An attempt to well-define OCF as a number notation/Up to SDO

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This section is so long that it deserves its own blog post.

Valid expression[]

All valid expressions are the form of:

f_A(n)

,where A is an erdinal and n is a natural number excluding 0 and 1 (maybe it works for 0 and 1, but not needed. Who would plug in ZERO in the argument of FGH?).

And the definition of an "erdinal" is as follows:

0
A+B, where A and B are erdinals and A cannot be written as X+Y for erdinals X and Y
ψ_A(B), where A and B are erdinals and A is legural (of course this psi symbol can be replaced by a trident mark)
C(|{n}o), where n is a natural number including 0 (I made the 1 and 0 slightly look different to distinguish them from the "1" in the syntax sugar)

Let I=C(|{0}0), W=ψ_I(0), 1=ψ_W(0), and w=ψ_W(1).

Expansion[]

If A=0: f_0(n) = n+1
Else, if Cough(A)=1: f_A(n) = f_〈A〉[0](f_〈A〉[0](...(f_〈A〉[0](n))...)), with n f's
Else: f_ψ_A(B)(n) = f_〈ψ_A(B)〉[Cast(n)](n)

This definition is not perfect because I have not defined Cough, Cast, and 〈X〉[n]. I will define those below.

!!! SPAGHETTI MONSTER ALERT !!!

EVERYTHING BELOW IS INTELLIGENT-DESIGNED

Before dealing with an erdinal expression, there must be a few phases before the actual computation. Or, more formally, these phases can be thought to be at the end of every computation.

Unoverrecursivizement[]

This phase removes something like $ \psi(\psi(\Omega)) $.

Procedure: If you see ψ_A(ψ_B(C)) where Cmp(A,B)=0 and Cmp(Cough(B),C)=-1 or 0, change it to ψ_A(C). The change is made from left to right according to the position of ψ corresponding to the A.

Desubroundification[]

This phase removes all the expressions ψ_B(0) except ψ_I(0) and ψ_W(0). Also it removes something like $ \psi_{\psi_M(M^{M^M})}(I) (= \psi_M(M^{M^I})) $.

Procedure: If you see ψ_B(C) (let it A),

If B's Tier is 0 or lower, desubroundificate B and C, and the process is done.
Else, if B=C(|{n}o) and Cmp(B,C)=+1 and n>0: Change A to ψ_C(|{m}o)(C), where m is the smallest natural number such that Cmp(C(|{m}o),C)=+1.
Else:
1. Desubroundificate B. Now B must be able to be written as ψ_D(E).
  1. If Cmp(Cough(D),C)=+1: Let F the desubroundification of ψ_D(〈E〉[C]). Change A to F, and the process is done.
  2. Else, change nothing.

The change is made from left to right according to the position of ψ corresponding to the B.

And now the main phase.

Tier[]

I define the Tier of an erdinal A as follows:

If A is 0, it is Tier -4.
If A is D+E:
1. If D=0: A is the same Tier as E.
2. Else, if E=0: A is the same Tier as D.
3. Else, A is the same tier as E.
If A is ψ_D(E):
1. If Cmp(D,W)=0 and E=0: A is Tier -3.
2. Else, if D's Tier >=1 and Cmp(D, Cough(E))=0 or -1: A's Tier is D's Tier -1.
3. Else, if D's Tier >=1 and Cmp(D, Cough(E))=+1 and E's Tier >=-2: A's Tier is E's Tier.
4. Else, if D's Tier is -1:
  1. If E's Tier is -4:
    1. D must be W because desubroundification is assumed. So A=ψ_W(0)=1's Tier is -3.
  2. Else, if E's Tier is -3 or -2: A's Tier is -2.
  3. Else, if Cmp(D, Cough(E))=+1: A's Tier is E's Tier.
  4. Else: A's Tier is -2.
5. Else, if D's Tier is 0:
  1. If E's Tier is -4 or -3: GOTO 3.7
  2. Else, if Cmp(D, Cough(E))=+1: A's Tier is E's Tier.
  3. Else: A's Tier is -2.
6. Else, if E's Tier is -4 or -3: GOTO 3.7
7. If you get to here, D's Tier must be 0 or higher and E's Tier is -4 or -3.
8. In that case, A=ψ_D(E)'s Tier is -1.
If A is C(|{n}o), it is Tier n.

Intuitively:

Tier -4 is zero
Tier -3 is successor
Tier -2 is countable limit
Tier -1 is accessible cardinals
Tier 0 is inaccessibles, including hypers
Tier 1 is Mahlos
Tier 2 is stages
Tier 3 is metastages
Tier 4 is metametastages
etc.

Coughinality[]

I define Cough(A) for an erdinal A as follows:

If A=0: Cough(0)=0
If A=D+E:
1. If D=0: Cough(0+E)=Cough(E)
2. Else, If E=0: Cough(D+0)=Cough(D)
3. Else: Cough(D+E)=Cough(E)
If A=C(|{n}o):
1. Cough(A)=A
If A=ψ_D(E):
1. If A is Tier -3: Cough(A)=1
2. Else, if A is Tier -2: Cough(A)=w
3. Else: Cough(A)=A

If an erdinal A satisfies Cmp(Cough(A),A)=0, A is said to be legural.

Association[]

I define A⊕B as follows:

If B=0: A⊕B=A.
If B cannot be written as D+E:
1. If A cannot be written as F+G: A⊕B=A+B.
2. Else: Let A=F+G and A⊕B=F+(G⊕B).
Else: Let B=D+E and A⊕B=(A⊕D)⊕E.

I need this because I let A+B+C is always A+(B+C) in the definition.

Comparison[]

I define Cmp(A,B) for erdinals A and B as follows. It returns either -1, 0, or +1 (these are meaningless symbols). Cmp(A,B)=+1 is supposed to imply A>B, Cmp(A,B)=0 is supposed to imply A=B, and Cmp(A,B)=-1 is supposed to imply A<B.

If A=0:
1. If B=0: Cmp(0,0)=0
2. Else: Cmp(0,B)=-1
If A=D+E:
1. If D=0: Cmp(0+E,B)=Cmp(E,B)
2. Else, if E=0: Cmp(D+0,B)=Cmp(D,B)
3. Else, if B=0: Cmp(D+E,0)=+1
4. Else, if B=F+G:
  1. If F=0: Cmp(D+E, 0+G)=Cmp(D+E,G)
  2. Else, if Cmp(D,F) is not 0: Cmp(D+E, F+G)=Cmp(D,F)
  3. Else: Cmp(D+E, F+G)=Cmp(E,G)
5. Else, if Cmp(D,B)=0: Cmp(D+E,B)=+1
6. Else: Cmp(D+E,B)=Cmp(D,B)
If A=ψ_D(E):
1. If B=0: Cmp(A,0)=+1
2. Else, if B=F+G:
  1. If Cmp(A,F)=0: Cmp(A,F+G)=-1
  2. Else: Cmp(A,F+G)=Cmp(A,F)
3. Else, if B=ψ_F(G): // I'm not sure if it works well!
  1. If the Tiers of D and F are the same:
    1. If Cmp(D,F) is not 0: Cmp(A,B)=Cmp(D,F)
    2. Else: Cmp(A,B)=Cmp(E,G)
  2. Else, if Cmp(D,F)=0: Cmp(A,B)=Cmp(E,G)
  3. Else:
    1. If Tier of D is less than Tier of F: Cmp(A,B)=-Cmp(B,A) where -(-1)=+1, -(0)=0, and -(+1)=-1
    2. Else:
      1. If F's Tier is -1:
        Let H as follows.
        If there exists n such that Cmp(D, C(|{n}o))=0: H=ψ_D(E⊕1)
        
        Else: take the smallest n such that Cmp(D, C(|{n}o))=-1. Then H=ψ_C(|{n}o)(A⊕1).
        
        If Cmp(H,F) is not 0: Cmp(A,B)=Cmp(H,F).
        
        Else, if G is not 0: Cmp(A,B)=-1.
        
        Else: Cmp(A,B)=0.
      2. Else:
        Let D's Tier h and F's Tier i.
        
        Let J[l], where i<=l<=h-1, as follows:
        Let J[h-1]=ψ_D(E⊕D). // ψ_J[h-1](0) should be equal to A.
        
        for k from h-2 downto i:
        Let J[k]=ψ_J[k+1](J[k+1]). // ψ_J[k](0) should be equal to A for al h-2>=k>=i.
        
        If Cmp(J[i],F) is not 0: Cmp(A,B)=Cmp(J[i],F).
        
        Else, if G is not 0: Cmp(A,B)=-1.
        
        Else: Cmp(A,B)=0.
4. Else, if B=C(|{n}o):
  1. If Cmp(D,B)=-1 or 0: Cmp(A,B)=-1
  2. Else, if there exists m such that Cmp(D,C(|{m}o))=0:
    1. If Cmp(D,E)=+1: Cmp(A,B)=Cmp(E,B)
    2. Else, if Cmp(D,E)=0 and m=n+1: Cmp(A,B)=0
    3. Else: Cmp(A,B)=+1
  3. Else: Cmp(A,B)=+1
If A=C(|{n}o):
1. If B=0: Cmp(A,0)=+1
2. Else, if B=F+G:
  1. If Cmp(A,F)=0: Cmp(A,F+G)=-1
  2. Else: Cmp(A,F+G)=Cmp(A,F)
3. Else, if B=ψ_F(G):
  1. Cmp(A,B)=-Cmp(B,A) where -(-1)=+1, -(0)=0, and -(+1)=-1 JUST LOOK AT ANOTHER SECTION
4. Else, if B=C(|{m}o):
  1. If n>m: Cmp(A,B)=+1
  2. Else, if n=m: Cmp(A,B)=0
  3. Else: Cmp(A,B)=-1

Fundamental Sequence[]

I define 〈X〉[n], where X and n are erdinals, as follows. This is the main part of this notation.

If X=0: 〈X〉[n]=0
If X=A+B:
1. If A=0: 〈0+B〉[n]=〈B〉[n]
2. Else, if B=0: 〈A+0〉[n]=〈A〉[n]
3. Else: 〈A+B〉[n]=A+〈B〉[n]
If X=ψ_A(B):
1. If X is Tier -3: 〈X〉[n]=0
2. If X is Tier -2:
  1. If Cmp(Cough(B),1)=0:
    1. If n=0: 〈X〉[n]=0
    2. If n=1: 〈X〉[n]=ψ_A(〈B〉[0])
    3. Else, if n=D+1: 〈X〉[n]=〈X〉[D]⊕〈X〉[1]
    4. Else (when n is neither 0 nor 1+1+...+1): 〈X〉[n]=0
  2. Else, if Cmp(A,Cough(B))=0:
    1. If n=0: 〈X〉[n]=ψ_A(〈B〉[0])
    2. If n=1: 〈X〉[n]=〈X〉[0+1]
    3. Else, if n=D+1: 〈X〉[n]=ψ_A(〈B〉[〈X〉[D]])
    4. Else (when n is neither 0 nor 1+1+...+1): 〈X〉[n]=0
  3. Else: 〈X〉[n]=〈ψ_A(〈B〉[ψ_Cough(B)(B)])〉[n]
3. If X is Tier -1 or higher: 〈X〉[n]=n
If X=C(|{n}o): 〈X〉[n]=n

Cast[]

I will define Cast(n) as follows. It takes a natural number and returns an erdinal.

Cast(0)=0
Cast(1)=1
Cast(n+1)=Cast(n)⊕1 for n>0

EVERYTHING BELOW IS MORE JOKE THAN EVERYTHING ABOVE

Syntax Sugar[]

"Once you have these rules in your heads you can write a million different numbers by mixing them up!"

"But it doesn't MEAN anything!"

"So we use alphabets. One letter for one expression."

N = ψ_T(ψ_ψ_X(X+1)(T))
K = ψ_T(ψ_ψ_X(X+1)(ψ_ψ_X(X+1)(T)))
4 = 1+1+1+1
3 = 1+1+1
X = C(|{3}o)
2 = 1+1
6 = 1+1+1+1+1+1

5 = 1+1+1+1+1
8 = 1+1+1+1+1+1+1+1
7 = 1+1+1+1+1+1+1
9 = 1+1+1+1+1+1+1+1+1
M = C(|{1}o)
Y = C(|{4}o)
T = C(|{2}o)

Googology[]

Limit[]

Assuming everything above WERE well-defined, I will define the limit function as follows:

f_SDO(n) = f_C(|{n}o)(n)

Ordinals(NOT Erdinals)[]

Fix one fundamental sequence each for EVERY limit ordinal up to $ \omega_1^{CK} $ inclusive(yes, I understand what I wrote here). Now FGH up to $ \omega_1^{CK} $ inclusive is well-defined because the fundamental sequence is defined.

Consider the following condition on $ \alpha $:

$$ \forall m \in \mathbb{N}, \exists N \in \mathbb{N}, \forall x \in \mathbb{N}, \left( x > N \Rightarrow {FGH}^m_{\alpha}(x) < f_{SDO}(x) \right) $$

I will define SANA(SDO Analogue from NIECF Analysis) as the smallest $ \alpha $ such that the condition above is FALSE. I did not end the acronym with O because SANA depends on how you choose the fundamental sequences, but SANA is believed to be ψψIO with "the standard" fundamental sequences.

The smallest possible SANA is $ \omega $ by taking $ \omega[n]=f_{SDO}(n) $(note that this is not circular definition), so I call $ \omega $ SSO for the first fixed point of α→"Small α Ordinal" after "Small SANA Ordinal". In other words, SSO stands for "Small SSO Ordinal".