User blog:Dchew89/Basic Notation for Reaching FGH epsilon Level

Just made this for fun and for the challenge. There may be some typos.

"\({...}^n\)" implies \(n\) of a repeating pattern, for example, \(1+1{...}^n+1=n\). Otherwise, "\(...\)" just means "something".

In any circumstance in the following definitions where it is unclear of what "\(n\)" to use to expand any expression, the most powerful "\(n\)" is used. For example:

\(\{n\}[n]^n=\{n\}[a]^b=\{\{{...}^n\{n\}[a]^{b-1}{...}^n\}[a]^{b-1}\}[a]^{b-1}\)

But "\(\{a\}[b]^c\)" must equal "\(\{a\}[n]^n\)" to equal "\(\{a\}[n,1]^0\)". This is why "\(n\)" is always used instead of more letters, as to simplify the expression of so many equivalencies with so many "\(n\)"'s.

\(Bracket \space Notation\)

\([n]^0=n+1=f_0(n)\)

\([n]^1=[[{...}^n[n]^0{...}^n]^0]^0=f_1(n)\)

\([n,1]^0=[n]^n=[[{...}^n[n]^{n-1}{...}^n]^{n-1}]^{n-1}=f_\omega(n)\)

\([n,2]^0=[n,1]^n=[[{...}^n[n,1]^{n-1}{...}^n]^{n-1},1]^{n-1}=f_{\omega2}(n)\)

\([n,1,0]^0=[n,n]^0=[n,n-1]^n=f_{\omega^2}(n)\)

\([n,1,1]^0=[n,1,0]^n=[[{...}^n[n,1,0]^{n-1}{...}^n]^{n-1},1,0]^{n-1}=f_{\omega^2+\omega}(n)\)

\([n,2,0]^0=[n,1,n]^0=[[{...}^n[n,1,n-1]^{n-1}{...}^n]^{n-1},1,n-1]^{n-1}f_{\omega^22}(n)\)

\([n,1,0,0]^0=[n,n,0]^0=[n,n-1,n]^0=[n,n-1,n-1]^n=[[{...}^n[n,n-1,n-1]^{n-1}{...}^n]^{n-1},n-1,n-1]^{n- 1}=f_{\omega^3}(n)\)

\([n,1,0,1]^0=[n,1,0,0]^n=f_{\omega^3+\omega}(n)\)

\([n,1,1,0]^0=[n,1,0,n]^0=f_{\omega^3+\omega^2}(n)\)

\([n,2,0,0]^0=[n,1,n,0]^0=f_{\omega^32}(n)\)

\([n,1,0,0,0]^0=f_{\omega^4}(n)\)

\([n,...,a,0,0,...]^0=[n,...,a-1,n,0,...]^0\)

\([n,...,a]^0=[n,...,a-1]^n\)

\(\{n\}=[n,1,0,0,{...}^n]^0=f_{\omega^\omega}(n)\)

\(\{n\}[n]^0=\{n\}+1=\{\{{...}^n\{n\}{...}^n\}\}=f_{\omega^\omega+1}(n)\)

\(\{n\}[n]^n=\{\{{...}^n\{n\}[n]^{n-1}{...}^n\}[n]^{n-1}\}[n]^{n-1}=\{n\}[n,1]^0=f_{\omega^\omega+\omega}(n)\)

\(\{n\}\{n\}=\{n\}[n,1,0,0,{...}^n]^0=f_{\omega^\omega2}(n)\)

\(\{n\}\{n\}[n]^n=\{\{{...}^n\{n\}\{n\}[n]^{n-1}{...}^n\}\{n\}[n]^{n-1}\}\{n\}[n]^{n-1}=\{n\}\{n\}[n,1]^0=f_\omega(f_{\omega^\omega2}(n))\)

\(\{n\}\{n\}\{n\}=\{n\}\{n\}[n,1,0,0,{...}^n]^0=f_{\omega^\omega3}(n)\)

\(\{n,1\}=\{n\}\{n\}{...}^n\{n\}=f_{\omega^{\omega+1}}(n)\)

\(\{n,2\}=\{n,1\}\{n,1\}{...}^n\{n,1\}=f_{\omega^{\omega+2}}(n)\)

\(\{n,n\}=\{n,1,0\}=f_{\omega^{\omega2}}(n)\)

\(\{n,1,1\}=\{n,1,0\}\{n,1,0\}{...}^n\{n,1,0\}=f_{\omega^{\omega3}}(n)\)

\(\{n,2,0\}=\{n,1,n\}=f_{\omega^{\omega^2}}(n)\)

\(\{n,2,1\}=\{n,2,0\}\{n,2,0\}{...}^n\{n,2,0\}=f_{\omega^{\omega^2+\omega}}(n)\)

\(\{n,3,0\}=\{n,2,n\}=\{n,2,n-1\}\{n,2,n-1\}{...}^n\{n,2,n-1\}=f_{\omega^{\omega^22}}(n)\)

\(\{n,n,0\}=\{n,1,0,0\}=f_{\omega^{\omega^3}}(n)\)

\(\{n,n,0,0\}=\{n,1,0,0,0\}=f_{\omega^{\omega^4}}(n)\)

\(\{n\}^1=\{n,1,0,0,{...}^n\}=f_{\omega^{\omega^\omega}}(n)\)

\(\{n\}^1\{n\}^1=f_{\omega^{\omega^\omega2}}(n)\)

\(\{n,1\}^1=\{n\}^1\{n\}^1{...}^n\{n\}^1=f_{\omega^{\omega^\omega+1}}(n)\)

\(\{n\}^2=\{n,1,0,0,{...}^n\}^1=f_{\omega^{\omega^{\omega2}}}(n)\)

\(\{/n\}^1=\{n\}^n=\{n,1,0,0,{...}^n\}^{n-1}=f_{\omega^{\omega^{\omega^2}}}(n)\)

\(\{/n\}^2=\{/n,1,0,0,{...}^n\}^1=f_{\omega^{\omega^{\omega^2+\omega}}}(n)\)

\(\{/n\}^n=\{//n\}^1=f_{\omega^{\omega^{\omega^22}}}(n)\)

\(\{//n\}^n=\{///n\}^1=f_{\omega^{\omega^{\omega^23}}}(n)\)

\(\{/n/n\}^1=\{/^nn\}^n=f_{\omega^{\omega^{\omega^3}}}(n)\)

\(\{/n//n\}^1=\{/n/n\}^n=f_{\omega^{\omega^{\omega^3+\omega^2}}}(n)\)

\(\{/n/^nn\}^1=\{//n/n\}^1=f_{\omega^{\omega^{\omega^32}}}(n)\)

\(\{/n/n/n\}^1=\{/^nn/n\}^1=f_{\omega^{\omega^{\omega^4}}}(n)\)

\(\{/_1n\}^1=\{/n/n{...}^n/n\}^1=f_{\omega^{\omega^{\omega^\omega}}}(n)\)

\(\{/_1/n\}^1=\{/_1n\}^n=f_{\omega^{\omega^{\omega^\omega+\omega^2}}}(n)\)

\(\{/_1/n/n\}^1=\{/_1/^nn\}^n=f_{\omega^{\omega^{\omega^\omega+\omega^3}}}(n)\)

\(\{/_1/_1n\}^1=\{/_1/n/n{...}^n/n\}^1=f_{\omega^{\omega^{\omega^\omega2}}}(n)\)

\(\{/_1n/_1n\}^1=\{/_1^nn\}^1=f_{\omega^{\omega^{\omega^{\omega+1}}}}(n)\)

\(\{/_1n/_1/_1n\}^1=f_{\omega^{\omega^{\omega^{\omega+1}+\omega^\omega}}}(n)\)

\(\{/_1/_1n/_1n\}^1=f_{\omega^{\omega^{\omega^{\omega+1}2}}}(n)\)

\(\{/_1n/_1n/_1n\}^1=\{/_1^nn/_1n\}^1=f_{\omega^{\omega^{\omega^{\omega+2}}}}(n)\)

\(\{/_2n\}^1=\{/_1n/_1n{...}^n/_1n\}^1=f_{\omega^{\omega^{\omega^{\omega2}}}}(n)\)

\(\{/_2/_2n\}^1=\{/_2/_1n/_1n{...}^n/_1n\}^1=f_{\omega^{\omega^{\omega^{\omega2}2}}}(n)\)

\(\{/_2n/_2n\}^1=\{/_2^nn\}^1=f_{\omega^{\omega^{\omega^{\omega2+1}}}}(n)\)

\(\{/_nn\}^1=\{/_{(1,0)}n\}^1=f_{\omega^{\omega^{\omega^{\omega^2}}}}(n)\)

\(\{/_{\{n,0\}}/_{(1,0)}n\}^1=f_{\omega^{\omega^{\omega^{\omega^2}2}}}(n)\)

\(\{/_{\{n,0\}}n/_{(1,0)}n\}^1=f_{\omega^{\omega^{\omega^{\omega^2+1}}}}(n)\)

\(\{/_{\{n,1\}}n\}^1=f_{\omega^{\omega^{\omega^{\omega^2+\omega}}}}(n)\)

\(\{/_{\{n,1,0\}}n\}^1=f_{\omega^{\omega^{\omega^{\omega^22}}}}(n)\)

\(\{/_{\{n,1,0,0\}}n\}^1=f_{\omega^{\omega^{\omega^{\omega^3}}}}(n)\)

\(\{/_{\{n,1,0,0,0\}}n\}^1=f_{\omega^{\omega^{\omega^{\omega^4}}}}(n)\)

\(\{/_{\{n\}^1}n\}^1=f_{\omega^{\omega^{\omega^{\omega^\omega}}}}(n)\)

\(\{/_{\{/_{\{n\}^1}n\}^1}n\}^1=f_{\omega^{\omega^{\omega^{\omega^{\omega^{\omega^2}}}}}}(n)\)

\(\{`n\}^1=\{/_{\{/_{{...}^n\{n\}^1{...}^n}n\}^1}n\}^1=f_{\varepsilon_0}(n)\)