FANDOM


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:

Chronolegends's Egg Notation

Deedlit's notation

Buchholz's function

Fast-growing hierarchy

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 ultimatively short 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 Takeuti-Feferman-Buchholz ordinal

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) \( a|b(c()_{d+1} e)_f g = a|b(c(c(\cdots(c(\underbrace{)_d e)_d \cdots e)_d e)_d}_{a \quad d's} )_f g\),

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 and d+1>f.

Note: the most external pair of parentheses has zero subscript. 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.

Version on my site

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