Talk:Bird's array notation

Plugging array-like notations to functions
I have the plan how to adopt Bird's array notation (or probably any recursive notation) to any other function. Just change some main rules:

Rule M1: (no separators)

\(\{a\} = \Sigma(a)\)

Rule M2: (first entry is 1)

\(\{1,\#\} = 1\)

Rule M6: (string of 1's from 2rd entry to nth)

\(\{a,1,1,\cdots,1,1,c \#\} = \{a,a,a,\cdots,a,\{a-1,1,1,\cdots,1,1,c \#\},c-1 \#\}

Rule M7: (otherwise)

\(\{a,b #\} = \{\{a-1,b \#\},b-1 \#\}\)

All other rules remain unchanged. The limit ordinal of this notation is \(\omega_1^{CK}+\theta(\varepsilon_{\Omega+1})\), in other words, Bachmann-Howard ordinal-typed recursion around busy beaver function. There will be also more powerful variant of H(n) function (that grows on par with \(f_{\omega_1^{CK}+\theta(\varepsilon_{\Omega+1})}(n)\)).

In general, we can plug any so-far-defined recursive notation around any so-far-defined function, if we replace the terminating rule of the notation to computing function and probably change some other rules. If the function has ordinal level \(\alpha\) and the notation \(\beta\), then super-notation will has the ordinal level \(\alpha+\beta\). Ikosarakt1 (talk ^ contribs) 12:28, May 25, 2013 (UTC)