1)\[2]
   1. = /𝛚《𝛚²》(𝛚+1)\[2]
   2. = /𝛚《𝛚2》(𝛚+1)\[2]
   3. = /𝛚《𝛚+2》(𝛚+1)\[2]
   4. = /(𝛚《𝛚+2》𝛚)《𝛚+1》𝛚\[2]
   5. = /(𝛚《𝛚+2》𝛚)《𝛚+1》2\[2]
   6. = /((𝛚《𝛚+2》𝛚)《𝛚+1》1)《𝛚》(𝛚《𝛚+2》𝛚)\[2]
   7. = /(𝛚《𝛚+2》𝛚)《𝛚》(𝛚《𝛚+2》2)\[2] (Note: (𝛚《𝛚+2》𝛚)《𝛚+1》1 = 𝛚《𝛚+2》𝛚)
   8. = /(𝛚《𝛚+2》𝛚)《𝛚》((𝛚《𝛚+2》1)《𝛚+1》𝛚)\[2]
   9. = /(𝛚《𝛚+2》𝛚)《𝛚》(𝛚《𝛚+1》2)\[2]
   10. = /(𝛚《𝛚+2》𝛚)《𝛚》((𝛚《𝛚+1》1)《𝛚》𝛚)\[2]
   11. = /(𝛚《𝛚+2》𝛚)《𝛚》(𝛚《𝛚》𝛚)\[2]
   12. = /(𝛚《𝛚+2》𝛚)《𝛚》(𝛚《2》2)\[2]
   13. = /(𝛚《𝛚+2》𝛚)《𝛚》(𝛚2)\[2]
   14. = /(𝛚《𝛚+2》𝛚)《2》(𝛚2)\[2]
   15. = /(𝛚《𝛚+2》𝛚)𝛚2\[2]
   16. = /(𝛚《𝛚+2》𝛚)𝛚 + (𝛚《𝛚+2》𝛚)𝛚\[2]
   17. = /(𝛚《𝛚+2》𝛚)𝛚 + (𝛚《𝛚+2》𝛚)2\[2]
   18. = /(𝛚《𝛚+2》𝛚)𝛚 + 𝛚《𝛚+2》𝛚 + 𝛚《𝛚+2》𝛚\[2]
   19. = /(𝛚《𝛚+2》𝛚)𝛚 + 𝛚《𝛚+2》𝛚 + 𝛚《𝛚》𝛚\[2]
   20. = /(𝛚《𝛚+2》𝛚)𝛚 + 𝛚《𝛚+2》𝛚 + 𝛚《2》2\[2]
   21. = /(𝛚《𝛚+2》𝛚)𝛚 + 𝛚《𝛚+2》𝛚 + 𝛚2\[2]
   22. = /(𝛚《𝛚+2》𝛚)𝛚 + 𝛚《𝛚+2》𝛚 + 𝛚 + 𝛚\[2]
   23. = /(𝛚《𝛚+2》𝛚)𝛚 + 𝛚《𝛚+2》𝛚 + 𝛚 + 2\[2]
   24. = /(𝛚《𝛚+2》𝛚)𝛚 + 𝛚《𝛚+2》𝛚 + 𝛚 + 1\[/(𝛚《𝛚+2》𝛚)𝛚 + 𝛚《𝛚+2》𝛚 + 𝛚 + 1\[2]]
   25. = /(𝛚《𝛚+2》𝛚)𝛚 + 𝛚《𝛚+2》𝛚 + 𝛚 + 1\[/(𝛚《𝛚+2》𝛚)𝛚 + 𝛚《𝛚+2》𝛚 + 𝛚\[/(𝛚《𝛚+2》𝛚)𝛚 + 𝛚《𝛚+2》𝛚 + 𝛚\[2]]]
   26. = ...