User blog:Denis Maksudov/Comparison of notations

Approximations
Up to \(\varepsilon_0\)

Up to \(\Gamma_0\)

Up to Omega fixed point

Note:

IN denotes I-notation;

EA+CHF denotes Extended Arrows with ordinals \(\alpha\le\varepsilon_0\) written in Cantor normal form;

EA+BF denotes Extended Arrows with ordinals \(\alpha\le\Lambda\) written in normal form for Buchholz function whose fundamental sequences are assigned according ruleset for this function;

BAN denotes Bird's array notation;

E^ denotes Cascading-E notation;

xE^ denotes Extended Cascading-E Notation;

Appendix:  Extended Arrows's Definition

We define for non-zero natural numbers \(n\), \(b\) and for ordinal number \(\alpha\geq 0\):

1) \(n\uparrow^\alpha b= nb \text{ if }\alpha=0\),

2) \(n\uparrow^{\alpha+1} b = \left\{\begin{array}{lcr} n \text{ if }b=1\\ n\uparrow^{\alpha}(n\uparrow^{\alpha+1} (b-1))\text{ if }b>1\\ \end{array}\right.\),

3) \(n\uparrow^{\alpha} b=n\uparrow^{\alpha[b]} n\) iff \(\alpha\) is a limit ordinal,

where \(\alpha [b]\) denotes the b-th element of the fundamental sequence assigned to the limit ordinal \(\alpha\).