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:

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

This is possible because every time multiple \(n\)'s appear, a new representation with only a single \(n\) is defined. Therefore every new \(n\) uncovered when expanding a function with only one \(n\) is actually only a component of the larger \(n\) in that larger function. This function can use other variables in it larger than the \(n\) used in the most powerful position, but the functions created would be considered nonstandard.

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.

\(Bracket \space Notation\)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\(\{n,2,0\}=\{n,1,n\}=f_{\omega^{\omega3}}(n)\)

\(\{n,2,1\}=\{n,2,0\}\{n,2,0\}{...}^n\{n,2,0\}=f_{\omega^{\omega3+1}}(n)\)

\(\{n,3,0\}=\{n,2,n\}=\{n,2,n-1\}\{n,2,n-1\}{...}^n\{n,2,n-1\}=f_{\omega^{\omega4}}(n)\)

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

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

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

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

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

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

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

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

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

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

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

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

\(\{/_1//{...}^n/n\}^1=\{/_1/_1n\}^1=f_{\omega^{\omega^{\omega3}}}(n)\)

\(\{/_2n\}^1=\{/_1/_1{...}^n/_1n\}^1=f_{\omega^{\omega^{\omega^2}}}(n)\)

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

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

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

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

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

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

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

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

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

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

\(\{/_{\{n,1,1\}}n\}^1=\{/_{\{n,1,0\}}/_{\{n,1,0\}}{...}^n/_{\{n,1,0\}}n\}^1=f_{\omega^{\omega^{\omega^{\omega2+1}}}}(n)\)

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

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

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

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

\(\{/_{\{n,2,0,0\}}n\}^1=f_{\omega^{\omega^{\omega^{\omega^22}}}}(n)\)

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

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

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

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

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

\(\{/^1n,1\}^1=\{/^1n\}^1\{/^1n\}^1{...}^n\{/^1n\}^1=f_{\varepsilon_0\omega}(n)\)

\(\{/^1/^1n\}^1=\{/^1/_{\{/_{{...}^n\{n\}^1{...}^n}n\}^1}n\}=f_{\varepsilon_0^2}(n)\)

\(\{/^2n\}^1=\{/^1_{\{/^1_{{...}^n\{n\}^1{...}^n}n\}^1}n\}^1=f_{\varepsilon_0^{\varepsilon_0}}(n)\)

\(\{/^nn\}^1=\{/^{\{n,1\}}n\}^1=f_{\varepsilon_0^{\varepsilon_0\omega}}(n)\)

\(\{/^{\{n,2\}}n\}^1=f_{\varepsilon_0^{\varepsilon_0\omega2}}(n)\)

\(\{/^{\{n,1,0,0\}}n\}^1=f_{\varepsilon_0^{\varepsilon_0\omega^3}}(n)\)

\(\{/^{\{/^1n\}^1}n\}^1=f_{\varepsilon_0^{\varepsilon_0^2}}(n)\)

\(\{/^{\{/^{\{/^1n\}^1}n\}^1}n\}^1=f_{\varepsilon_0^{\varepsilon_0^{\varepsilon_0^2+1}}}(n)\)

\(\{/^/n\}^1=f_{\varepsilon_1}(n)\)

\(\{/^//^/n\}^1=f_{\varepsilon_1^2}(n)\)

\(\{/^{/1}n\}^1=f_{\varepsilon_1^{\varepsilon_1}}(n)\)

\(\{/^{//}n\}^1=f_{\varepsilon_2}(n)\)

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

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

\(\{/^{/_1/_1}n\}=f_{\varepsilon_{\omega2}}(n)\)

\(\{/^{/_2}n\}=f_{\varepsilon_{\omega^2}}(n)\)

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

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

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

\(\{/^{/^1}n\}=f_{\varepsilon_{\varepsilon_0}}(n)\)

\(\{/_/n\}=\{/^{/^{{...}^{n^/}}}n\}=f_{\zeta_0}(n)\)