Cat's Number 10

yay i've finally won the game! :3

## Introduction to Cat's numbers.

**Cat's numbers** use different oracle systems of **Cat's Function**, first one defined as

C

K
(

ψ
k

(
ω
)
)

1

(
α
)

{\displaystyle C^{1}\_{K(\psi\_k(\omega))}( \alpha)}

.

> *Uh,that would be so nice to have something like cat's function as the biggest FG ordinal ever,who doesn`t love cats!* *Also dont forget to assume that*
>
>
>
> α
>
>
> {\displaystyle \alpha}
>
>  *and N* *here is an output of all logical definitions of \*LTL(n). :3*
> *Any other theories made with C(n) or ones that will try to extend it will be included in \*LTL(N) subset theory , so you don't have to hard try. >:3*
>
>

## Introduction to LTL set theory.

If the least ordinal to the last ordinal theory exists:

We define it as LTL(N) *=>* LTL(N) , where N ≤ 0 = 0

Otherwise: LTL(N) = L1→L2 , where L1 is the ordinal defined in LTL(N-1), and L2 is the last ordinal that can be defined in its fundamental sequence.

Any ordinal that include "infinitely decreasing","infinitely increasing","dimensions" numbers, "above all", "exists outside" and etc. will be included in LTL using infinite rule/proof enumerator. *No matter what they have in their definition, since LTL(n) is a powerset of ordinals in it's fundamental sequence using it's system fundamental rules.*

If some rules of LTL isn't applyable to some FS,you make a separate list,where you do nothing with current ordinal defined in it's FS,instead look for a fixed point if it's possible,paste it into next separate list and etc., otherwise find the limit of the function.

If some ordinal doesn't allow you to do it,or you can't extend it,make a different list for it and apply it's own rules into it.

*Note: LTL can mimic any ordinal or FS,no matter if they have smth LTL-typo rules or same rules,if an ordinal have a specific rules,that can avoid it from being in the LTL's list,it would be included as not-embeddable list in LTL(n) set theory. Right now you can't say that your ordinal can't be included in LTL's list or set theory,or that it's bigger,cus otherwise it will be included in stronger theory aka \*LTL(n) bigger than every FG or other googologys ordinals combined..*

*(haha i beat the game :3 , now prove that your ordinal isn't applyable to any of these rules.)*

**List encoded as a binary string:**

> You define: 0 as smaller , 1 as bigger.
>
>
> First you start with an ordinal N and give it a 0
>
>
> 010101... - some ordinal is not embeddable into LTL set theory,but otherwise is included.
>
>
> 000000... - some ordinal is infinitely smaller or infinitely decreasing LTL set theory,but otherwise is included.
>
>
> 011111... - some ordinal is infinitely bigger than LTL set theory,but otherwise is included.
>
>
>
> *00100010001,0111011001, and every possible or either impossible binary strings are included in LTL set theory.*
>
>

## Analysis of LTL set theory.

LTL(0) = 0

LTL(1) = 0,1,2,3,4,...,ω ; LTL(1) = ω

LTL(2) = ω,ε0,ζ0,*...*,φ(ω,0),*...*,ψ(Ωω),... *=>* Last googology ordinal.

LTL(3) = Least ordinal coming after last googology ordinal. = Least FG ordinal.

LTL(4) = Last FG ordinal.

LTL(5) = Least ordinal coming after last fictional googology ordinal. = Least ? ordinal.

## Introduction to Kitten's function & Cat's psi.

Let's define LTL set theory as small kitten ordinal aka **k**.

Therefore k(n) = LTL(n) , notice that LTL is a powerset of all Googology,FG and etc. ordinals.

Strong kitten ordinal aka **K** = subset theory of all possible LTL(n) definitions and beyond. *=>* \*LTL(n)

## \*LTL(n) theory.

Let's firstly define what's \*LTL is:

**\*LTL(n)** is a subset theory of all possible LTL(n) definitions and beyond,but what's that?

Let's take LTL(4) as an example, since it's defined in LTL(n) set theory, \*LTL(n)'s subset contains it, LTL(LTL(4)) is defined in LTL(n) set theory too,so \*LTL(n)'s subset contains it, and etc.

If some ordinal N outgrows LTL(N) it will be contained in \*LTL(n) subset theory.

If some ordinal N will start infinitely decreasing or increasing \*LTL(n) subset theory , \*LTL(n) will become fixed point of infinitely decreasing or increasing,and anyway will be contained in normal \*LTL(n).

Okay so let's define n for \*LTL(n) subset theory:

\*LTL(0) = Fixed point of all possible definitions for LTL(n)

\*LTL(1) = Fixed point of all possible definitions for \*LTL(0)'s = K(1)

\*LTL(2) = ? = K(2)

## Cat's Psi.

Cat's psi is a function based on \*LTL(n) subset theory,but what makes it better,is that access \*\*LTL(n),\*\*\*LTL(n) and other's subset theory's.

Note: n here is *an output of all logical definitions of LTL(n).*

Therefore

ψ
k

(
0
)

{\displaystyle \psi\_k (0)}

 is a subset of all possible \*LTL(n)`s , \*\*LTL(n) aka fixed point of \*LTL(n) subset theory , \*\*\*LTL(n) truly undescribeable for now, and etc.

ψ
k

(
1
)

{\displaystyle \psi\_k (1)}

 then is fixed point of

ψ
k

(
0
)

{\displaystyle \psi\_k (0)}

 subsequences including *=>

ψ

k
1

(
0
)

{\displaystyle \psi\_{k+1} (0)}

,

ψ

k
2

(
0
)

{\displaystyle \psi\_{k2} (0)}

,

ψ

ψ
k

(
0
)

(
0
)

{\displaystyle \psi\_{\psi\_k (0)} (0)}

,

ψ

ψ

ψ
k

(
0
)

(
0
)

(
0
)

{\displaystyle \psi\_{{\psi}\_{\psi\_k (0)}(0)} (0)}

,* and etc.

Otherwise:

ψ
k

(
ω
)

{\displaystyle \psi\_k (\omega)}

 is a fixed point of Cat's Psi function.

## Little introduction to Cat's Sequence.

CaS mimics LTL's psi expansion:

Aka

ψ

∗

L
T
L

(
α
)

{\displaystyle \psi\_{^\*LTL} (\alpha)}

 =

ψ

∗

L
T

L

α
−
1

(
α
)

{\displaystyle \psi\_{^\*LTL\_{\alpha-1}} (\alpha)}

 , where

ψ

∗

L
T
L

0

(
α
)

{\displaystyle \psi\_{^\*LTL{}\_0} (\alpha)}

 = fixed point of

K
(

ψ
k

(
ω
)
)
(
α
)

{\displaystyle K(\psi\_k(\omega))( \alpha)}

 with respect to Kitten's Sequence and Cat's Psi.

## Oracle Systems used in Cat Numbers.

∀
C
∀
R
(
m
<
n
:

C
m

∈

C
n

⇔

C
1

∈
R
:
[
ψ

]
α

∈
[
ψ

]
R

⇔

R
m

∈

R
n

)

{\displaystyle \forall C \forall R (m < n : C\_m \in C\_n \Leftrightarrow C\_1 \in R : [ \psi ]\_\alpha \in [ \psi ]\_R \Leftrightarrow R\_m \in R\_n )}

Where

[
ψ
]

{\displaystyle [\psi ]}

 is the fixed point of the index,

[
ψ

]
α

{\displaystyle [\psi ]\_\alpha}

 is the fixed point of an ordinal type, which is contained in the fixed point of some

R

{\displaystyle R}

.

Basically saying that

C
1

{\displaystyle C\_1}

 is contained in

R

{\displaystyle R}

, which is basically saying

C
1

{\displaystyle C\_1}

 is

R

{\displaystyle R}

.

C
α

{\displaystyle C\_\alpha}

 can be contained in some fixed point of

R

{\displaystyle R}

, but,it basically defines every

C
α

{\displaystyle C\_\alpha}

 ordinal as well as the fixed point of

R

{\displaystyle R}

, and

C

{\displaystyle C}

.

## And finally... Cat's Oracle Systems.

C

K
(

ψ
k

(
ω
)
)

1

(
α
)

{\displaystyle C^{1}\_{K(\psi\_k(\omega))}( \alpha)}

 is defined as First Cat's Oracle System, can be shortened to

C
(
α
)

{\displaystyle C(\alpha)}

.

Where

C

{\displaystyle C}

 is a Cat's Function , *yes finally we've started with the main function...*

C

{\displaystyle C}

 itself is

R
[
α
# ]

R

α
−
1

[
α
]

{\displaystyle R[\alpha] = R\_{\alpha-1}[\alpha]}

 , where R mimics different oracle system's ,

R
0

[
α
]

{\displaystyle R\_0[\alpha]}

 is a fixed point.

First Oracle System:

C
(
α
)

{\displaystyle C(\alpha)}

 , where R mimics Cat's Sequence.

Second Oracle System:

C
(
C
(
α
)
)

{\displaystyle C( C(\alpha) )}

 , where R mimics fixed point of Cat's Sequence. (

ψ

∗
α

L
T
L

(
α
)

{\displaystyle \psi\_{^{\*^\alpha}LTL} (\alpha)}

 ) ,

α

{\displaystyle \alpha}

 here - is an output of

C
(
α
)

{\displaystyle C(\alpha)}

.

...

Tenth Oracle System:

C
↑↑
10
(
α
)

{\displaystyle C\uparrow\uparrow10 ( \alpha )}

 , where R mimics

C
↑↑
9
(
α
)

{\displaystyle C\uparrow\uparrow9( \alpha)}

 fixed point using

C
↑↑
9
(
α
)

{\displaystyle C\uparrow\uparrow9( \alpha)}

 Logic fixed point and Star Function Logic fixed point made by fictional googologist Wifty x3.