BMS vs sth

# Analysis

and when we add a "(0)",the ordinal increases 1.

(0)(1)=(0)(0)(0)(0)...

so we add a (0)(1) behind = +1+1+1+1+...=ω to the ordinal

after (0)(1)(1)=(0)(1)(0)(1)(0)(1)...=ω+ω+ω+...=ω^2 we will start our analysis trip happily

### 2.1.2:(0)(1)(1) to (0)(1)(2)

if we add a (0)(1)(1) behind a standard BMS,what will happened?

because (0)(1)(1)=ω^2,so we add a ω^2 to the ordinal

when we write (0)(1)(1) ω times,we got a (0)(1)(1)(1),that means ω^2+ω^2+...=ω^3

when we write (0)(1)(1)(1) ω times,we got a (0)(1)(1)(1)(1),that means ω^4

1,ω,ω^2,ω^3,ω^4,...

look at this ordinals,they each are ω times of the ordinal before it

so we can have something in common:

**1. when we add a "(0)..." at the end of a standard BMS,we make the ordinal +(0)...**

**2. when we add a "(1)" at the end of a standard BMS without (0) at middle or end,we make the ordinal ×ω**

how can we claim it? because (1) will has the bad root "(0)" at the first forever

then the analysis is clearly to (0)(1)(2)

### 2.1.3:(0)(1)(2) to (0)(1,1)

we can clearly see that:

when we have a "(1)",that means ×ω

when we have two "(1)"s,that means ×ω^2

when we have a "(1)(2)",that means ×ω^ω

when we have a "(1)(2)(1)",that means ×ω^(ω+1)=×ω^((0)(1)(0))

so (1)(2)(1)(2)=×ω^((0)(1)(0)(1)),(1)(2)(2)=×ω^((0)(1)(1)),(1)(2)(3)=×ω^((0)(1)(2)),...

## Part2:PSS

### 2.2.1:(0)(1,1) to (0)(1,1)(1,1)

**let ε0=ωω,εn+1=ωεn,then (0)(1,1)=ε0.**
we get a "(0)(1,1)",which seems a little hard to deal with.

but we can use what we used before

like we can add a "(1)",add "(1)(2)" means ×(0)(1),and add "(1)(2,1)" means ×(0)(1,1)

when we add ω many (1)(2,1)'s,we get (0)(1,1)(1)(2,1)(2)=ε0×ε0×ε0×...=ε0ω

when we add (0)(1)(0) many (1)(2,1)'s,we get (0)(1,1)(1)(2,1)(2)(1)(2,1)

when we add (0)(1)(1) many (1)(2,1)'s,we get (0)(1,1)(1)(2,1)(2)(2)

when we add (0)(1)(2) many (1)(2,1)'s,we get (0)(1,1)(1)(2,1)(2)(3)

when we add (0)(1,1) many (1)(2,1)'s,we get (0)(1,1)(1)(2,1)(2)(3,1)=ε0ε0

when we add (0)(1,1)(1) many (1)(2,1)'s,we get (0)(1,1)(1)(2,1)(2)(3,1)(2)

when we add (0)(1,1)(1)(2,1) many (1)(2,1)'s,we get (0)(1,1)(1)(2,1)(2)(3,1)(2)(3,1)

when we add (0)(1,1)(1)(2,1)(2) many (1)(2,1)'s,we get (0)(1,1)(1)(2,1)(2)(3,1)(3)

...

### 2.2.2:(0)(1,1)(1,1) to (0)(1,1)(2,1)

add "(1)(2,1)" means ×(0)(1,1),add "(1)(2,1)(2,1)" means ×(0)(1,1)(1,1)=×ε1

we can guess that we will get (0)(1,1)(1,1)(1,1)=ε2,and then we can guess how "(1,1)" works.

**when we add a "(1,1)" at the end of a standard BMS without (0) at middle or end,we make the ordinal next ε point**

### 2.2.3:(0)(1,1)(2,1) to (0)(1,1)(2,1)(3,1)

let

**φn+1(m+1)=next φn(a) fixed point after φn+1(m)**
**φ(α,n+1)=sup(φ(α[x],φ(α,n)))**
**ζn=φ2(n),ηn=φ3(n),then (0)(1,1)(2,1)=ζ0.**
after (0)(1,1)(2,1),we can still add (1,1) aft to get next epslion point,

how about add a "(1,1)(2,1)"?

we can analysis to next (1,1)(2,1) first,and then we can get further

### 2.2.4:(0)(1,1)(2,1)(3,1) to (0)(1,1)(2,2)

after (0)(1,1)(2,1)(3,1),φ is a little hard to understand,so we will start to use B.O.C.F.

we use a huge ordinal Ω and a function ψ to collapse it.

**ψ(a+1)=ψ(a)+ψ(a)+ψ(a)+...,ψ(0)=1**
**let &=+,× or ^,then ψ(h(Ω)&Ω)=min a that a=ψ(h(Ω)&a)** //h(Ω) is sth like Ω&Ω&Ω&... example:ΩΩ×ω+Ω4×3+Ω3

ψ(0)=1

ψ(0)+ψ(0)=2

ψ(1)=ω

ψ(2)=ω^2

ψ(Ω)=min a->ψ(a)=ψ(ψ(ψ(...)))=ε0

ψ(Ω^2)=min a->ψ(Ω\*a)=ζ0

ψ(Ω^Ω)=min a->ψ(Ω^a)=min a->φ\_a(0)=Γ0,that's (0)(1,1)(2,1)(3,1).

BOCF is similar to PSS Hydra,so we can analysis easily

(we will put a row of φ too:φ(a,b)=φa(b)

because (x,1) like Ω,(x) like ω,also:

if we cut the 1 on the second row,(0)(1)(2)(3)(4)=4ω,that (0)(1,1)(2,1)(3,1)(4,1)=ψ(3Ω).

so we predict (0)(1,1)(2,2)=ψ(ωΩ),let's analysis

### 2.2.5:(0)(1,1)(2,2) to (0)(1,1,1)

B.O.C.F is not end there,we can put some ordinals more larger in it,like Ωn

**Ωn+1 is Ωn can't reached by recursive operations,so it can collapse to some complex recursive operations of Ωn by some simple operations**
**we also use ψΩn,which fold Ωn to somewhere below(ψ is short for ψΩ,and we use ψn is short for ψΩn+1)**
ψ(Ω)=ωω,while ψ1(Ω2)=ωΩ
BOCF is similar to PSS Hydra,so we can analysis easily(also because no φ there lol)

## Part3:TSS

### 2.3.1:(0)(1,1,1) to (0)(1,1,1)(2,1)

welcome to TSS part! things are becoming complex and difficult to analysis there.

(after (0)(1,1,1)(2,1))

so 2.3.1's analysis is easy

first,maybe we need to guess what (1,1,1) equal to by analysis

wow,it seems that (1,1,1)=+Ωω in ψ,but really?

### 2.3.2:(0)(1,1,1)(2,1) to (0)(1,1,1)(2,1)(1,1,1)

(0)(1,1,1)(2,1)(1,1,1) is well-known beacuse of its upgrade.

first,what it expand to?

(0)(1,1,1)(2,1)(1,1,1)=(0)(1,1,1)(2,1)(1,1)(2,2,1)**(3,2)**(2,2)(3,3,1)**(4,3)**...

ehh...,we can find (3,2) (4,3) ... in it

will them make this BMS stronger?

as you see,(1,1,1) still Ωω,but not +it

(2,1) is ×Ω at first,but(1,1,1) let (2,1) to (3,2)->×Ω2,(4,3)->×Ω3,... and finally we get (0)(1,1,1)(2,1)(1,1,1)=ψ(Ωω2)

(0)(1,1,1)(2,1)(1,1,1)=p(Ωω×p1(Ωω×p2(...))) in extend B-Hydra with Ω,which equal to ψ(Ωω2) in OCF.

### 2.3.3:(0)(1,1,1)(2,1)(1,1,1) to (0)(1,1,1)(2,1,1)

next we will meet (0)(1,1,1)(2,1)(1,1,1)(1,1,1),will it has an upgrade too?

not,(0)(1,1,1)(2,1)(1,1,1)(1,1)(2,2,1)(3,2)(2,2,1) is just ψ(Ωω2+ψ1(Ωω2)),OCF can block this.

after there,something very difficult happened.

### 2.3.3:(0)(1,1,1)(2,1,1) to (0)(1,1,1)(2,1,1)(3,1)(2)

like (1,1,1),(2,1,1) have some strong upgrade.

they may also base on (x,1) but really powerful()

let's analysus.