Online calculator for |-notation (up to \(\varepsilon_0\))
Online calculator for |-notation (up to \(\psi_0(\varepsilon_{\Omega_9+1})\))
Expression written in |-notation is exactly equal to output of function of fast growing hierarchy indexed by ordinal number generated by Buchholz's function.
What inspired me:
a|b corresponds to \(f_b(a)\) where \(f_b\) is a function of fast-growing hierarchy
To the right of the sign "|" :
1) () corresponds to 1, (()) corresponds to \(\omega\) and (...) always corresponds to a countable ordinal number,
2) \(()_b\) corresponds to \(\Omega_b\) where \(\Omega_b=\aleph_b=\psi_b(0)\) denotes b-th uncountable ordinal,
3) \((...)_b\) corresponds to \(\psi_b(...)\) where \(\psi_b\) denotes Buchholz's function.
|-notation allows to obtain the shortest possible ruleset for well-defined simulation of both rulesets: for fast-growing hierarchy and for fundamental sequences of ordinals generated by Buchholz's function.
Definition up to ψ(Ω_ω)
1) \(a|=a+1\), where \(a\) is a natural number,
2) \(a|b()=a\underbrace{|b|b\cdots|b|b}_{a\quad |'s}\) where a is a natural number and b is the rest part of expression,
3) \(a|b(c())_d e=a|b \underbrace{(c)_d (c)_d \cdots (c)_d}_{a \quad d's} e\) where a, d are natural numbers; b,c can include left and right parentheses with any subscripts; e can include only right parentheses with any subscripts,
4) \( a|b(c()_{d+1} e)_f g = a|b(c(c(\cdots(c(\underbrace{)_d e)_d \cdots e)_d e)_d}_{a \quad d's} e)_f g\),
where a, d, f are natural numbers; b,c can include left and right parentheses with any subscripts; g can include only right parentheses with any subscripts; e can include only right parentheses with subscript d+1 or more; d+1>f.
Note: if some pair of brackets is not enclosed in any other then subscript of this pair should be zero. Parentheses with zero subscript can be written without subscript: \((...)=(...)_0\).
The limit of this notation is \(a|(()_{a})=f_{\psi_0(\psi_\omega(0))}(a)=f_{\psi_0(\Omega_\omega)}(a)\).
Examples of applying of rules 1,2:
\(3|=3+1=4=f_0(3)\)
\(3|()=3|||=4||=5|=6=f_1(3)\)
\(3|()()=3|()|()|()=6|()|()=12|()=24=f_2(3)\)
Example of applying of rule 3:
\(f_{\psi_0(\Omega^{\omega^\omega+\omega^3})}(3)=\) \( f_{\psi_0(\psi_1(\psi_1(\psi_0(\psi_0(\psi_0(0)))+\psi_0(3)))}(3)=\)
\(3|(((((()))(()()()))_1)_1)=\)
\(3|(((((()))(()())(()())(()()))_1)_1)=\)
\(f_{\psi_0(\Omega^{\omega^\omega+\omega^2\cdot 3})}(3)\)
Example of applying of rule 4:
\(f_{\psi_0(\Omega_3+\Omega_1 \cdot \psi_0(\Omega_2^{\Omega_2}+\Omega_2^{\Omega_2})}(2)=\)
\(f_{\psi_0(\psi_3(0)+\psi_1(\psi_0(\psi_2(\psi_2(\psi_2(0)))+\psi_2(\psi_2(\psi_2(0)))))))}(2)=\)
\(2|(()_3((((()_2)_2)_2((()_2)_2)_2))_1)=\)
\(2|(()_3((((()_2)_2)_2(((((()_2)_2)_2((()_1)_2)_2)_1)_2)_2))_1)=\)
\(f_{\psi_0(\Omega_3+\Omega_1\cdot\psi_0(\Omega_2^{\Omega_2}+\Omega_2^{\psi_1(\Omega_2^{\Omega_2}+\Omega_2^{\Omega_1})})}(2)\)
Analysis
\(()=1\)
\(()()=2\)
\((())=\omega\)
\((())()=\omega+1\)
\((())()()=\omega+2\)
\((())(())=\omega+\omega=\omega2\)
\((())(())()=\omega+\omega+1\)
\((())(())(())=\omega3\)
\((()())=\omega^2\)
\((()())()=\omega^2+1\)
\((()())(())=\omega^2+\omega\)
\((()())(())()=\omega^2+\omega+1\)
\((()())(()())=\omega^2+\omega^2\)
\((()()())=\omega^3\)
\((()()()())=\omega^4\)
\(((()))=\omega^\omega\)
\(((()))()=\omega^\omega+1\)
\(((())())=\omega^{\omega+1}\)
\(((()()))=\omega^{\omega^2}\)
\((((())))=\omega^{\omega^\omega}\)
\((()_1)=((\cdots(())\cdots))=\psi(\Omega)=\varepsilon_0\),
\((()_1)()=\psi(\Omega)+1=\varepsilon_0+1\),
\((()_1())=(()_1)(()_1)\cdots(()_1)=\psi(\Omega+1)=\varepsilon_0\omega=\omega^{\varepsilon_0+1}\),
\((()_1()())=\psi(\Omega+2)=\varepsilon_0 \omega^2=\omega^{\varepsilon_0+2}\),
\((()_1(()_1))=\psi(\Omega+\psi(\Omega))=\omega^{\varepsilon_0+\varepsilon_0}=\omega^{\varepsilon_0 2}\)
\((()_1(()_1()))=\psi(\Omega+\psi(\Omega+\psi(0)))=\omega^{\varepsilon_0+\varepsilon_0 \omega}=\omega^{\varepsilon_0 (1+\omega)}=\omega^{\varepsilon_0 (\omega)}=\omega^{\omega^{\varepsilon_0+1}}\)
\((()_1()_1)=\psi(\psi_1(0)+\psi_1(0))=\psi(\Omega2)=\varepsilon_1\)
\((()_1()_1()_1)=\psi(\psi_1(0)+\psi_1(0)+\psi_1(0))=\psi(\Omega3)=\varepsilon_2\)
\(((())_1)=\psi(\psi_1(\psi(0)))=\psi(\Omega\omega)=\varepsilon_\omega\)
\((((()_1))_1)=\psi(\psi_1(\psi(\psi_1(0)))=\psi(\Omega\psi(\Omega))=\varepsilon_{\varepsilon_0}\)
\(((()_1)_1)=\psi(\psi_1(\psi_1(0)))=\psi(\Omega^2)=\zeta_0\)
\(((()_1)_1())=\psi(\Omega^2+1)=\psi(\Omega^2)\cdot \omega=\zeta_0 \cdot\omega=\omega^{\zeta_0}\cdot\omega=\omega^{\zeta_0+1}\)
\(((()_1)_1()())=\psi(\Omega^2+2)=\omega^{\zeta_0+2}\)
\(((()_1)_1((()_1)_1()))=\psi(\Omega^2+\psi(\Omega^2+1))=\omega^{\zeta_0+\omega^{\zeta_0+1}}=\omega^{\zeta_0(1+\omega)}=\omega^{\zeta_0\omega}=\omega^{\omega^{\zeta_0+1}}\)
\(((()_1)_1()_1)=\psi(\Omega^2+\Omega)=\omega^{...^{\omega^{\omega^{\zeta_0+1}}}}=\varepsilon_{\zeta_0+1}\)
\(((()_1)_1()_1())=\psi(\Omega^2+\Omega+1)=\varepsilon_{\zeta_0+1}\cdot\omega=\omega^{\varepsilon_{\zeta_0+1}+1}\)
\(((()_1)_1()_1()())=\psi(\Omega^2+\Omega+2)=\omega^{\varepsilon_{\zeta_0+1}+2}\)
\(((()_1)_1()_1((()_1)_1()_1()))=\psi(\Omega^2+\Omega+\psi(\Omega^2+\Omega+1))=\omega^{\varepsilon_{\zeta_0+1}+\omega^{\varepsilon_{\zeta_0+1}+1}}=\omega^{\omega^{\varepsilon_{\zeta_0+1}+1}}\)
\(((()_1)_1()_1()_1)=\psi(\Omega^2+\Omega2)=\omega^{...^{\omega^{\omega^{\varepsilon_{\zeta_0+1}+1}}}}=\varepsilon_{\zeta_0+2}\)
\(((()_1)_1(((()_1)_1()_1))_1)=\psi(\Omega^2+\Omega\cdot\psi(\Omega^2+\Omega))=\varepsilon_{\zeta_0+\varepsilon_{\zeta_0+1}}=\varepsilon_{\varepsilon_{\zeta_0+1}}\)
\(((()_1)_1(()_1)_1)=\psi(\Omega^22)=\psi(\Omega^2+\Omega^2)=\varepsilon_{..._{\varepsilon_{\varepsilon_{\zeta_0+1}}}}=\zeta_1\)
\(((()_1((()_1)_1))_1)=\psi(\Omega^2\cdot\psi(\Omega^2))=\zeta_{\zeta_0}\)
\(((()_1()_1)_1)=\psi(\psi_1(\psi_1(0)+\psi_1(0)))=\psi(\Omega^3)=\zeta_{..._{\zeta_{\zeta_0}}}=\eta_0=\varphi(3,0)\)
\((((())_1)_1)=\psi(\psi_1(\psi_1(\psi(0))))=\psi(\Omega^\omega)=\varphi(\omega,0)\)
\((((()_1)_1)_1)=\psi(\psi_1(\psi_1(\psi_1(0))))=\psi(\Omega^\Omega)=\varphi(1,0,0)=\Gamma_0=\theta(\Omega,0)\)
\((((()_1()_1)_1)_1)=\psi(\psi_1(\psi_1(\psi_1(0)+\psi_1(0))))=\psi(\Omega^{\Omega^{2}})=\theta(\Omega^2,0)=\varphi(1,0,0,0)\)
\((((()_1()_1()_1)_1)_1)=\psi(\psi_1(\psi_1(\psi_1(0)+\psi_1(0)+\psi_1(0))))=\psi(\Omega^{\Omega^{3}})=\theta(\Omega^3,0)=\varphi(1,0,0,0,0)\)
\(((((())_1)_1)_1)=\psi(\psi_1(\psi_1(\psi_1(1))))=\psi(\Omega^{\Omega^{\omega}})=\theta(\Omega^\omega,0)\) - Small Veblen ordinal
\(((((()_1)_1)_1)_1)=\psi(\psi_1(\psi_1(\psi_1(\psi_1(0)))))=\psi(\Omega^{\Omega^{\Omega}})=\theta(\Omega^\Omega,0)\) - Large Veblen ordinal
\((()_2)=\psi(\psi_2(0))=\psi(\Omega_2)=\psi(\varepsilon_{\Omega_1+1})=\theta(\varepsilon_{\Omega_1+1},0)\)
\((()_2)_1=\psi_1(\Omega_2)=\varepsilon_{\Omega+1}=\Omega^{\Omega^{\Omega^{...}}}=\varphi(1,\Omega+1)\)
\((()_2())_1=\psi_1(\Omega_2+1)=\varepsilon_{\Omega+1}\omega=\omega^{\varepsilon_{\Omega+1}+1}\)
\((()_2()())_1=\psi_1(\Omega_2+2)=\varepsilon_{\Omega+1}\omega^2=\omega^{\varepsilon_{\Omega+1}+2}\)
\((()_2(()_2())_1)_1=\psi_1(\Omega_2+\psi_1(\Omega_2+1))=\omega^{\varepsilon_{\Omega+1}+\varepsilon_{\Omega+1}\omega}=\omega^{\omega^{\varepsilon_{\Omega+1}+1}}\)
\((()_2()_2)_1=\psi_1(\Omega_2+\Omega_2)=\varepsilon_{\Omega+2}\)
\((()_2()_2())_1=\psi_1(\Omega_2+\Omega_2+1)=\varepsilon_{\Omega+2}\omega=\omega^{\varepsilon_{\Omega+2}+1}\)
\((()_2()_2(()_2()_2())_1)_1=\psi_1(\Omega_2+\Omega_2+\psi_1(\Omega_2+\Omega_2+1))=\omega^{\varepsilon_{\Omega+2}+\varepsilon_{\Omega+2}\omega}=\omega^{\omega^{\varepsilon_{\Omega+2}+1}}\)
\((()_2()_2()_2)_1=\psi_1(\Omega_2 3)=\psi_1(\Omega_2+\Omega_2+\Omega_2)=\varepsilon_{\Omega+3}\)
\((((()_2)_1)_2)_1=\psi_1(\psi_2(\psi_1(\psi_2(0))))=\psi_1(\Omega_2 \psi_1(\Omega_2))=\varepsilon_{\Omega+\varepsilon_{\Omega+1}}=\varepsilon_{\varepsilon_{\Omega+1}}\)
\(((()_2)_2)_1=\psi_1(\psi_2(\psi_2(0)))=\psi_1(\Omega_2^2)=\psi_1(\Omega_2\Omega_2)=\zeta_{\Omega+1}=\varphi(2,\Omega+1)\)
\((((())_2)_2)_1=\psi_1(\psi_2(\psi_2(1)))=\psi_1(\Omega_2^\omega)=\varphi(\omega,\Omega+1)\)
\((((()_1)_2)_2)_1=\psi_1(\psi_2(\psi_2(\psi_1(0))))=\psi_1(\Omega_2^\Omega)=\Gamma_{\Omega+1}\)
\((((()_1)_2)_2)=\psi(\Omega_2^\Omega)=\psi(\Gamma_{\Omega+1})=\theta(\Omega_2,0)\)
\((()_{i+1})=\psi(\Omega_{i+1})=\psi(\varepsilon_{\Omega_i+1})=\theta(\varepsilon_{\Omega_i+1},0)\)
\((((()_1)_i)_i)=\psi(\psi_i(\psi_i(\psi_1(0))))=\psi(\Omega_i^\Omega)=\theta(\Omega_i,0)\)
Other versions
To obtain fast iteration hierarchy instead fast growinghierarchy, just insert any increasing function in first rule, for example:
1) \(a|=10^a\), where \(a\) is a natural number.
To obtain Hardy hierarchy instead fast growing hierarchy, rewrite rules 1,2 as follows:
1) \(a|=a\), where \(a\) is a natural number,
2) \(a|b()=c|b\) where b is the rest part of expression, c,a are natural numbers and \(c=a+1\).
For extension of this notation up to omega fixed point, write subscripts of parentheses as ordinals (i.e. also as combinations of parentheses) and rewrite rules 3,4 for this case.