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.
γ2γ = multiplication, e.g. π2, ππ
γ3γ = power or β, e.g. π3, ππ
My original idea was to make use of higher hyper operators (γ4γ and above) to make the ordinal expansion go much further than just these three, but I was stuck at πββ...ββπ+a. Then yesterday the "aha!" moment came along and I had this systematic way to solve it.
1- Rules and Description[]
- For positive integer b:
- πγb+1γ(π+a+1) = (πγb+1γ(π+a))γbγπ
- or in general form:
- Ordγb+1γ(π+a+1) = (Ordγb+1γ(π+a))γbγOrd (where Ord is any ordinal).
- Obviously, other rules are the same for higher hyper operators as in current ones.
- The ordinal arithmetic operation is always from right to left.
2- Fundamental Sequence[]
i- Cantor Normal Form[]
Every ordinal number π° can be uniquely written as
- π° = + + ... +
where k is a natural number, , , ... , are positive integers, and > > ... > 0 are ordinal numbers.
ii- NOHO Form[]
CNF's π° can be named as 1st level NOHO ordinal and be rewritten as
- = ((πγ3γ)γ2γ)γ1γ((πγ3γ)γ2γ)γ1γ ... γ1γ((πγ3γ)γ2γ)
and in general form, b-th level ordinal is expressed as
- = ((πγb+2γ)γb+1γ)γbγ((πγb+2γ)γb+1γ)γbγ ... γbγ((πγb+2γ)γb+1γ)
where , , ... , can either be ordinals or positive integers.
iii- Fundamental Sequence[]
Ordinal πγπγπ can be expressed in this sequence:
- {πγ1γ1, πγ2γ2, πγ3γ3, πγ4γ4, πγ5γ5, πγ6γ6, ... }
or in conventional form:
- {π+1, π2, πβ3, πββ4, πβββ5, πββββ6, ... }
- = {π+1, π2, π3, ππππ, , , ... } (some of them were shown in examples below)
A- Examples[]
- πββ(π+1) = πγ4γ(π+1) = (πγ4γπ)γ3γπ =
- πββ(π+2) = πγ4γ(π+1+1) = (πγ4γπ+1)γ3γπ =
- πββπ2 = πγ4γ(π+π) =
- πββπ2 = πγ4γ(ππ) =
- πββππ = πγ4γ(ππ) =
- πββ(πββπ) = πγ4γ(πγ4γπ) =
- πββ(πββ(πββπ)) = πγ4γ(πγ4γ(πγ4γπ)) =
- It can be said that πββ(π+A) = (where A denotes an ordinal or a positive integer)
- πβββπ = πγ5γπ = = (Note: I am not sure why Sbiis Sabian reserved πβββπ for )
- πβββ(π+1) = πγ5γ(π+1) = (πγ5γπ)γ4γπ = =
- πβββπ2 = πγ5γ(π+π) =
- πβββ(π2+1) = (πγ5γπ2)γ4γπ = =
- πβββπ3 = (πγ5γ(π2+π)) =
- πβββ(πβπ) = πγ5γ(πγ3γπ) =
- πβββ(πββπ) = πγ5γ(πγ4γπ) =
- πβββ(πβββ(πββπ)) = πγ5γ(πγ5γ(πγ4γπ)) =
- πββββπ = πγ6γπ =
- πββββ(π+1) = πγ6γ(π+1) = (πγ6γπ)γ5γπ
- = (πββββπ)ββ(πββββπ)ββ ... (πββββπ)ββ(πββββπ)
- = ββββ ... ββ or γ4γγ4γ ... γ4γ
Side Note[]
The way I see it, this rule can be applied to any larger ordinal, not just π, i.e.
- Oγb+1γ(Ord+a+1) = (Oγb+1γ(Ord+a))γbγO (where O denotes any ordinal)
Hence the last two terms of Example 17 above can be expressed as
- (πββββπ)ββ(πββββπ) = (πββββπ)γ4γ(O+b+2) (where O denotes combinations of epsilon and omega)
- = ((πββββπ)γ4γ(O+b+1))γ3γ(πββββπ)
- = ((πββββπ)γ4γ(O+b+1))γ3γ(O+b+2)
- = ((πββββπ)γ4γ(O+b+1))(O+b+1)((πββββπ)γ4γ(O+b+1))
- = ((πββββπ)γ4γ(O+b+1))(O+b+1)(((πββββπ)γ4γ(O+b))γ3γ(O+b+2))
- = ((πββββπ)γ4γ(O+b+1))(O+b+1)((πββββπ)γ4γ(O+b))(O+b+1) ... ((πββββπ)γ4γ(π+1))(O+b+1)((πββββπ)γ4γπ)(O+b+1)((πββββπ)γ4γπ)
- = (γ4γ(O+b+1))(O+b+1)(γ4γ(O+b))(O+b+1) ... (γ4γ(π+1))(O+b+1)(γ4γπ)(O+b+1)(γ4γπ)
- (Note: this is the form explained in Section 2.ii, with in place of π)
3- Ordinal inside Hyper Operator[]
- Only two additional rules are needed:
- For non-negative integers a and b,
- OγD+b+1γ(Q+a+1) = (OγD+b+1γ(Q+a))γD+bγO (where D, O and Q are ordinals)
- For positive integer c,
- OγD+b+1γ(c+1) = (OγD+b+1γc)γD+bγO
- For non-negative integers a and b,
- Other rules remain the same.
i- Fundamental Sequence[]
The π based sequence can be expressed as
- πγγπγγπ = {πγπγπ, πγπγπγπγπ, πγπγπγπγπγπγπ, ... } (with central π pairs of γ γ's)
In general form: πγγπ, where is from Section 2.iii above. All rules of πγγπγγπ shall follow those of πγπγπ.
B- Examples[]
Define /k\[n] as of FGH, then
- /πγππγ(π+1)\[2]
- = /πγπ2γ(π+1)\[2]
- = /πγπ2γ(π+1)\[2]
- = /πγπ+2γ(π+1)\[2]
- = /(πγπ+2γπ)γπ+1γπ\[2]
- = /(πγπ+2γπ)γπ+1γ2\[2]
- = /((πγπ+2γπ)γπ+1γ1)γπγ(πγπ+2γπ)\[2]
- = /(πγπ+2γπ)γπγ(πγπ+2γ2)\[2] (Note: (πγπ+2γπ)γπ+1γ1 = πγπ+2γπ)
- = /(πγπ+2γπ)γπγ((πγπ+2γ1)γπ+1γπ)\[2]
- = /(πγπ+2γπ)γπγ(πγπ+1γ2)\[2]
- = /(πγπ+2γπ)γπγ((πγπ+1γ1)γπγπ)\[2]
- = /(πγπ+2γπ)γπγ(πγπγπ)\[2]
- = /(πγπ+2γπ)γπγ(πγ2γ2)\[2]
- = /(πγπ+2γπ)γπγ(π2)\[2]
- = /(πγπ+2γπ)γ2γ(π2)\[2]
- = /(πγπ+2γπ)π2\[2]
- = /(πγπ+2γπ)π + (πγπ+2γπ)π\[2]
- = /(πγπ+2γπ)π + (πγπ+2γπ)2\[2]
- = /(πγπ+2γπ)π + πγπ+2γπ + πγπ+2γπ\[2]
- = /(πγπ+2γπ)π + πγπ+2γπ + πγπγπ\[2]
- = /(πγπ+2γπ)π + πγπ+2γπ + πγ2γ2\[2]
- = /(πγπ+2γπ)π + πγπ+2γπ + π2\[2]
- = /(πγπ+2γπ)π + πγπ+2γπ + π + π\[2]
- = /(πγπ+2γπ)π + πγπ+2γπ + π + 2\[2]
- = /(πγπ+2γπ)π + πγπ+2γπ + π + 1\[/(πγπ+2γπ)π + πγπ+2γπ + π + 1\[2]]
- = /(πγπ+2γπ)π + πγπ+2γπ + π + 1\[/(πγπ+2γπ)π + πγπ+2γπ + π\[/(πγπ+2γπ)π + πγπ+2γπ + π\[2]]]
- = ...