User blog:Denis Maksudov/I-notation

Expression written in |-notation is exactly equal to output of function of fast growing hierarchy indexed by ordinal number generated by Buchholz's function.

Where is my inspiration comes from:

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) \(a|b(c)_d e=b \underbrace{(c)_d (c)_d \cdots (c)_d}_{a \quad d's} e\) 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 and it 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 rule 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 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 rule 4:

\(f_{\psi_0(\Omega_3+\Omega_1 \cdot \psi_0(\Omega_2^{\Omega_2}+\Omega_2^{\Omega_2})}(2)=\)

\(f_{\psi_0(\psi_3(0)+\psi_1(\psi_0(\psi_2(\psi_2(\psi_2(0)))+\psi_2(\psi_2(\psi_2(0)))))))}(2)=\)

\(2|(_3((((_2)_2)_2((_2)_2)_2))_1)=\)

\(2|(_3((((_2)_2)_2(((((_2)_2)_2((_1)_2)_2)_1)_2)_2))_1)=\)

\(f_{\psi_0(\Omega_3+\Omega_1\cdot\psi_0(\Omega_2^{\Omega_2}+\Omega_2^{\psi_1(\Omega_2^{\Omega_2}+\Omega_2^{\Omega_1})})}(2)\)

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.