LGH and L-range

LGH or Limititerational Growth Hierarchy is a hierarchy similar to the Fast Growing Hierarchy, Hardy Hierarchy, and Slow Growing Hierarchy and many other variations. This is intended to be used to classify the size of numbers.

## Definition

Define g: Ord->Ord as the following:

g
(
α
1
# )
g
(
α
)

{\displaystyle g(\alpha +1)=g(\alpha) }

g
(
α
# )
α

{\displaystyle g(\alpha )= \alpha }

 if

α
∈

{\displaystyle \alpha \in}

 Lim

g
(
1
# )
1

{\displaystyle g(1)=1 }

L

0

(
x
,
y
# )
1

{\displaystyle L\_{0}(x,y)=1 }

L

1

(
x
,
y
# )
y
⋅
x

{\displaystyle L\_{1}(x,y)=y \cdot x }

L

α
1

(
x
,
y
# )

L

g
(
α
)

(

L

α

(
x
,
y
)
,
y
)

{\displaystyle L\_{\alpha +1}(x,y) = L\_{g(\alpha)}(L\_{\alpha}(x,y),y) }

L

α

(
x
,
y
# )

L

α
[
x
]

(
1
,
y
)

{\displaystyle L\_{\alpha}(x,y) = L\_{\alpha [x]}(1,y) }

 if

α
∈
L
i
m

{\displaystyle \alpha \in Lim }

L

α

(
x
# )

L

α

(
x
,
10
)

{\displaystyle L\_{\alpha}(x)=L\_{\alpha}(x,10) }

## Usage

We can use LGH to define a range that will be called the L range, which is a range of natural numbers, and from there, we can get the L-level of any natural numbers given a proper fundamental sequence system upto some ordinal.

## Definition of L-range and L-level

L-range:

L

α

=
{
x
∈

N

:

L

α

(
1
)
≤
x
<

L

α
1

(
1
)
}

{\displaystyle L^{\alpha} = \{x \in \mathbb{N}: L\_{\alpha}(1) \le x < L\_{\alpha+1}(1)\} }

An natural number k have a L-level of

κ

⟺

κ

{\displaystyle \kappa \iff \kappa }

 is the maximum ordinal such that k

∈

L

κ

{\displaystyle \in L^{\kappa} }

## Some specific L-ranges

```
**Failed to parse (unknown function "\textemdash"): {\displaystyle L^0 = 1 \textemdash 9 }**

**Failed to parse (unknown function "\textemdash"): {\displaystyle L^1 = 10 \textemdash 99 }**

**Failed to parse (unknown function "\textemdash"): {\displaystyle L^2 = 100 \textemdash 999 }**

**Failed to parse (unknown function "\textemdash"): {\displaystyle L^3 = 1000 \textemdash 9999 }**

**Failed to parse (unknown function "\textemdash"): {\displaystyle L^4 = 10^4 \textemdash 10^5-1 }**

**Failed to parse (unknown function "\textemdash"): {\displaystyle L^5 = 10^5 \textemdash 10^6-1 }**

**Failed to parse (unknown function "\textemdash"): {\displaystyle L^6 = 10^6 \textemdash 10^7-1 }**

**Failed to parse (unknown function "\textemdash"): {\displaystyle L^7 = 10^7 \textemdash 10^8-1 }**

**Failed to parse (unknown function "\textemdash"): {\displaystyle L^8 = 10^8 \textemdash 10^9-1 }**

**Failed to parse (unknown function "\textemdash"): {\displaystyle L^9 = 10^9 \textemdash 10^10-1 }**

**Failed to parse (unknown function "\textemdash"): {\displaystyle L^10 = 10^10 \textemdash 10^11-1 }**

**Failed to parse (unknown function "\textemdash"): {\displaystyle L^{\omega} = 10 \textemdash 10^10-1 }**

**Failed to parse (unknown function "\textemdash"): {\displaystyle L^{\omega+1} = 10^10 \textemdash 10^10^10-1 }**

**Failed to parse (unknown function "\textemdash"): {\displaystyle L^{\omega+2} = 10^10^10 \textemdash 10^10^10^10-1 }**

**Failed to parse (syntax error): {\displaystyle L^{\omega+n} = 10^^(n+1) \textemdash 10^^(n+2)-1 }**

**Failed to parse (syntax error): {\displaystyle L^{\omega \cdot 2} = 10^^2 \textemdash 10^^10^^2 }**

**Failed to parse (syntax error): {\displaystyle L^{\omega \cdot 2 +1} = 10^^10^^2 \textemdash 10^^10^^10^^2 }**

```