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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.